2024

[1]Mario Zitelli

[1]Department of Information Engineering, Electronics and Telecommunications,
Università degli Studi di Roma Sapienza, Via Eudossiana 18, Rome, 00184, RM,
Italy

# Optical solitons in multimode fibers: recent advances

<EMAIL_ADDRESS>*

###### Abstract

Optical solitons in multimode fibers have been predicted 40 years ago and
extensively investigated theoretically. Transmission experiments in nonlinear
multimode fibers have gained renewed interest, motivated by their potential to
extend the capacity of long-distance transmission systems; only in the last
few years, new experiments have revealed unexpected properties of optical
solitons propagating in graded-index and step-index multimode fibers,
partially re-writing the existing theory. Here we provide an overview of the
recent experimental, numerical and theoretical studies which revealed those
new properties. It will be shown that multimode fiber solitons form with
specific pulse width and energy dependent on the wavelength, and that they
naturally evolve toward fundamental-mode Raman solitons. New soliton fission
mechanisms, governed by the modal dispersion, will be explained. Possible
applications in space-division multiplexed systems will be discussed. A recent
thermodynamic approach to soliton condensation will be described.

###### keywords:

multimode fibers, solitons, nonlinear optics

111The license does not apply to third-party figures in this article, which
remains subject to the terms of the respective rights holder.

##  1 Introduction

Solitons are stable solutions of the nonlinear Schrödinger equation (NLSE),
governing the wave propagation in single-mode as well in multimode fibers
(MM).

Historically, MM fibers have been used in the 1970s and 1980s, before the
development of single-mode fibers 6772470 , in telecom systems of the first
generation, which made use of LED sources and GaAs semiconductor lasers at 850
nm wavelength, and of the second generation, which used InGaAsP lasers
operating at 1300 nm. The cross-talk among modes, together with the modal
delay responsible for temporal delays among modes, severely limited the
bandwidth of such systems to below 100 Mb/s. Hence, MM fibers were soon
replaced by the single-mode fibers which extended the channel capacity to 2
Gbit/s already in the early 1980s. During the first decade of 2000s, optical
transport networks based on single-mode fibers adopted advanced technologies,
such as wavelength-division multiplexing (WDM), coherent detection, advanced
amplitude and phase modulation formats, Raman amplification and digital signal
processing (DPS) for chromatic and polarization dispersion compensation
doi:https://doi.org/10.1002/0471221147.ch5 . Hence, the Shannon capacity limit
for a single mode fiber channel was reached 5420239 . The availability of new
multiple-input multiple-output (MIMO) technologies, able to recover the modal
cross-talk, allowed designers to reconsider the use of multimode fibers in
order to add a further degree of data multiplexing in the space domain (SDM).

Single-mode fiber solitons and dispersion managed solitons had been seriously
considered in the 1990s for developing high speed transport networks
ESSIAMBRE2002232 . However, the advent of digital compensation convinced
designers to use low-power transmission systems, where nonlinear effects were
negligible, while linear detrimental effects were compensated electronically.

Multimode fibers, however, are capable of transmitting several tens Watts of
power, against less than 500 mW for a single-mode fiber; the propagation of
solitons in MM fibers is therefore appealing for designing, for example,
unrepeatered systems able to reach long distances (up to 400-500 km), thanks
to the high peak power that multimode solitons can assume.

In this review work, we will retrace the steps that led to the discovery of
solitons in multimode fiber, and will describe their peculiar properties such
as modal condensation and pulse width invariance; we will see how those
properties are essential for the design of SDM soliton systems. New details
will be provided in the process of multimode soliton fission. A brief
description will be given of the recently introduced thermodynamic theory of
multimode soliton condensation.

##  2 Linear and soliton propagation in multimode fibers

A graded-index (GRIN) multimode fiber (MMF) is a cylindrical symmetric
waveguide where the refractive index decreases with the radial distance $r$ as
$n(r)=n_{0}\sqrt{1-2\Delta(r/a)^{\delta}}$, being $n_{0}$ the index at the
center of the core, $\Delta\approx(n_{0}-n_{c})/n_{0}$ the relative index
difference, $n_{c}$ the cladding index, $a$ the core radius and $\delta$ a
shape factor agrawal_2023 .

In commercial parabolic OM4 GRIN fibers, it is $\delta\approx 2.08$; hence,
the fiber assumes parabolic profile. This type of fiber is capable of
propagating $Q$ groups of Laguerre-Gauss degenerate modes; each group includes
$g_{j}$ modes and polarizations, $j=1,..,Q$; the number of modes and
polarizations in the j-th group equals $g_{j}=2j$. The total number of
propagated modes and polarizations is $M=Q(Q+1)=V^{2}/4$, with
$V=n_{0}k_{0}a\sqrt{2\Delta}$ a normalized frequency, and
$k_{0}=2\pi/\lambda$. The degenerate modes of a group have substantially equal
propagation constant Olshansky:75
$\beta^{(j)}=n_{0}k_{0}\sqrt{1-2\Delta(j/Q)^{2\delta/(\delta+2)}}$, which are
equi-spaced among modal groups.

In commercial step-index MM fibers, the shape factor $\delta>>10$, and the
refractive index profile assumes rectangular shape; such fibers are capable of
propagating $V^{2}/2$ modes whose field solution involves the Bessel functions
agrawal_2023 ; the propagation constants have non regular spacing in step-
index fibers.

An optical field propagating in a MMF supporting $M$ modes can be expressed as

$E(x,y,z,t)=\sum_{p=1}^{M}F_{p}(x,y)A_{p}(z,t)\exp{(in_{0}k_{0}z)},$ (1)

where $F_{p}(x,y)$ are the transverse field profiles of the modes and
$A_{p}(z,t)$ the complex amplitudes, accounting for modal power, phase, modal
dispersion and the temporal shape of the transmitted pulse.

In the case of parabolic GRIN fibers, the general wave equation for
$E(x,y,z,t)$ can be solved under the paraxial and slowly varying envelope
approximations, assuming $A_{p}(z,t)$ is a slowly varying function of $z$.
This brings to the (3D+1) generalized NLSE model Hasegawa:80 ; agrawal_2023 ,
which propagates all modes on a single polarization component $A_{x}$ or
$A_{y}$ of the total optical field; for polarization $x$ it is (the dual is
for $y$)

$\frac{\partial A_{x}(x,y,z,t)}{\partial
z}=\frac{i}{2k}\Big{(}\frac{\partial^{2}A_{x}}{\partial
x^{2}}+\frac{\partial^{2}A_{x}}{\partial
y^{2}}\Big{)}+i\sum_{n=2}^{4}\frac{\beta_{n}^{(x)}}{n!}\big{(}i\frac{\partial}{\partial
t}\big{)}^{n}A_{x}-\frac{\alpha_{x}(\lambda)}{2}A_{x}+\\\
+\frac{ik}{2}\Big{[}\frac{n^{2}(x,y)}{n_{0}^{2}}-1\Big{]}A_{x}+i\gamma\Big{[}(1-f_{R})A_{x}\Big{(}\lvert
A_{x}\rvert^{2}+\frac{2}{3}\lvert A_{y}\rvert^{2}\Big{)}+\\\ +f_{R}A_{p}\int
d\tau h_{R}(\tau)\Big{(}\lvert A_{x}(t-\tau)\rvert^{2}+\frac{2}{3}\lvert
A_{y}(t-\tau)\rvert^{2}\Big{)}\Big{]}.$ (2)

In Eq. 2, $\beta_{n}^{(x)}$ is the n-th order chromatic dispersion term for
polarization $x$, $\alpha_{x}(\lambda)$ the wavelength-dependent loss
coefficient, $\gamma=n_{2}\omega_{0}/(c\pi w^{2})$ accounts for the Kerr
nonlinearity, with $n_{2}$ (m2/W) the nonlinear index coefficient and $w$ the
beam waist, $h_{R}(\tau)$ is the Raman response Stolen1989 .

In the linear propagation regime, Eq. 2 includes only the linear terms, and
can be solved using the variational method Karlsson:92 , obtaining a
propagating beam characterized by an effective waist $w(z)$ oscillating with
distance $z$ around an input value $w_{0}$, according to the self-imaging law
Hansson2020

$w(z)=w_{0}\Big{[}\cos{\Big{(}\frac{\pi
z}{z_{p}}\Big{)}}^{2}+C\sin{\Big{(}\frac{\pi
z}{z_{p}}\Big{)}}^{2}\Big{]}^{1/2},$ (3)

with $z_{p}=\pi a/\sqrt{2\Delta}$ the self-imaging period (typically of the
order of 0.55 mm), and $C=z_{p}^{2}/(\pi\beta_{0}a_{0}^{2})^{2}$ the
oscillation extent, being $a_{0}=w_{0}/\sqrt{2}$ a beam width; self-imaging is
caused by the coherent interference among modes.

If a pulse is propagating instead of a continuous wave, the modal groups
separate in time because of the modal dispersion. Modal separation is depicted
in Fig. 1a, where 3 modes with axial symmetry, $LP_{01}$, $LP_{02}$ and
$LP_{03}$ (sometimes referred to as (0,0), (1,0) and (2,0) when addressing to
the radial and azimuthal indexes), are propagated over 1 km of GRIN fiber at
low power; the 3 modes experience pulse broadening induced by the chromatic
dispersion, as they separate.

Figure 1: Numerical simulation, showing the propagation of a 3-mode bullet at
the input and after 50 m and 1 km of GRIN fiber. a) 3D bullets in the linear
regime, b) soliton regime, c) modal instantaneous power at soliton energy.

The (3D+1) model of Eq. 2 propagates all modes on a single polarization
component $A_{x}$ or $A_{y}$ of the total optical field; it is very accurate
in describing the propagation of a large number of modes, and it is also
suitable for analyzing the propagation in fibers with complex index $n(x,y)$;
however, it is not suitable for describing the process of multimode soliton
formation, because it is lacking of the modal dispersion terms. As it will be
shown in the following sections, multimode optical soliton propagation is
characterized by an equilibrium among modal dispersion, Kerr and Raman
nonlinearity and chromatic dispersion; for the correct analysis of soliton
formation, it is of fundamental importance to include the effect of modal
dispersion.

