# Feasibility of the experimental study of $D_{s}^{\ast}$ ${\to}$
${\phi}{\pi}$ decay

Yueling Yang  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China  Kang Li  Institute of Particle and Nuclear
Physics, Henan Normal University, Xinxiang 453007, China  Zhenglin Li
Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang
453007, China  Jinshu Huang  School of Physics and Electronic Engineering,
Nanyang Normal University, Nanyang 473061, China  Junfeng Sun  Institute of
Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China

###### Abstract

The current knowledge on the $D_{s}^{\ast}$ meson are very limited. Besides
the dominant electromagnetic decays, the $D_{s}^{\ast}$ weak decays are legal
and offer the valuable opportunities to explore the wanted $D_{s}^{\ast}$
meson. In this paper, the $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decay was
studied with the factorization approach. It is found that the branching ratio
${\cal B}(D_{s}^{\ast}{\to}{\phi}{\pi})$ ${\sim}$ ${\cal O}(10^{-7})$, which
corresponds to several thousands of events at the $e^{+}e^{-}$ collider
experiments including STCF, SuperKEKB, CEPC and FCC-ee, and several millions
of events at the hadron collider experiments, such as LHCb@HL-LHC. It is
feasible to experimentally study the $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ weak
decay in the future, even considering the identification efficiency.

Eur. Phys. J. C 82, 555 (2022)

##  I Introduction

The first evidence for the charmed strange mesons $D_{s}^{\ast}$ was observed
in the exclusive reaction $e^{+}e^{-}$ ${\to}$ $F\bar{F}^{\ast}$ by the DASP
collaboration in the year of 1977 Phys.Lett.B.70.132 , where the symbols of
$F$ and $F^{\ast}$ were formerly used to denote the $D_{s}$ and $D_{s}^{\ast}$
particles, respectively. According to the $SU(4)$ quark model assignments, the
vector mesons $D_{s}^{\ast}$ are assumed to have the same quark compositions
as their twin pseudoscalar partners $D_{s}$. Both the $D_{s}^{{\ast}+}$ and
$D_{s}^{+}$ mesons are consisting of a quark-antiquark pair $c\bar{s}$, and
have the same additive quantum numbers of Charm, Strangeness and Charge, i.e.,
$C$ $=$ $S$ $=$ $Q$ $=$ $+1$. The different spin configurations of interquark
potential make the mass of the ground spin-triplet $1^{3}S_{1}$ state for the
$D_{s}^{\ast}$ mesons to be above that of the ground spin-singlet $1^{1}S_{0}$
state for the $D_{s}$ mesons pdg2020 .

Compared with the pseudoscalar meson $D_{s}$, the experimental information on
the properties of the vector meson $D_{s}^{\ast}$ is still very limited by now
pdg2020 . Although there were many measurements of the mass of the
$D_{s}^{\ast}$ meson (such as in Refs. Phys.Lett.B.70.132 ; Phys.Lett.B.80.412
; Phys.Lett.B.146.111 ; PhysRevLett.53.2465 ; PhysRevLett.58.2171 ;
Phys.Lett.B.207.349 ; PhysRevD.50.1884 ; PhysRevLett.75.3232 ), only one
measurement was solemnly quoted by the Particle Data Group (PDG) until now
pdg2020 . The measurement was carried out by the Mark III collaboration in
1987 PhysRevLett.58.2171 , thirty-five years ago. And the errors of the
measurement of mass, $m_{D_{s}^{\ast}}$ $=$ $2109.3{\pm}2.1{\pm}3.1$ MeV
PhysRevLett.58.2171 , are significantly larger than those of current values of
the $D_{s}$ meson, $m_{D_{s}}$ $=$ $1968.35{\pm}0.07$ MeV pdg2020 . For the
full width of the $D_{s}^{\ast}$ meson, only the upper limit was given by
different experimental groups pdg2020  and the latest and minimal upper limit
on the decay width of the $D_{s}^{\ast}$ meson was given by the CLEO
Collaboration in 1995 PhysRevLett.75.3232 , twenty-seven years ago. The
natural spin-parity of the $D_{s}^{\ast}$ meson was analyzed to be most likely
$J^{P}$ $=$ $1^{-}$ PhysRevLett.75.3232 , but has not been unambiguously
determined experimentally pdg2020 .