However, as a first approximation, we can consider the multimode soliton as
already formed and apply a trial function to Eq. 2 Renninger2013

$E(x,y,z,t)=\sqrt{\frac{E_{p}}{2\pi
w(z)^{2}T_{0}}}\text{sech}\Big{(}\frac{t}{T_{0}}\Big{)}\exp{\Big{[}-\frac{x^{2}+y^{2}}{w(z)^{2}}+i(\theta
t^{2}+\alpha(x^{2}+y^{2})+\phi)\Big{]}}\mathchar 24635\relax\;$ (4)

in Eq. 4, $w(z)$ is the beam waist, $E_{p}$ the pulse energy,
$\theta,\alpha,\phi$ are chirp and phase parameters, and
$T_{0}=T_{FWHM}/1.763$ is related to the full-width at half-maximum temporal
pulse width. In RAGHAVAN2000377  and Ahsan:18  it was demonstrated that Eq. 4
is a solution of the (3D+1) equation in the variational approximation, for
negligible loss and Raman nonlinearity, and provided that the dispersion
length $L_{D}=T_{0}^{2}/\lvert\beta_{2}\rvert$, the nonlinearity length
$L_{NL}=2T_{0}/(\gamma E_{p})$, the diffraction length
$L_{diff}=\beta_{0}w_{0}^{2}$ and the self-imaging period $z_{p}$ are related
as $L_{D}/L_{NL}=z_{p}/(\pi L_{diff}$) , with $\beta_{0}$ the propagation
constant of the fundamental mode and $w_{0}$ the input beam waist,
respectively. According to the solution of Eq. 4, the multimode soliton
propagates with constant $sech$ temporal shape, and the beam waist undergoes
periodic oscillations with distance, with period $z_{p}$, around the value
$w_{0}$.

The stability of spatiotemporal solitons in GRIN fibers was theoretically
discussed and numerically analyzed in PhysRevA.97.013841 . It was observed
that the effective two-dimensional potential formed by the graded refractive
index prevents three-dimensional collapse into singularity as is known to
occur in uniform three-dimensional media. Stable fundamental and dipole-mode
spatiotemporal solitons were numerically found (Fig. 2).

Figure 2: Representation of stable spatiotemporal solitons propagating in a
graded-index optical fiber. Two types of spatial cross-section profiles are
shown on the right, namely, fundamental (b) and dipole-mode (c) solitons,
respectively. With permission from ref PhysRevA.97.013841 , under a Creative
Commons licence CC BY 4.0, American Physical Society.

A full description of the process of soliton formation must account for the
delay among modal groups, or modal walk-off. A common numerical model
accounting for this effect, used for pulse propagation in multimode fibers, is
the coupled GNLSE Horak2012 ; Wright2018a  for mode and polarization $p$,
modified to include modal and wavelength-dependent losses, and linear random-
mode coupling (RMC)

$\frac{\partial A_{p}(z,t)}{\partial
z}=i(\beta_{0}^{(p)}-\beta_{0})A_{p}-(\beta_{1}^{(p)}-\beta_{1})\frac{\partial
A_{p}}{\partial
t}+i\sum_{n=2}^{4}\frac{\beta_{n}^{(p)}}{n!}\big{(}i\frac{\partial}{\partial
t}\big{)}^{n}A_{p}-\frac{\alpha_{p}(\lambda)}{2}A_{p}+\\\
+i\sum_{m}q_{mp}A_{m}+in_{2}k_{0}\sum_{l,m,n}S_{plmn}\big{[}(1-f_{R})A_{l}A_{m}A_{n}^{*}+f_{R}A_{l}[h_{R}*(A_{m}A_{n}^{*})]\big{]}.$
(5)

In Eq. 5, $\beta_{n}^{(p)}$ is the n-th order dispersion term (modal and
chromatic) for mode $p$, $\alpha_{p}(\lambda)$ the modal and wavelength-
dependent loss coefficient, $n_{2}$ (m2/W) the nonlinear index coefficient
multiplying the Kerr and Raman terms. $S_{plmn}$ is an overlap integral among
modes, accounting for inter-modal four wave mixing (IM-FWM) and inter-mode
stimulated Raman scattering (IM-SRS)

$\displaystyle S_{plmn}=\frac{\iint
F_{p}^{*}F_{l}F_{m}F_{n}^{*}\mathrm{d}x\mathrm{d}y}{\sqrt{\iint\lvert
F_{p}\rvert^{2}\mathrm{d}x\mathrm{d}y\iint\lvert
F_{l}\rvert^{2}\mathrm{d}x\mathrm{d}y\iint\lvert
F_{m}\rvert^{2}\mathrm{d}x\mathrm{d}y\iint\lvert
F_{n}\rvert^{2}\mathrm{d}x\mathrm{d}y}}.$ (6)

$q_{mp}$ is the linear RMC coupling coefficient, from mode $m$ to $p$, arising
from randomly distributed index profile imperfections $\Delta n(x,y,z)$
HO2013491

$q_{mp}=\frac{k_{0}}{2n_{m}(I_{m}I_{p})^{1/2}}\iint\Delta
n^{2}(x,y,z)F_{p}^{*}(x,y)F_{m}(x,y)\mathrm{d}x\mathrm{d}y,$ (7)

where $k_{0}$ is the vacuum wavenumber, $n_{m}$ the modal effective index, and
$I_{m},I_{p}$ the modal intensity transverse profile. Degenerate modes are not
accounted for in Eq. 7, because their coupling is so fast that power
equipartition can be assumed into groups.

Common parameters at $\lambda=1550$ nm for a GRIN OM4 MMF are: $\beta_{2}=-27$
ps2/km, $\beta_{3}=0.14$ ps3/km for the fundamental mode, $\alpha=0.21$ dB/km,
negligible modal loss, $n_{2}=2.7\text{x}10^{-20}$ m2/W, $f_{R}=0.18$, index
parabolic factor $g=2.08$, relative index difference $\Delta=0.010$, core
radius $a=25\mu$m.

##  3 Modal attractors in multimode solitons

The solution proposed in Eq. 4 for a GRIN fiber, describing a self-imaging
soliton which preserves the modal content over long distance, assumes that a
multimode soliton has already formed, without considering the role of the
modal dispersion.

However, experimental results reported for MMF solitons reveal properties not
accounted for by the periodical solution of Eq. 4.

A first experimental evidence of a multimode soliton generated by the
stimulated Raman scattering (SRS) was reported in Grudinin:88 ; an ultrashort
pulse with 90 fs duration at 1650 nm was generated by injecting a 150 ps pump
pulse at 1064 nm, with 600 kW peak power, into 500 m of parabolic GRIN fiber.

In Renninger2013 , it was reported the generation of a multimode soliton at
telecom wavelength. 300-fs pulses with energy up to 3 nJ were injected into
100 m of GRIN fiber; an output mode-field-diameter of 17.9 $\mu$m was obtained
when injecting an 11.5 $\mu$m input beam; at 0.5 nJ input energy, it was
obtained an output pulse width comparable to the input. For larger energies,
it was observed a soliton affected by wavelength red-shift produced by the
SRS; it was also reported a residual wave with no wavelength shift.

In Wright:15 , experimental results (Fig. 3), compared to numerical
simulations, confirmed the generation of Raman multimode solitons; 500-fs
pulses at $\lambda=1550$ nm were launched into 25 m of GRIN fiber with
$2a=62.5\mu$m. The initial excitation was controlled in order to stimulate the
higher-order modes at the input fiber end. Raman solitons were observed,
affected by wavelength red-shift up to $\lambda=2100$ nm for input energy
increasing from 3.2 nJ up to 34 nJ. The presence of a large amount of
broadband dispersive wave Wright2015a , related to the spatiotemporal
oscillations of the soliton, was also detected.

In Wu:21 , numerical and experimental results reported the near-field,
spectrum and pulse width of multimode solitons, when 500 fs pulses at 1550 nm
were launched into a 15 cm GRIN fiber with a 50 $\mu$m core diameter and 0.2
numerical aperture. Spatial beam self-cleaning Krupa2017  was observed when
there was significant overlap of the launched beam profile with the
fundamental mode of the fiber. Output pulse width decreased from 480 fs to 30
fs when the input peak power was increased from 20 kW to 410 kW.

In Zhu:16 , it was reported the observation of multimode solitons in few-mode
fibers (FMF), a GRIN fiber supporting only modes $LP_{01}$, $LP_{11a}$ and
$LP_{11b}$. 280-fs pulses at $\lambda=1550$ nm were launched into 25 m of FMF
at variable energy, controlling the input coupling so that the $LP_{01}$ mode
(case IC1) or the $LP_{11}$ modes (case IC2, IC3) had larger power. Figures 4a
and 4b in Zhu:16  show the simulated and measured output pulse width,
respectively, for increasing input energy and in the three coupling
conditions. Experimental data reported an output pulse width decreasing for
growing input energy, converging to 120 fs for energy between 2 and 3 nJ. It
will be shown in Sec. 4 that this is a standard pulse width duration for
multimode solitons at telecom wavelength.

In Zitelli:22  it was experimentally reported the generation of a soliton in a
multimode step-index fiber; 70-fs pulses at 1450 nm were launched into 10 m of
a commercial step-index fiber with $2a=50\mu$m. A multimode soliton was
observed at the output after injecting 21 nJ input energy. The soliton
captured the modes $LP_{01}$, $LP_{11a}$ and $LP_{11b}$ while forming, and it
evolved to a fundamental mode Raman soliton; the residual wave included the
higher-order modes, and was lacking of the fundamental mode.

Figure 3: Experimental evidence of Raman soliton generation after 25 m of GRIN
fiber. With permission from ref. Wright:15 , Optica Publishing Group.

In Zitelli2021  and Zitelli2021a  they were reported numerical simulations and
experimental results using GRIN fibers, describing a complex scenario for
soliton formation, where the modal walk-off length
$L_{W}=T_{0}/\Delta\beta_{1}$ comes into play; the quantity

$\Delta\beta_{1}=\frac{\sum_{j}P_{j}(\beta_{1j}-\beta_{10})}{\sum_{j}P_{j}},$
(8)

in (s/m), is the differential first-order dispersion coefficient with respect
to the fundamental mode, averaged among the propagating modes with power
$P_{j}$.

Figure 4 shows a numerical simulation performed in Zitelli2021  using the
model of Eq. 5, where a 70-fs pulse at $\lambda=1550$ nm is launched at the
input of a GRIN OM4 fiber with 3 radial modes $LP_{01}$, $LP_{02}$ and
$LP_{03}$, at a total peak power $P_{0}=28$ kW. Modal pulses form a multimode
soliton after several meters distance. The power profile of the 3 modes at the
fiber input end is represented in Fig. 4a. During the first few meters of
propagation, the Kerr nonlinearity causes a wavelength shift of the spectra of
the modes (inset of Fig. 4d), in such a way that the chromatic dispersion
compensates for the modal dispersion. Once the modal velocities are equalized,
the 3 overlapping pulses propagate exchanging power by IM-FWM and IM-SRS; the
nonlinear process promotes the fundamental mode, which acts as an attractor
the energy of the other modes YANKOV1997147  (Fig. 4c).

Figure 1b depicts the propagating bullets in the soliton regime, as power is
gradually transferred to the fundamental mode. Figure 1c shows the temporal
profiles of the 3 propagating modes. The basic difference respect to the
linear regime of Fig. 1a, is that the mode groups do not separate in time, and
propagate while mutually trapped; once the mode $LP_{01}$ has attracted all
the pulse energy, the soliton propagates as a fundamental mode bullet.

At the same time, stimulated Raman scattering induces a wavelength red-shift
to the modal pulses (Fig. 4d), or soliton self-frequency shift (SSFS), whose
change per unit distance can be quantified as Gordon1986 ; Zitelli_9887813

$\frac{d\Delta\lambda}{dz}=\frac{4T_{R}}{15\pi
c\lambda^{2}(z)\lvert\beta_{2}(z)\rvert^{3}}\Big{(}\frac{n_{2}E_{s0}}{w_{e}^{2}}\Big{)}^{4},$
(9)

with $w_{e}$ the effective beam waist, and $T_{R}=f_{R}\int\tau
h_{R}(\tau)d\tau$ the mean Raman response time, equal to approximately 3 fs in
silica fibers. Experimental evidence of the SSFS is given in Fig. 3.

Since the higher-order modes have lower wavelength with respect to $LP_{01}$,
the IM-SRS boosts the transfer of power to the fundamental mode. After 100 m
distance, nearly all the energy has been transferred to the $LP_{01}$ mode,
which propagates as a fundamental mode Raman soliton Zitelli2021 . Fiber loss
reduces the pulse energy; as a consequence, the soliton must increase the
pulse width according to the soliton equation relating the pulse energy and
the pulse width Renninger2013

$E_{p}=\frac{\lambda\lvert\beta_{2}(\lambda)\rvert w_{e}^{2}}{n_{2}T_{0}}.$
(10)

After 1 km propagation, a single-mode larger soliton remains (Fig. 4b) with
some residual wave still conserving the input wavelength.

Figure 4: Numerical simulation, showing a 70-fs pulse at $\lambda=1550$ nm and
28 kW peak power, including 3 radial modes propagating over 1 km of GRIN OM4
fiber. a), b): modal instantaneous power at the input and output,
respectively. c), d): Modal energy and wavelength vs. distance, respectively.
With permission from ref. Zitelli2021 , Optica Publishing Group.

Figure 5 shows the experimental results from Zitelli2021a , when propagating
67 or 245 fs pulses at $\lambda=1300$ to $1550$ nm over 120 m of GRIN OM4
fiber. The insets on the top of Fig. 5a illustrate the measured
autocorrelation traces and spectra for input pulse width 67 fs and wavelength
1550 nm. For increasing input pulse energy $E_{p}=1$ to $8$ nJ, the output
pulse width has a minimum of 260 fs at an optimal energy of 2 nJ.
Correspondingly, the soliton pulse wavelength experiences an increasing red-
shift with respect to the small residual wave, whose wavelength is conserved.
Figure 5a also shows that the minimum pulse width does not change
significantly for different input pulse width or wavelength, with the
exception of $\lambda=1300$ nm, near the zero-chromatic dispersion wavelength.

The insets above Fig. 5b illustrate the corresponding near-fields measured for
a 67 fs pulse at 1550 nm. For increasing energy, the output waist reduces as
the soliton is forming; at an optimal 2 nJ input energy, the output beam waist
approaches to the theoretical value of the fundamental mode ($w_{0}=7.8\mu$m);
for higher energies, the output beam waist assumes larger values, denoting the
presence of a large amount of dispersive wave spatially overlapped to the
soliton near-field. When a different input pulse width (245 fs) is tested at
the same wavelength, the same values of the case with 67 fs input pulse width
are found for the minimum waist and the corresponding energy. The 2 nJ energy
value corresponding to minimum waist was also found for the minimum pulse
width.

The experiment confirms the dynamics of soliton formation already seen in Fig.
4. As a consequence of IM-FWM and IM-SRS, multimode soliton evolves, at an
optimal energy, to a fundamental mode soliton experiencing Raman red-shift; a
residual wave is eventually observed, which is generally minimized at an
optimal input energy.

Figure 5: Experimental results propagating 67 or 245 fs pulses at
$\lambda=1300$ to $1550$ nm over 120 m of GRIN OM4 fiber. a), b): output pulse
width and beam waist at 120 m distance, respectively. The insets above the
plots are the measured spectra, autocorrelation traces and beam near-filed, at
$\lambda=1550$ and at the indicated pulse energies. Adapted with permission
from ref. Zitelli2021a , under a Creative Commons licence CC BY 4.0, Springer
Nature Limited.

##  4 Pulse width invariance

In Zitelli_9887813 , the experimental data of Fig. 6 were reported, which must
be compared with the experimental results of Fig. 5 in order to have a better
insight into the process of multimode soliton formation.

In the figure, it is shown the output pulse width vs. pulse energy, measured
at $\lambda=1450,1550$ or $1650$ nm for a 70 fs input pulse width, over
different lengths of GRIN OM4 fiber. At 6 m distance, pulse width rapidly
reaches a minimum of 100 fs for increasing energy; it is not possible to find,
at this short distance, an optimal soliton energy, because it has not fully
formed yet.

For increasing length, the existence of an optimal soliton energy becomes
clear; at 830 m distance, a sharp minimum of the output pulse width is found
at $E_{s}=1.5,2$ or $2.5$ nJ input energy for the 3 wavelengths, respectively.
Hence, the initial pulsewith $T_{s0}$ of the soliton must be measured at short
distance, in correspondence to the optimal energy measured after long
distance. In the specific case of Fig. 6, it is $T_{s0}=70,120$ and $170$ fs
for the three wavelengths.

Figure 6: Experimental pulse width vs. soliton energy for a pulse propagating
over 6 m or 830 m of GRIN fiber. Input parameters are: 1450 nm and 60 fs, 1550
nm and 70 fs, 1650 nm and 60 fs; 30 $\mu$m beam diameter. Red marks are
calculated from Eq. 11. With permission from ref. Zitelli_9887813 , Optica
Publishing Group.

In Zitelli2021a , soliton propagation was experimentally investigated with
cut-backs from 120 m to 1 m of GRIN fiber, for different input pulse width and
wavelength; the experimental data of Fig. 7 were obtained. Here, the initial
pulse width of the formed soliton $T_{s0}$ is reported vs. the input
wavelength for different input pulse widths, which do not seem affecting the
value of $T_{s0}$. Experimental data were collected at the optimal soliton
energy $E_{s0}$ corresponding to minimum long-distance pulse width; we recall
that this is generally larger that $T_{s0}$, because of linear losses and the
red-shift induced by the Raman soliton self-frequency shift (SSFS) Grudinin:88
; Gordon1986 .

Figure 7: Theoretical curve of soliton pulse duration vs. wavelength, compared
with measured and simulated soliton pulse width. Experiments carried out with
1 m of GRIN fiber, with input wavelength and pulse width: 1300 nm and 61 fs,
1350 nm and 61 fs, or 1420 nm and 70 fs, respectively. Experiment carried out
with 6 m of GRIN fiber, with input wavelength and pulse width: 1550 nm and 67
fs, 1680 nm and 96 fs, respectively. With permission from ref. Zitelli2021a ,
under a Creative Commons licence CC BY 4.0, Springer Nature Limited.

In the same figure, numerical simulations obtained using the model of Eq. 5
are reported. Simulations show that two pulses with different input pulse
width (67 or 235 fs), and same wavelength (1350, 1550 or 1680 nm), produce the
same multimode soliton after 3 to 6 m distance. The pulse width $T_{s0}$
immediately after the soliton formation looks affected only by the input
wavelength.

This property of pulse width invariance appears as a discriminating point for
multimode solitons with respect to the single-mode counterpart
Zakharov1970ExactTO ; Agrawal2013 . Here is where the modal walk-off length
$L_{w}$ comes into play. Solitons in single-mode fiber form when the chromatic
dispersion length $L_{D}$, measuring the distance where pulses have broadened
by a factor $\sqrt{2}$, balance the nonlinearity length $L_{NL}$, indicating
where the nonlinear phase shift induced by the self-phase modulation (SPM)
equals 1 rad Agrawal2013 .

In multimode fibers, other distances to consider are the random mode coupling
and birefringence correlation lengths, $L_{cm}$ and $L_{cp}$, defined as the
characteristic lengths associated with the linear coupling between modes or
between polarizations, respectively HO2013491 ; Xiao2014TheoryOI .

In order to explain the observations of Fig. 7, it must be assumed that
nonlinearity acts over distances shorter than those associated with random
mode coupling and birefringence, i.e., for $L_{NL}<L_{cm},L_{cp}$. Besides, in
multimode solitons the modal dispersion must balance the chromatic dispersion
as well; modal pulses need to propagate at the same speed before interacting
by IM-FWM and IM-SRS; hence, the modal walk-off length must be comparable to
the dispersion and nonlinearity lengths. A condition for temporal trapping of
the optical modes was originally predicted, for the j-th mode, as Hasegawa:80
; Crosignani:81  $L_{wj}\geq L_{D}=L_{NL}$. This condition was restricted in
Zitelli2021a  to $L_{D}=L_{NL}=\delta L_{w}$, with $\delta$ an experimental
adjustment constant close to unity. By replacing the respective expressions,
it was found for the initial soliton pulse width

$T_{s0}(\lambda)=\delta\cdot
1.763\frac{\lvert\beta_{2}(\lambda)\rvert}{\Delta\beta_{1}(\lambda)}.$ (11)

Equation 11 fairly predicts the value of multimode solitons pulse width, given
the wavelength and the fiber dispersion parameters. Because of the fundamental
role of the walk-off length in the process, the name of walk-off solitons was
proposed for this type of pulses. In Fig. 7, the black curve is calculated
from Eq. 11, showing good agreement with the experimental and numerical data.
In Fig. 6 the theoretical soliton pulse width from Eq. 11 are reported as red
marks over the short distance curves, for the three tested wavelengths.

According to the observations of Figs. 6 and 7, the walk-off solitons form at
a specific pulse energy calculated from Eqs. 11 and 10 (with
$T_{0}=T_{s0}/1.763$). For a GRIN OM4 fiber at $\lambda=1550$ nm, it is
$T_{s0}=120$ fs and $E_{s}=1.5$ nJ, plus an amount of input energy eventually
remaining in the residual wave.

The invariance property for the pulse width and the energy of a multimode
soliton implies that, when injecting a pulse with proper energy $E_{s0}$ and
pulse width larger or shorter than $T_{s0}$, the travelling modes adjust their
pulse width to the soliton value, Eq. 11, depending on the fiber dispersion
and the input wavelength. In the case of input pulse width much larger than
$T_{s0}$ (for example, few picoseconds at $\lambda=1550$ nm), the multimode
soliton does not form; this property is important for the design of soliton
space-division multiplexed systems (SDM), as it will be seen in Sec. 7. If, on
the other hand, the input energy is much larger than $E_{s0}$, a soliton pulse
may originate with proper pulse width and energy, emerging from a large amount
of dispersive wave.

##  5 Soliton fission and interaction

In those cases where the modal content of the input pulse is complex, for
example if the higher-order modes possess larger energy respect to the
fundamental, or a large number of modes is propagating, or the input pulse
energy is greater than $E_{s0}$ from Eq. 10, multiple solitons can originate
from a single input pulse.

In Wright:15 , experimental evidence of multiple soliton generation was
provided using 95 m of GRIN fiber, and launching 500-fs pulses at
$\lambda=1550$ nm. A main higher energy Raman soliton was observed at the
output, with red-shifted 1620 nm wavelength and 10 $\mu$m beam waist. A
smaller secondary soliton spectrum was also detected, together with a large
amount of dispersive wave.

Cherenkov spectral peaks, generated by multimode soliton fission processes in
nonlinear MMFs, were observed in Eftekhar2021 . It was reported the formation
of a multimode soliton, that forces all the modes involved to first coalesce
in the temporal domain, and in doing so, individually shift their spectral
content. A phase-matching condition of solitons after fission occurs, which is
the cause of the Cherenkov dispersive waves extending from the 1550 nm input
wavelength down to the normal dispersion region of a MMF.

In Buch:16 ; Sun:22  it was numerically and experimentally investigated the
collision interaction of multimode solitons. When propagating solitons into
degenerate modes, it was found that each soliton transfers some of its power
to the other mode through IM-FWM. In spite of this intermodal power transfer,
each soliton keeps propagating as a bimodal soliton and interacts with the
other bimodal soliton as if they were propagating inside a single-mode fiber.
The collision outcome is dependent on the input relative phase of the two
modes. Fig. 3 of Buch:16  shows the collision between modes $LP_{11a}$ and
$LP_{11b}$ with different values of relative phase between the two input
beams; a complete transfer of power between the two modes is observed for
relative phase $\pi/8$, which reduces to a partial transfer for phase $\pi/4$
and is absent for phase $\pi/2$.

In Zitelli:20 , it was numerically and experimentally investigated the soliton
fission in short spans (30 cm) of GRIN fiber at the telecom wavelength. For
input energy larger than 15 nJ, the presence of nonlinear losses had to be
taken into account. In this regime of extreme nonlinear effects, it was
observed the generation of a train of solitary pulses with common pulse width
(50-60 fs); pulse duration was observed to be nearly constant for input energy
in the range 15-50 nJ.

Recently, it was proposed an alternative interpretation of the fission
mechanism for multimode solitons, involving the interplay of modal dispersion,
random-mode coupling (RMC) and IM-FWM zitelli2023statistics . The process is
described by Fig. 8 showing a numerical simulation, performed using the model
of Eq. 5, of a 250 fs pulse at 1400 nm, propagating in a long span of GRIN
fiber; pulse is composed by 28 Laguerre-Gauss modes having uniform power
distribution at the input. Figure 8a illustrates the propagation with
negligible nonlinearity (input energy 0.02 nJ); pulses composed by the
individual groups of degenerate modes separate in time and broaden as a
consequence of modal and chromatic dispersion, respectively. If RMC was not
present, the modal content of the pulses would be limited to the respective
groups; instead, RMC transfers energy to the other groups, in a way that each
pulse will also include modes from the lower-order and higher-order groups and
from the fundamental mode (the black curve in the figure); the result is a
multimodal background noise into each propagating pulse. When the input pulse
is increased to soliton level (5.0 nJ in Fig. 8b), the self-phase modulation
(SPM) counteracts the dispersion-induced broadening; IM-FWM transfers power
among the modes of a same pulse; the fundamental mode, which is present into
each pulse as a consequence of the RMC, is an attractor from the higher-order
modes; after hundreds meters distance, a train of fundamental mode solitons is
obtained, each one assuming the pulse width given by Eq. 11. Solitons are
affected by Raman self-frequency shift, and IM-SRS provides a contribution to
the process of condensation to the fundamental mode.

Figure 8: Modal instantaneous power for a 250 fs pulse at 1400 nm, propagating
in a long span of GRIN fiber; pulse is composed by 28 Laguerre-Gauss modes
with uniform power distribution at the input. a): linear regime, b): soliton
power.

##  6 Thermodynamic approach of multimode solitons

Optical thermodynamics of multimode systems Wu2019 ;
doi:10.1080/23746149.2023.2228018 ; pourbeyram2022direct  is a powerful tool
for analyzing complex multimode interactions. When a physical process is
responsible for a chaotic power transfer among fiber modes, such as the IM-
FWM, and provided the pulse energy and peak power are preserved, the power
distribution at the output of a multimode fiber can be described by a
Rayleigh-Jeans or by a Bose-Einstein law.

Multimode soliton propagation is an example of energy and peak power
conservation. It was experimentally demonstrated zitelli2023statistics  that
the process of multimode soliton generation can be described as a modal
condensation in thermodynamic terms and, during its formation, the modal power
content assumes a Bose-Einstein distribution. 250-fs pulses at 1400 nm were
launched into 830 m of OM4 GRIN fiber; the power distribution of the modal
groups was measured starting from the peaks of the detected pulses,
corresponding to the different modal groups, similar to what is shown in Fig.
8. It was found that, at soliton power, the mean modal power fraction of the
degenerate modes with same eigenvalue (the differential propagation constant
$\epsilon_{i}=\beta_{i}-\beta_{M},i=1,..,M$ , being $M$ the number of
propagating modes) followed a Bose-Einstein distribution when plotted against
$\epsilon_{i}$. More than half of the total power was attracted by the
fundamental mode (the first detected pulse). The corresponding output near
field showed the fundamental mode dominating over a residual of higher-order
modes. Consequently, it was found that the output mean modal power fraction
condensates to the fundamental mode. The Bose-Einstein law fitted the modal
distribution up to the 9-th modal group (45 modes per polarization),
demonstrating that the thermodynamic approach is an effective way to describe
the process of multimode soliton formation and condensation to the fundamental
mode.

##  7 Optical solitons in SDM systems

The inability of picosecond pulses to form a multimode soliton, discussed in
Sec. 4, is the basis of soliton SDM, where the individual channels are coupled
into single modes or groups of degenerate modes. In Mecozzi:12  it was
demonstrated that, although using large pulses, individual groups of
degenerate or quasi-degenerate modes are able to generate intra-group Manakov
solitons; RMC among degenerate modes, which is responsible for a disordered
transfer of energy among the modes of a same group, does not prevent the
formation of those solitary solutions predicted by the coupled Manakov
equations Mumtaz:13 ; to the contrary, disorder causes a fast energy exchange
among degenerate modes, which minimize the modal dispersion within one group
and promotes the formation of Manakov solitons.

By properly distributing the input power between the mode groups, one soliton
channel for each group can be generated 10034667 ; the solitons of different
groups cannot merge into a single multimode soliton, because the input pulse
width does not satisfy the walk-off condition represented by Eq. 11. Hence,
soliton channels propagate at different velocities, and experience elastic
collisions with the other modal groups in a similar way to what it is observed
in wavelength-division multiplexed systems. This property permits the design
of stable, high-speed multimode soliton SDM systems.

##  8 Conclusions

Multimode fiber solitons demonstrate unique properties, such as pulse width
invariance and energy condensation, which are not found in the single-mode
counterpart. When picosecond pulses are propagated, compatible with telecom
applications, the disorder caused by fiber imperfections promotes the
formation of Manakov solitons within the individual modal groups. However, the
picosecond soliton pulses generated within the groups are inhibited from
merging into a single multimode soliton, because Eq. 11 is not satisfied.
Hence, stable modal soliton channels are possible, performing elastic
collisions with other channels; this lays the foundation for the design of
soliton SDM transmission systems.

The ability of MM fibers to transport tens of Watt of optical power, makes it
possible to design high peak power soliton systems, between a few Watts and a
few tens of kW, able to propagate over long distance with no amplification, or
using Raman amplification pumped from the transmitter or the receiver side.
Unrepeatered systems of this type can potentially double the distance that is
commonly obtained with single-mode fiber.

Furthermore, the capability of condensating all the launched power into the
fundamental mode of a MM soliton, improves the quality of beam power delivery
systems employing multimode fibers.

##  9 Backmatter

Funding This work was supported by:

Project ECS 0000024 Rome Technopole, Funded by the European Union -
NextGenerationEU.

Disclosures The author declares no conflicts of interest.

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# On the normality of monoid monomorphisms

Nelson Martins-Ferreira  Instituto Politécnico de Leiria, Leiria, Portugal
<EMAIL_ADDRESS>and  Manuela Sobral  CMUC and Departamento de
Matemática, Universidade de Coimbra, 3001–501 Coimbra, Portugal
<EMAIL_ADDRESS>
###### Abstract.

In the category of monoids we characterize monomorphisms that are normal, in
an appropriate sense, to internal reflexive relations, preorders or
equivalence relations. The zero-classes of such internal relations are first
described in terms of convenient syntactic relations associated to them and
then through the adjunctions associated with the corresponding normalization
functors. The largest categorical equivalences induced by these adjunctions
provide an equivalence between the categories of relations generated by their
zero-classes and the ones of monomorphisms that we suggest to call _normal
with respect to_ the internal relations considered. This idea, although being
transverse to the literature in the field, has not in our opinion been
presented and explored in full generality. The existence of adjoints to the
normalization functors permits developing a theory of normal monomorphisms,
thus extending many results from groups and protomodular categories to monoids
and unital categories.

###### Key words and phrases:

Internal binary relations. Zero-classes. Clots. Positive cones. Normal
subojects. Syntactic relations

###### 2010 Mathematics Subject Classification:

18B10, 06F05, 18A40, 08A30

This work is supported by Fundação para a Ciência e a Tecnologia FCT/MCTES
(PIDDAC) through the following Projects: Associate Laboratory ARISE
LA/P/0112/2020; UIDP/04044/2020; UIDB/04044/2020; PAMI - ROTEIRO/0328/2013 (Nº
022158); MATIS (CENTRO-01-0145-FEDER-000014 - 3362); Generative.Thermodynamic;
by CDRSP and ESTG from the Polytechnic Institute of Leiria.

The second author was partially supported by the Centre for Mathematics of the
University of Coimbra – UID/MAT/00324/2020

##  1\. Introduction

In the category of groups every internal reflexive relation is an internal
equivalence relation, i.e. a congruence. In addition, the zero-classes of such
congruences, that are exactly the normal subgroups, determine all the other
classes and so the congruence itself. As it is well known, this is not the
case in the category $\mathbf{Mon}$ of monoids where we have strict inclusions

$\mathbf{Eq}(\mathbf{Mon})\subset\mathbf{Ord}(\mathbf{Mon})\subset\mathbf{RRel}(\mathbf{Mon})$
(1)

of the full subcategories of the category $\mathbf{Rel}(\mathbf{Mon})$ of
internal relations in monoids consisting of reflexive relations, preorders and
equivalence relations. Furthermore, the corresponding zero-classes do not
determine the relations, in general.

Clots [1] are the zero-classes of internal reflexive relations and positive
cones is the name of the zero-classes of preorders. The normal submonoids are
the zero-classes of congruences. In [12], internal transitive and reflexive
relations in monoids, i.e. the preordered monoids, denoted there by
$\mathbf{Ord}\mathbf{Mon}$, were studied and an equivalence was established
between the full subcategory $\mathbf{Ord}\mathbf{Mon}^{*}$ of all preordered
monoids whose preorder is induced by its positive cone $P$ (where
$a\leq_{P}b\Leftrightarrow b\in P+a$) and the full subcategory of
monomorphisms in monoids that consists of what were called there the right
normal mononorphisms (those inclusions $M\to A$ for which $a+M\subseteq M+a$
for every $a\in A$). Here we are going to describe the larger equivalence of
categories induced by the adjunction $F\dashv N$, where $N$ is the
normalization functor, which, in particular, provides a categorical definition
of positive cone.

The categorical definition of a clot was given in [9] for semi-abelian
categories and then in [10] in pointed regular categories with finite
coproducts. In Section 3, we consider clots in what seems to be the more
general context where the definition makes sense and its correspondence to
internal reflexive relations, called semicongruences there, can be studied.

Inspired by the so-called syntactic congruence and syntactic preorder for
subsets $M$ of a free monoid $A$ (see [5, 15]), we define what we call the
M-congruence and the M-preorder, when $M$ is a submonoid of an arbitrary
monoid $A$. The name is justified by the fact that these relations are
completely determined by $M$. Furthermore, they characterize the submonoids of
$A$ that are the zero classes of congruences and of preorders, respectively
(Propositions 1 and 2).

We also define, for each submonoid $M$ of $A$, a reflexive relation $R_{M}$.
Whenever $R_{M}$ is an internal relation, in which case we call it the
$M$-reflexive relation of the submonoid $M$, we obtain a completely analogous
characterization of the submonoids that are clots (Proposition 3). There is a
large class of monoids for which these reflexive relations are indeed internal
that includes the finite and the Dedekind finite monoids. However, it is an
open problem to decide whether this holds or not for all monoids as stated in
Example J (see Section 4).

So natural (and useful) these relations may be, they are not functorial: there
is no functor with these functions of objects as we show in Example H. In
contrast, in Section 3, we describe such functorial relations as left adjoints
to the three normalization functors

$N\colon\mathcal{R}\to\mathbf{Mono}(\mathbf{Mon}),$

where $\mathcal{R}$ is one of the categories of internal relations referred to
above as instances of a more general situation (Theorem 1).

The largest equivalence of categories induced by each of these adjunctions
$F\dashv N(\eta,\varepsilon)$, i.e. $Fix(\varepsilon)\sim Fix(\eta)$, between
the full subcategories with all the objects for which the counit or the unit
are isomorphisms, respectively, have a nice interpretation. In each case, the
full subcategory $Fix(\varepsilon)$ consists of all the corresponding
relations that are generated by the zero-class (in the sense that such a
relation is the smallest one with the same zero-class) and $Fix(\eta)$ is the
full subcategory $\mathbf{NMono}(\mathbf{Mon})$ of
$\mathbf{Mono}(\mathbf{Mon})$ consisting of what we call the normal
monomorphisms with respect to objects of $\mathcal{R}$ that are exactly the
clots, the positive cones and the normal monomorphisms, respectively.

The main purpose of this paper is to provide a systematic study of normal
monomorphisms relative to specific types of internal binary (endo)relations
and it is is organized as follows. In Section 2, specific congruences,
internal preorders and reflexive relations on a monoid $A$, associated with
each submonoid $M$ of $A$, are introduced in order to characterize their zero-
classes. In Section 3, we define left adjoints to the corresponding
normalization functors as a specialization of a more general situation in the
context of a pointed, wellpowered and finitely complete category $\mathbf{C}$
with arbitrary intersections of subobjets and categories $\tau(\mathbf{C})$ of
binary relations of a fixed type $\tau$ on objects of $\mathbf{C}$, satisfying
some conditions. In pointed varieties $\mathcal{V}$, the smallest internal
relations on an object $A$ of $\mathcal{V}$ whose zero class contains a
subobject $M$ of $A$ can be described as special subobjects of $A\times A$, as
we do here when $\mathcal{V}=\mathbf{Mon}$, providing an alternative way to
describe these left adjoints, a straightforward procedure that contributes to
better exhibit the differences between the corresponding zero-classes. We put
together our observations in three theorems that, in particular, give a
systematic way to construct for each monomorphism that is normal to one of the
internal relations considered above the smallest internal relation to which it
is normal to.

In the last section, we present several examples illustrating our previous
claims.

##  2\. Syntactic relations

In this section we define binary relations from the syntactical point of view
that will play a role in the characterizations of clots, positive cones and
normal submonoids presented in Theorems 2–4.

The syntactic monoid is widely used in computation theory due to its
applications in automata theory and regular languages (see [5, 8, 13, 15]).
According to Eilenberg ([5], p.75), the first clear-cut and systematic
exposition of the syntactic monoid can be found in [13]. The syntactic
congruence and syntactic preorder are usually defined for subsets $M$ of a
free monoid $\Sigma^{*}$. For our purposes we will consider an arbitrary
monoid $A$ instead of a free monoid and submonoids $M$ of $A$ rather than
subsets. However, both the syntactic congruence and the syntactic preorder are
internal relations regardless of whether $M$ is a submonoid or just a subset
of $A$. The relevance of this aspect in our study contrasts with the one
usually needed in computer science. Instead of asking if a subset $M$ of a
monoid $A$ is recognizable (in the sense of [5], p. 68), we will be interested
in knowing whether $M$ is or not the zero-class of the syntactic congruence or
the syntactic preorder, in which cases $M$ is necessarily a submonoid of $A$.

###  2.1. The syntactic congruence

Given a subset $M$ of a monoid $A$, the syntactic M-congruence is defined by
$a\sim_{M}b$ if, for every $x,y\in A$,

$xay\in M\Leftrightarrow xby\in M.$

This binary relation is an equivalence relation. Furthermore, it is compatible
with the monoid operation: if $a\sim_{M}b$ and $a^{\prime}\sim_{M}b^{\prime}$
then, for every $x,y\in A$,

$xa(a^{\prime}y)\in M\Leftrightarrow xb(a^{\prime}y)=(xb)a^{\prime}y\in
M\Leftrightarrow(xb)b^{\prime}y\in M.$

In other words, the set $E=\\{(a,b)\in A\times A\mid a\sim_{M}b\\}$ is a
submonoid of $A\times A$, and being an equivalence relation, it is a
congruence on $A$. The syntactic monoid is the quotient $A/E$ and the zero-
class of $E$,

$[1]_{E}=\\{u\in A\mid(1,u)\in E\\},$

is a submonoid of $A$.

######  Proposition 1.

Let $M$ be a submonoid of the monoid $A$. Then $M$ is the zero-class of some
congruence in $A$ if and only if the following condition holds

(C) For every $x,y\in A$ and every $u\in M$,

$xy\in M\Leftrightarrow xuy\in M.$

In this case, $M$ is exactly the zero class of $\sim_{M}$.

###### Proof.

The zero class of $\sim_{M}$ is contained in $M$. Indeed, if $1\sim_{M}u$
then, for every $x,y\in A$,

$xy\in M\Leftrightarrow xuy\in M.$

So, in particular, for $x=y=1$, $1\in M\Leftrightarrow u\in M$.

Furthermore, $M\subseteq[1]_{\sim_{M}}$ whenever $(C)$ holds: if $u\in M$ then
it belongs to the zero class of the $M$-congruence when

$xy\in M\Leftrightarrow xuy\in M,$

that is exactly condition $(C)$.

If $M=[1]_{\sim}$ for some congruence $\sim$ on $A$ then if $1\sim u$, by
reflexibility and compatibility, we conclude that $xy\sim xuy$. Since $xy\in
M$ means that $1\sim xy$ then also $1\sim xuy$, by transitivity of the
relation, and so $xuy\in M$. Consequently, condition $(C)$ is fulfilled.

So, we conclude that $M$ is the zero class for some congruence on $A$ if and
only if it is the zero class of its $M$-congruence. ∎

######  Remark 1.

In [6] Facchini and Rodaro present (Thm 9), the following necessary and
sufficient condition for a submonoid $M$ of a monoid $A$ to be a kernel:

(F) For every $x,y\in A$,

$xMy\cap M\neq\emptyset\Rightarrow xMy\subseteq M,$

that is, obviously, equivalent to condition

(C) For every $x,y\in A$ and every $u\in M$,

$xy\in M\Leftrightarrow xuy\in M,$

as it is easy to prove:

(F) $\Rightarrow$ (C)

If $x,y\in A$ and $u\in M$ then

$xy\in M\Rightarrow xMy\cap M\neq\emptyset\Rightarrow xMy\subseteq
M\Rightarrow xuy\in M.$

And

$xuy\in xMy\cap M\Rightarrow xMy\subseteq M\Rightarrow x1y=xy\in M.$

(C) $\Rightarrow$ (F)

For $x,y\in A$, if there exists an element $u\in M$ such that $xuy\in M$ then,
for every $v\in M$,

$xuy\in M\Leftrightarrow xy\in M\Leftrightarrow xvy\in M$

that is $xMy\subseteq M$.

Note that kernels need not coincide with normal subobjects in non-exact
categories like for example the category of torsion free abelian groups (see
e.g. [4]).

###  2.2. The syntactic preorder

Every subset $M$ of a monoid $A$ induces an internal reflexive and transitive
relation $\leq_{M}$ on $A$ called the syntactic preorder (see [15]). It is
defined as $a\leq_{M}b$ if for every $x,y\in A$

$xay\in M\Rightarrow xby\in M.$

This binary relation is reflexive, transitive and compatible with the monoid
operation. Its positive cone consists of all $a\in A$ such that, for every
$x,y\in A$,

$xy\in M\Rightarrow xay\in M,$

which implies that it is a submonoid of $A$.

######  Proposition 2.

Let $M$ be a submonoid of a monoid $A$. Then $M$ is the positive cone of some
preorder in $A$ if and only if the following condition holds

(P) For every $x,y\in A$ and every $u\in M$.

$xy\in M\Rightarrow xuy\in M.$

In this case, $M$ is the positive cone of $\;\leq_{M}$.

###### Proof.

The proof is analogous to the one of the previous proposition and, in the same
way, we conclude that $M$ is the zero class of some preorder on $A$ if and
only if it is the zero class of its associated $M$-preorder. ∎

Example H in Section 4 shows that neither of the above syntactic constructions
is functorial.

Let us now investigate the case of clots, that is the zero-classes of internal
reflexive relations.

###  2.3. The syntactic reflexive relation

Every submonoid $M$ of a monoid $A$ induces a reflexive relation $R_{M}$ in
$A$ defined by $aR_{M}b$ if for every $x,y\in A$,

$xay=1\Rightarrow xby\in M.$

As we show next there is a large class of monoids where $R_{M}$ is an internal
reflexive relation, in which case we call it the syntactic $M$-reflexive
relation of the submonoid $M$. We point out that we don’t know if this is the
case for all monoids.

######  Proposition 3.

Let $M$ be a submomoid of a monoid $A$. Then $M$ is the zero-class of the
reflexive relation $R_{M}$ if and only if the following condition holds

(R) For every $x,y\in A$ and every $u\in M$

$xy=1\Rightarrow xuy\in M.$

If $M$ is a clot then $M=[1]_{R_{M}}$ and the converse holds when the relation
$R_{M}$ is internal.

###### Proof.

Like in the previous cases, we have that $[1]_{R_{M}}$ is a subset of $M$ and
that $M\subseteq[1]_{R_{M}}$ if and only if condition (R) is satisfied.
Consequently, in presence of condition (R), we have that $M=[1]_{R_{M}}$.

Let $M$ be the zero class of some internal reflexive relation $S$ on $A$. If
$1Su$, by the reflexibility and compatibility of the relation, $xySxuy$. Then,
if $xy=1$ we have that $1Sxuy$, that is $xuy\in M$, (R) holds and so
$M\subseteq[1]_{R_{M}}$.

The converse is true if $R_{M}$ is compatible. ∎

The relation $R_{M}$ is compatible with the monoid operation as soon as the
following extra condition is satisfied

$\forall x,y,s,t\in A,(xy=1,xs\in M,ty\in M)\Rightarrow ts\in M$ (2)

This follows from the condition saying that every left (or right) invertible
element in $A$ is invertible:

$\forall x,y\in A,(xy=1\Rightarrow yx=1)$ (3)

that characterizes the Dedekind finite monoids, also called Von Neumann finite
or directly finite monoids (See [7]). Dedekind finite monoids includes groups,
commutative monoids, finite monoids, cancellative monoids and others but not
all monoids. For example the monoid $\mathbf{Set}(X,X)$ for any infinite set
$X$ with respect to composition of functions as well as the bicyclic monoid
are not Dedekind finite.

In Dedekind finite monoids from $xs,ty\in M$ we get $tyxs\in M$ and, if $yx=1$
as soon as $xy=1$, then we conclude $ts\in M$.

The condition of being Dedekind finite is stronger than Condition (2) which is
not a necessary one for compatibility of $R_{M}$ as Example I shows (see
Section 4).

######  Remark 2.

In the category of groups, condition (R) means that M is a normal subgroup of
the group $A$ and the congruence $R_{M}$ coincides with the classical one
induced by the normal subgroup M, that is

$a\sim b\Leftrightarrow ab^{-1}\in M.$

We recall that $\sim$ is an equivalence relation on the underlying set of $A$
if and only if $M$ is a subgroup and that it is a congruence if and only if
$M$ is a normal subgroup of $A$.

##  3\. Left adjoints to the normalization functors

In this section we are going to define functors
$F\colon\mathbf{Mono}(\mathbf{Mon})\to\mathcal{R}$ which are left adjoints to
the corresponding normalization functors, as examples of a construction in a
more general context.

Let $\mathbf{C}$ be a pointed, well-powered and finitely complete category
with arbitrary intersections, that is with generalized pullbacks of
monomorphisms.

By $\mathcal{R}=\tau(\mathbf{C})$ we denote the category of internal binary
relations of a fixed type $\tau$ on $\mathbf{C}$-objects with morphisms the
morphisms in $\mathbf{C}$ compatible with the relations. In addition, we
assume that such relations are stable under pullback and closed under
arbitrary intersections for every object of $\mathbf{C}$, that is, we assume
that the functor $(\;\;)_{0}\colon\tau(\mathbf{C})\to\mathbf{C}$ that assigns
to each relation its base object is a fibration and that each fiber
$\tau(\mathbf{C})_{A}$ is a complete lattice. So, in particular, we assume
that, for each object $A$ in $\mathbf{C}$, the indiscrete relation
$\nabla_{A}$ belongs to $\tau(\mathbf{C})$.

######  Definition 1.

A monomorphism $m\colon M\to A$ is said to be normal w.r.t. $\tau(\mathbf{C})$
if it is part of a pullback in $\mathbf{C}$

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{m}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
0,1\rangle}$$\textstyle{A\times A}$

for a relation $(R,r)$ of type $\tau$ on $A$.

It is easy to prove that such morphisms are stable under pullback when the
same holds for the relations $\tau$. The above definition includes clots,
positive cones and normal subobjects in the classical sense in any pointed
variety but makes sense in every non-varietal category satisfying the
prescribed conditions, defining there the full subcategory
$\mathbf{NMono}(\mathbf{C},\tau)$ of the category $\mathbf{Mono}(\mathbf{C})$
consisting of what we call the normal monomorphisms with respect to $\tau$.

We point out that the categorical notion of normal monomorphism with respect
to a congruence was introduced in [2].

For each $\tau$, the normalization functor

$N\colon\tau(\mathbf{C})\to\mathbf{Mono}(\mathbf{C})$

is defined on objects by $N(A,R)=m$ as in the diagram of Definition 1. Indeed,
if $r_{i}=p_{i}\cdot r$ with $p_{i}$ the direct product projections for i = 1,
2, then $u$ is the kernel of $r_{1}$ and $m=r_{2}\cdot u$, the classical
definition of the normalization functor [2]. The universal property of
pullbacks provides the definition of $N$ on morphisms.

######  Theorem 1.

Let $\mathbf{C}$ be a pointed, well-powered and finitely complete category
with arbitrary intersections and $\tau(\mathbf{C})$ be a pullback-stable full
subcategory of the one of internal relations on $\mathbf{C}$ closed under
arbitrary intersections. Then the normalization functor
$N\colon\tau(\mathbf{C})\to\mathbf{Mono}(\mathbf{C})$ has a left adjoint

$F\colon\mathbf{Mono}(\mathbf{C})\to\tau(\mathbf{C})$

defined on objects by $F(m\colon M\to A)=(R,r)$ with
$(R,r)=\cap{\mathbf{S}}_{m}$, where ${\mathbf{S}}_{m}$ is the set of all
$\tau$-relations on $A$ through which $\langle 0,1\rangle m$ factors.

###### Proof.

We define $F(m)=(R,r)=\cap{\mathbf{S}}_{m}$, where $\mathbf{S}_{m}$ is not
empty because it contains $\nabla_{A}$ since $\tau(\mathbf{C})$ is closed
under arbitrary intersections.

The fact that the relations $\tau$ on $\mathbf{C}$ are stable under pullback,
enables us to assign to each $(f,\overline{f})\colon m\to m^{\prime}$ a
morphism

$(f\times f,g)\colon F(m)=(R,r)\to F(m^{\prime})=(R^{\prime},r^{\prime})$

where $g\colon R\to R^{\prime}$ is the morphism $t\cdot l$,
$(t,\overline{r^{\prime}})$ being the pullback of $r^{\prime}$ along $f\times
f$, which gives the relation $({\overline{R^{\prime}}},\overline{r^{\prime}})$
that belongs to ${\mathbf{S}}_{m}$ and so there exists a morphism $l$ such
that $\overline{r^{\prime}}\cdot l=r$:

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\scriptstyle{r}$$\textstyle{\overline{R^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{r^{\prime}}}$$\scriptstyle{t}$$\textstyle{R^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{\prime}}$$\textstyle{A\times
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\times
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\times
f}$$\textstyle{A^{\prime}\times A^{\prime}}$ (4)

This defines a functor $F$ which, in addition, is left adjoint of $N$ with
unit $\eta_{m}=(1_{A},s)\colon m\to NF(m)$

---
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{m}$$\scriptstyle{s}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(m)}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{NF(m)}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
0,1\rangle}$$\textstyle{A\times A}$ (5)

where $P$ is the pullback of $F(m)$ along $\langle 0,1\rangle$. ∎

In the case of congruences in Barr-exact categories there is another way to
describe the left adjoint to the normalization functor.

######  Corollary 1.

Let $\mathbf{C}$ be a Barr-exact category and $\tau(\mathbf{C})$ denote the
category $\mathbf{Eq}(\mathbf{C})$ of internal equivalence relations on
$\mathbf{C}$. Then $F(m)$ is the kernel pair of the cokernel of $m$.

###### Proof.

If $F(m)=(R,r)$ and $(R,r_{1},r_{2})$ is the kernel pair of some morphism in
$\mathbf{C}$ then it is the kernel pair of the coequalizer $q\colon A\to B$ of
$r_{1},r_{2}\colon R\to A$ and so, up to isomorphism, $(R,r_{1},r_{2})$
coincides with $(A\times_{B}A,\pi_{1},\pi_{2})$.

It remains to prove that $q$ is the cokernel of $m$, that is, it is the
coequalizer of $(m,0)$. If $fm=f0$ then we have that the kernel pair of $f$
belongs to $S_{m}$ and so we conclude that $fr_{1}=fr_{2}$. Consequently,
there is a unique morphism $f^{\prime}$ such that $f^{\prime}q=f$. ∎

Recall that an internal binary relation $(R,r_{1},r_{2})$ on an object $A$ of
$\mathbf{C}$ is

  * •

reflexive if there exists a morphism $e\colon A\to A$ such that
$r_{1}e=r_{2}e=1_{A}$, that is, if $1_{A}\leq R$.

  * •

symmetric if there exists a morphism $s\colon R\to R$ such that $r_{1}s=r_{2}$
and $r_{2}s=r_{1}$, that is, if $R^{o}\leq R$.

  * •

transitive if there exists a morphism $t\colon R\times_{A}R\to R$ such that
$r_{1}t=r_{1}\pi_{1}$ and $r_{2}t=r_{2}\pi_{2}$, that is, if $RR\leq R$
whenever $\mathbf{C}$ is a regular category.

If, in Theorem 1, $\tau$ denotes reflexive relations, preorders or equivalence
relations in $\mathbf{C}$ (but not orders, because the existence of the order
$\nabla_{A}$ would imply that $A$ is singular) then $\tau(\mathbf{C})$ satisfy
the prescribed conditions and so we obtain adjunctions

(8) (11) (14)

denoting the functor $F$ by $R,P$ and $E$, respectively. This is the case when
$\mathbf{C}=\mathbf{Mon}$.

Whenever $\mathbf{C}$ is a pointed variety we have another classical procedure
to define these left adjoints that consists of assigning to each monomorphism
$m\colon{M\to A}$, that, without lost of generality, we assume to be an
inclusion, the smallest zero-class in $A$ that contains $M$, by describing
appropriate submonoids of $A\times A$. The explicit descriptions of such
submonoids, that we present next, enable us to better exhibit different
aspects of the zero classes of each of the internal relations as shown in
Example B of Section 4.

In the rest of this section, for simplicity of exposition, we use additive
notation but we do not assume that the monoids are commutative.

By $R=R(m)$ we denote the smallest internal reflexive relation on a monoid $A$
whose zero class contains $M$, that is, the submonoid of $A\times A$ generated
by the set $(\\{0\\}\times M)\cup\Delta_{A}$. This means that $xRy$ if and
only if there exists a natural number $n\in\mathbb{N}$ together with sequences
$a_{i}\in A$ and $u_{i}\in M$, with $i=1,\ldots,n$, such that

$x=a_{1}+\ldots+a_{n}\text{ and }y=a_{1}+u_{1}+\ldots+a_{n}+u_{n}.$ (15)

The relation $P=P(m)$ is the smallest reflexive and transitive relation (a
preorder) on $A$ whose zero class contains all the elements in $M$. In this
case it is the transitive closure of the above internal reflexive relation
which is again internal, as it is easy to check. We conclude that $xPy$ if and
only if there exist natural numbers $m,n\in\mathbb{N}$ together with double
indexed sequences $a_{ij}\in A$ and $u_{ij}\in M$, with $i=1,\ldots,m$ and
$j=1,\ldots,n$, such that

$\displaystyle x$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n}a_{1j}$
$\displaystyle\sum_{j=1}^{n}\left(a_{1j}+u_{1j}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}a_{2j}$ $\displaystyle\cdots$
$\displaystyle\sum_{j=1}^{n}\left(a_{(i-1)j}+u_{(i-1)j}\right)$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{n}a_{ij}$ $\displaystyle\cdots$
$\displaystyle\sum_{j=1}^{n}\left(a_{(m-1)j}+u_{(m-1)j}\right)$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{n}a_{mj}$
$\displaystyle\sum_{j=1}^{n}\left(a_{mj}+u_{mj}\right)$ $\displaystyle=$
$\displaystyle y.$

The relation $E=E(m)$ is the smallest internal equivalence relation on $A$
whose zero class contains $M$. We start by considering the submonoid of
$A\times A$ generated by $(M\times M)\cup\Delta_{A}$ that is the smallest
internal reflexive and symmetric relation whose zero-class contains $M$. Then
the transitive closure of this relation gives $E$. Thus we have that $xEy$ if
and only if there exist natural numbers $m,n\in\mathbb{N}$ together with
double indexed sequences $a_{ij}\in A$ and $u_{ij},v_{ij}\in M$, with
$i=1,\ldots,m$ and $j=1,\ldots,n$, such that

$\displaystyle x$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left(a_{1j}+v_{1j}\right)$ $\displaystyle\
\sum_{j=1}^{n}\left(a_{1j}+u_{1j}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left(a_{2j}+v_{2j}\right)$ $\displaystyle\cdots$
$\displaystyle\sum_{j=1}^{n}\left(a_{(i-1)j}+u_{(i-1)j}\right)$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{n}\left(a_{ij}+v_{ij}\right)$
$\displaystyle\cdots$
$\displaystyle\sum_{j=1}^{n}\left(a_{(m-1)j}+u_{(m-1)j}\right)$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{n}\left(a_{mj}+v_{mj}\right)$
$\displaystyle\sum_{j=1}^{n}\left(a_{mj}+u_{mj}\right)$ $\displaystyle=$
$\displaystyle y.$

######  Proposition 4.

If we denote by $N$ the normalization functor from the categories of internal
reflexive relations, preorders and equivalence relations in the category
$\mathbf{Mon}$ of monoids, we have the following adjunctions

(18) (21) (24)

where the functors $R$, $P$ and $E$ are defined on objects as described above.

The largest equivalence $\mathbf{Fix}(\varepsilon)\sim\mathbf{Fix}(\eta)$
induced by each of these adjunctions, consists of $\mathbf{Fix}(\varepsilon)$,
the category of all relations that are generated by their zero classes in the
sense that they are the smallest relations with these zero-classes, and
$\mathbf{Fix}(\eta)$, the category of all the corresponding zero classes. In
the three particular cases of our study, $\mathbf{Fix}(\eta)$ consist of all
clots, all positive cones and all normal submonoids.

Let us denote by $\mathbf{Fix}(\eta_{R})$, $\mathbf{Fix}(\eta_{P})$ and
$\mathbf{Fix}(\eta_{E})$ the full subcategories of
$\mathbf{Mono}(\mathbf{Mon})$ consisting of all monomorphisms $m\colon{M\to
A}$ such that the corresponding adjunction unit $\eta_{m}$ is an isomorphism.
Similarly, let us denote by $\mathbf{Fix}(\varepsilon_{E})$,
$\mathbf{Fix}(\varepsilon_{P})$ and $\mathbf{Fix}(\varepsilon_{R})$ the
respective subcategories of relations that are determined by their respective
zero-classes.

The following theorems sum up our observations with respect to normal
monomorphism relative to the internal binary relations we are dealing with in
$\mathbf{Mon}$. There, for every monomorphism $m\colon M\to A$ we always
identify $m(M)$ with $M$.

######  Theorem 2 (Normal subobject).

A monoid monomorphism $m\colon{M\to A}$ is a normal subobject if one, and so
all, of the following equivalent conditions is satisfied:

  1. (a)

$M$ is the zero class of some congruence;

  2. (b)

$M$ is the zero-class of its syntactic congruence;

  3. (c)

$M$ is the zero-class of the congruence $E(m)$;

  4. (d)

$M$ is the zero-class of the smallest congruence whose zero-class contains
$M$;

  5. (e)

$m$ belongs to $\mathbf{Fix}(\eta_{E})$, that is, $m$ is normal w.r.t.
$\mathbf{Eq}(\mathbf{Mon})$;

  6. (f)

$E(m)$ belongs to $\mathbf{Fix}(\varepsilon_{E})$;

  7. (g)

for every $x,y\in A$ and every $u\in M$.

$xy\in M\Leftrightarrow xuy\in M.$

  8. (h)

for every natural numbers $m,n\in\mathbb{N}$ and every double indexed
sequences $a_{ij}\in A$, $u_{ij},v_{ij}\in M$, with $i=1,\ldots,m$ and
$j=1,\ldots,n$, if

$0=a_{11}+v_{11}+\ldots+a_{1n}+v_{1n}$

and

$\displaystyle a_{11}+u_{11}+\ldots+a_{1n}+u_{1n}$ $\displaystyle=$
$\displaystyle a_{21}+v_{21}+\ldots+a_{2n}+v_{21}$ $\displaystyle\cdots$
$\displaystyle a_{(i-1)1}+u_{(i-1)1}+\ldots+a_{(i-1)n}+u_{(i-1)n}$
$\displaystyle=$ $\displaystyle a_{i1}+v_{i1}+\ldots+a_{in}+v_{in}$
$\displaystyle\cdots$ $\displaystyle
a_{(m-1)1}+u_{(m-1)1}+\ldots+a_{(m-1)n}+u_{(m-1)n}$ $\displaystyle=$
$\displaystyle a_{m1}+v_{m1}+\ldots+a_{mn}+v_{mn}$

then

$a_{m1}+u_{m1}+\ldots+a_{mn}+u_{mn}\in M.$

######  Theorem 3 (Positive cones).

A monoid monomorphism $m\colon{M\to A}$ is a positive cone if one, and so all,
of the following equivalent conditions is satisfied:

  1. (a)

$M$ is the zero-class of some preorder;

  2. (b)

$M$ is the zero-class of its syntactic $M$-preorder;

  3. (c)

$M$ is the zero-class of the preorder $P(m)$;

  4. (d)

$M$ is the zero-class of the smallest preorder whose zero-class contains $M$;

  5. (e)

$m$ belongs to $\mathbf{Fix}(\eta_{P})$, that is, $m$ is normal w.r.t.
$\mathbf{Ord}(\mathbf{Mon})$;

  6. (f)

$P(m)$ belongs to $\mathbf{Fix}(\varepsilon_{P})$;

  7. (g)

for every $x,y\in A$ and every $u\in M$.

$xy\in M\Rightarrow xuy\in M.$

  8. (h)

for every natural numbers $m,n\in\mathbb{N}$ and every two double indexed
sequences $a_{ij}\in A$, $u_{ij}\in M$, with $i=1,\ldots,m$ and
$j=1,\ldots,n$, if

$0=a_{11}+\ldots+a_{1n}$

and

$\displaystyle a_{11}+u_{11}+\ldots+a_{1n}+u_{1n}$ $\displaystyle=$
$\displaystyle a_{21}+\ldots+a_{2n}$ $\displaystyle\cdots$ $\displaystyle
a_{(i-1)1}+u_{(i-1)1}+\ldots+a_{(i-1)n}+u_{(i-1)n}$ $\displaystyle=$
$\displaystyle a_{i1}+\ldots+a_{in}$ $\displaystyle\cdots$ $\displaystyle
a_{(m-1)1}+u_{(m-1)1}+\ldots+a_{(m-1)n}+u_{(m-1)n}$ $\displaystyle=$
$\displaystyle a_{m1}+\ldots+a_{mn}$

then

$a_{m1}+u_{m1}+\ldots+a_{mn}+u_{mn}\in M.$

######  Theorem 4 (Clots).

A monoid monomorphism $m\colon{M\to A}$ is a _clot_ if one, and so all, of the
following equivalent conditions is satisfied:

  1. (a)

$M$ is the zero-class of some internal reflexive relation;

  2. (b)

$M$ is the zero-class of the reflexive relation $R(m)$;

  3. (c)

$M$ is the zero-class of the smallest internal reflexive relation whose zero-
class contains $M$;

  4. (d)

$m$ belongs to $\mathbf{Fix}(\eta_{R})$, that is, $m$ is normal w.r.t.
$\mathbf{RRel}(\mathbf{Mon})$;

  5. (e)

$R(m)$ belongs to $\mathbf{Fix}(\varepsilon_{R})$;

  6. (f)

For every natural number $n\in\mathbb{N}$ and for every two sequences
$a_{i}\in A$, $u_{i}\in M$, with $i=1,\ldots,n$, if

$a_{1}+\ldots+a_{n}=0$

then

$a_{1}+u_{1}+\ldots+a_{n}+u_{n}\in M.$

Furthermore, when $R_{M}$ is an internal relation (i.e., it is the syntactic
$M$-reflexive relation) the above conditions are also equivalent to the
following:

  1. (g)

$M$ is the zero-class of its syntactic reflexive relation;

  2. (h)

for every $x,y\in A$ and every $u\in M$.

$xy=1\Rightarrow xuy\in M.$

##  4\. Examples

Examples A–C illustrate the strict inclusions displayed in $(\ref{eq:stric
inclusions})$. Example D describes a normal submonoid which is not a subgroup.
Examples E–K are related with observations made in Section 2.

  1. (A)

A simple example of a submonoid that is not a clot is obtained as follows.
Take the monoid with four elements {1,2,3,4} and multiplication defined by

$\cdot$ | 1 | 2 | 3 | 4
---|---|---|---|---
1 | 1 | 2 | 3 | 4
2 | 2 | 2 | 3 | 3
3 | 3 | 2 | 3 | 2
4 | 4 | 2 | 3 | 1
(25)

and observe that $M=\\{1,2\\}$ is a submonoid. However, the smallest
compatible reflexive relation $R$ containing the pair $(1,2)$ is

$\\{(1,1),(1,2),(1,3),(2,2),(2,3),(3,2),(3,3),(4,2),(4,3),(4,4)\\}$

whose zero-class is the submonoid $N(R)=\\{1,2,3\\}$.

  2. (B)

An example of a clot which is not a positive cone is provided by the
monomorphism $m\colon{\mathbf{N}\to F(a,b)}$, from the natural numbers to the
free monoid on two generators $a$ and $b$, which associates $1\in\mathbf{N}$
to the word $ab$. Then $m(n)=(ab)^{n}$ and $\mathbf{N}$ is the clot of a
reflexive relation but it is not the positive cone of any reflexive and
transitive relation on $A=F(a,b)$.

We can prove that in two different ways:

    1. (i)

First we observe that in $R(m)$ we have that $1Rx$ if and only if $x=(ab)^{n}$
for some natural number $n$. This already shows that $\mathbf{N}$ is a clot.
But it is not a positive cone because the relation is not transitive. Indeed,
otherwise, $1Raabb$ should hold since $1Rab$ and $abRaabb$, but $aabb$ is not
of the form $(ab)^{n}$ for any natural n. The fact that $abRaabb$ follows from
reflexivity and compatibility of the relation: since $aRa$, $1Rab$ and $bRb$
we obtain $abRaabb$.

    2. (ii)

Another way to see that is to use the characterizations given in Theorem 4 and
Theorem 3. First we observe that condition $(\ref{cond f of thm 4})$ of
Theorem 4 is satisfied because $a_{1}+\ldots+a_{n}=0$ implies that all $a_{i}$
have to be zero. Secondly, we observe that condition $(\ref{cond h of thm 3})$
of Theorem 3 is not satisfied: taking $m=2$, $n=2$, $a_{1j}=0$, $u_{11}=(ab)$,
$u_{12}=0$, $a_{21}=(a)$, $a_{22}=(b)$, $u_{21}=(ab)$, $u_{22}=0$ we observe
that $(ab)=(a)+(b)$ but $(a)+(ab)+(b)=aabb$ is not of the form $(ab)^{n}$ and
hence it is not in the image of the monomorphism $m\colon{\mathbf{N}\to
F(a,b)}$.

  3. (C)

An example of a positive cone which is not a normal submonoid (with respect to
$\mathbf{Eq}(\mathbf{Mon})$) is provided by the inclusion of the natural
numbers in the group of the integers.

  4. (D)

The inclusion of the even numbers in the monoid of the natural numbers is an
example of a normal submonoid. Indeed it satisfies the condition (g) of
Theorem 2.

  5. (E)

For every finite monoid, the relation $R_{M}$ is an internal reflexive
relation (Section 2) because these monoids are Dedekind finite.

  6. (F)

The monoid $\mathbf{Set}(X,X)$ of endofunctions of a set $X$ for composition,
if $X$ is an infinite set, is not Dedekind finite. Indeed any surjective
function that is not injective has a right inverse but not a left inverse.

  7. (G)

The bicylic monoid is the monoid freely generated by two elements $b$ and $c$,
satisfying the equation $bc=1$. It is not Dedekind finite since $bc=1$ but
$cb\neq 1$.

  8. (H)

There is no functor
$\mathbf{Mono}(\mathbf{Mon})\rightarrow\mathbf{Eq}(\mathbf{Mon})$ assigning to
each submonoid $M$ of $A$ its syntactic $M$-congruence (Section 2). To show
that let us consider the composition of morphisms $S_{2}\rightarrow
S_{3}\rightarrow S_{2}$ of symmetric groups that gives the identity
$id_{S_{2}}$. It induces also the identity in $\mathbf{Mono}(\mathbf{Mon})$ of
the monomorphism $m\colon S_{2}\rightarrow S_{2}$, say $gf=id_{m}$. There is
no way to define a function sending this composition to the identity of the
equivalence $S_{2}\times S_{2}$ on $S_{2}$. Indeed it is enough to notice that
the zero class of the domain is $S_{2}$ and the one of $\sim_{S_{2}}$ in
$S_{3}$ is just the identity $\epsilon$:

$(13)\epsilon(13)\in S_{2}\mbox{ but}(13)(12)(13)=(23)\notin S_{2}.$

So every congruence class is a singleton.

Consequently there is no functor sending a submonoid $M$ of $A$ to its
$M$-preorder or to its $M$-reflexive relation.

  9. (I)

For every submonoid $M$ of $A$ we have that if $A$ is Dedekind finite then
condition $(\ref{eq: (*) extra condition normal reflexive})$ holds and that if
$A$ satisfy $(\ref{eq: (*) extra condition normal reflexive})$ then $R_{M}$ is
compatible but these two implications are strict. Indeed,

    1. (i)

If $M=A$ then condition $(\ref{eq: (*) extra condition normal reflexive})$
holds and $A$ may be one of the examples of a non Dedekind finite monoid.

    2. (ii)

There are internal reflexive relations $R_{M}$ that do not satisfy condition
$(\ref{eq: (*) extra condition normal reflexive})$. Let
$A=\mathbf{Set}(\mathbf{N},\mathbf{N})$ for composition of functions and
$M=\\{1_{\mathbf{N}}\\}$. Then $R_{M}$ is compatible since now $aR_{M}b$ if,
for every functions $x,y\colon\mathbf{N}\to\mathbf{N}$,

$xay=1\Rightarrow xby=1,$

and we can find examples of functions $x,y,s,t$ such that $xy=1,xs=1$ and
$ty=1$ but $ts$ is not equal to $1$. For example,

$x\colon\mathbf{N}\to\mathbf{N}$ taking zero to zero and $n>0$ to $n-1$;

$y\colon\mathbf{N}\to\mathbf{N}$ sending each $n>0$ to $n+1$;

$s\colon\mathbf{N}\to\mathbf{N}$ taking zero to zero and $n>0$ to $n+1$;

$t\colon\mathbf{N}\to\mathbf{N}$ taking 0 to 5 and $n>0$ to $n-1$.

Then $xy=1,xs=1$ and $ty=1$ but $ts(0)=5$.

  10. (J)

We do not know whether $R_{M}$ is or is not always an internal relation. The
following example brings some light on the problem but does not solve it.

Let $A$ be the monoid $\mathbf{Set}(\mathbf{N},\mathbf{N})$, of all
endofunctions of the natural numbers for composition, and $M$ be the submonoid
generated by the function $u=2\times-$. Then $M$ consists of all functions
$u^{n}=2^{n}\times-$ for some $n\in\mathbf{N}_{0}$.

Then $u^{n}$ for $n>0$ does not belong to the zero set of the relation
$R_{M}$. Indeed, for $f,g\in A$ defined by $g(x)=x+1$ and $f(x)=x-1$ if $x>1$
and $f(1)=1$ we have that $f\cdot g=1_{\mathbf{N}}$ but $f\cdot u^{n}\cdot
g\not\in M$ because $f\cdot u^{n}\cdot g(x)=2^{n}(x+1)-1$, for
$x\in\mathbf{N}$.

Thus, $[1]_{R_{M}}=\\{1_{\mathbf{N}}\\}$ and so $M$ is not the zero class of
any internal reflexive relation on $A$.

  11. (K)

For any submonoid $M$ of a monoid $A$ we have the following inclusions

$[1]_{R_{M}}\subseteq M\subseteq[1]_{R(m)},$

that, in the previous example, are both strict. Indeed, not only $[1]_{R_{M}}$
is a proper submonoid of $M$ but also

$f\cdot u^{n}\cdot g\in[1]_{R(m)}$

because $1=f\cdot g$ and $h=f\cdot u^{n}\cdot g$ and so $1R(m)h$ but $h$ does
not belong to $M$.

##  5\. Final remarks

In [9], the authors give a categorical definition of clot in semi-abelian
categories that they extend in [10] to the context of pointed regular
categories with finite coproducts and prove that clots are the same as zero
classes of internal reflexive relations, called there semicongruences. In this
work we consider clots in the context of pointed, wellpowered and finitely
complete categories with arbitrary intersections of subobjects that seems to
be the more general context where studying clots versus internal reflexive
relations makes sense. In particular, if $\mathbf{C}=\mathbf{Mon}$, the
semicongruence associated to each clot $m:M\to A$ in Theorem 2.2 of [10] is
the internal relation $R(m)$ defined above.

In [14], for regular categories $\mathbf{C}$ with pushouts of split
monomorphisms along arbitrary maps, G. Metere proves that the restriction of
the bifibration $(\;\;)_{0}\colon{\mathbf{RGraph}(\mathbf{C})\to\mathbf{C}}$
to $\mathbf{RRel}(\mathbf{C})$ is a bifibration and this, in particular,
enables him to construct a left adjoint to the normalization functor. When
$\mathbf{C}$ is the category of monoids, this gives another way to construct
the left adjoint (18) and, in [3], for Mal’tsev categories, this provides the
left adjoint to $N\colon\mathbf{Eq}(\mathbf{C})\to\mathbf{Mono}(\mathbf{C})$.

In order to cover ideals, as a generalization of clots [1, 11], we would need
to replace the study of binary internal relations by the study of jontly monic
spans of which one leg is surjective, which would require a different approach
to the one followed here.

In the present work we have shown that the notion of normal monomorphism can
be defined as the zero-class of certain types of internal binary relations.
This means that every subcategory $\mathcal{R}$ of internal binary relations
on a pointed category $\mathbf{C}$, for which the normalization functor admits
a left adjoint, gives rise to a class of normal monomorphisms as the zero
classes of relations in $\mathcal{R}$. We have studied the particular case
where $\mathbf{C}=\mathbf{Mon}$ and $\mathcal{R}$ consists of reflexive
relations, preorders, and congruences, from which the respective zero-classes
are clots, positive cones, and classical normal subobjects. It is clear that
other situations may be of interest too, namely reflexive and symmetric
relations.

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