The experimental data on the $D_{s}^{\ast}$ mesons are accumulating
increasingly. The quantitative study on the $D_{s}^{\ast}$ mesons is coming.
Inspired by the potential prospects of high-luminosity-frontier flavor
experiments, more and more data of the $D_{s}^{\ast}$ mesons will be
available, so more accurate information and more detailed knowledge of the
properties of the $D_{s}^{\ast}$ mesons will be accessible. In the
$e^{+}e^{-}$ colliders, it is promisingly expected that there will be a total
of about $5{\times}10^{10}$ $c\bar{c}$ pairs at the SuperKEKB PTEP.2019.123C01
, about $10^{11}$ $c\bar{c}$ pairs from $10^{12}$ $Z^{0}$ boson decays at the
Circular Electron Positron Collider (CEPC) cepc , about $6{\times}10^{11}$
$c\bar{c}$ pairs from $5{\times}10^{12}$ $Z^{0}$ boson decays at the Future
Circular Collider (FCC-ee) fcc , where the branching fraction for the $Z^{0}$
boson decay into the $c\bar{c}$ pair is ${\cal B}(Z^{0}{\to}c\bar{c})$ $=$
$(12.03{\pm}0.21)\%$ pdg2020 . Considering the fraction of the charmed quark
fragmenting into the $D_{s}^{\ast}$ meson $f(c{\to}D_{s}^{\ast})$ ${\simeq}$
$5.5\%$ epjc.76.397 , these high statistical $c\bar{c}$ pairs correspond to
some $6{\times}10^{9}$, $10^{10}$ and $6{\times}10^{10}$ $D_{s}^{\ast}$ mesons
at the SuperKEKB, CEPC and FCC-ee, respectively. In addition, about $10^{10}$
$D_{s}^{\ast}$ mesons are expected above the ${\psi}(4040)$ threshold (see
Fig. 6 of Ref. epjc.81.1110 ) at both the super ${\tau}$-charm factory (STCF)
in China STCF  and the super charm-tau factory (SCTF) in Novosibirsk, Russia
SCTF , based on an integrated luminosity of $10\,{ab}^{-1}$ STCF . In the
high-energy hadron colliders, about $4{\times}10^{13}$ $D_{s}^{\ast}$ mesons
epjc.81.1110  are expected to be obtainable with a data sample of target
luminosity $300\,fb^{-1}$ at the LHCb@HL-LHC experiments epjst.228.1109 , and
more $D_{s}^{\ast}$ mesons will be accumulated at ALICE and ATLAS epjc.81.1110
. The huge amount of experimental data provide a tremendous foundation and
valuable opportunities for studying and understanding the properties of
$D_{s}^{\ast}$ meson. A brilliant portrait of the characteristics of
$D_{s}^{\ast}$ mesons is going to be unfolded smoothly and completely.

The fit mass of $D_{s}^{\ast}$ meson is $m_{D_{s}^{\ast}}$ $=$
$2112.2{\pm}0.4$ MeV pdg2020 , just below the mass threshold of the
$D\overline{K}$ pair and above the mass threshold of the $D_{s}{\pi}$ pair and
, i.e., the mass relations $m_{D_{u,d}}$ $+$ $m_{K}$ $>$ $m_{D_{s}^{\ast}}$
$>$ $m_{D_{s}}$ $+$ $m_{\pi}$. Thus the hadronic decays $D_{s}^{\ast}$ ${\to}$
$D\overline{K}$ are strictly forbidden by the law of conservation of energy.
The hadronic decay $D_{s}^{\ast}$ ${\to}$ $D_{s}{\pi}$ is permissible
kinematically, but violates the the isospin conservation in the strong
interactions111Within the chiral perturbative theory, it is usually taken for
granted that the $D_{s}^{\ast}$ ${\to}$ $D_{s}{\pi}$ decay can also decay
through the strong interactions via the ${\pi}^{0}$-${\eta}$ mixing by
assuming a small isocalar ${\eta}$ meson component in the physical ${\pi}^{0}$
meson, because the ${\eta}$ meson can couple to the strange quark in the
charmed strange mesons PhysRevD.49.6228 ; Nucl.Phys.B.529.62 ;
Nucl.Phys.A.710.99 ; PhysRevD.101.054019 .. The absences of decay modes
induced by the strong interactions make the $D_{s}^{\ast}$ meson to be very
narrow. The natural width of the the $D_{s}^{\ast}$ meson is significantly
less than the best experimental resolution. Here, it should be noted that the
$D_{s}^{\ast}$ ${\to}$ $D_{s}{\pi}$ decay is suppressed not only by the
phenomenological Okubo-Zweig-Iizuka (OZI) rule ozi-o ; ozi-z ; ozi-i  but also
by the extremely limited phase spaces, due to $m_{D_{s}^{\ast}}$ $-$
$m_{D_{s}}$ $-$ $m_{\pi}$ $<$ $6$ MeV. Thus the electromagnetic decay
$D_{s}^{\ast}$ ${\to}$ $D_{s}{\gamma}$ is dominant, with the branching ratio
${\cal B}(D_{s}^{\ast}{\to}D_{s}{\gamma})$ $=$ $(93.5{\pm}0.7)\%$ exceeding
that of hadronic decay ${\cal B}(D_{s}^{\ast}{\to}D_{s}{\pi})$ $=$
$(5.8{\pm}0.7)\%$ pdg2020 . In addition, for the $D_{s}^{\ast}$ ${\to}$
$D_{s}{\pi}^{0}$, $D_{s}{\gamma}$ decays222The neutral pion decay
predominantly through ${\pi}^{0}$ ${\to}$ ${\gamma}{\gamma}$ with a branching
ratio of $98.8\%$ pdg2020 . , the final photons are seriously polluted by
those from bremsstrahlung radiation, which will significantly affect the
identification efficiency of the accident photon. Besides, the $D_{s}^{\ast}$
meson can also decay via the weak interactions, although with a very small
probability. The weak decays of the $D_{s}^{\ast}$ meson provide another
platform and opportunities to explore and understand the properties of the
$D_{s}^{\ast}$ mesons. In this paper, we will evaluate the feasibility of
experimentally investigating the $D_{s}^{\ast}$ meson through the weak decay
$D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$.

Theoretically, the charm-flavor-changing decay $D_{s}^{{\ast}+}$ ${\to}$
${\phi}{\pi}^{+}$ is actually induced by the quark transition $c$ ${\to}$ $s$
$+$ $W^{{\ast}+}$ at the tree level in the standard model (SM) of elementary
particles. Here, it is assumed that the vector ${\phi}$ meson consists of the
pure $s\bar{s}$ quark pair with neither possible $u\bar{u}$ nor $d\bar{d}$
components, i.e., that the mixing between the ${\phi}$-${\omega}$ system is
ideal. Clearly, this decay mode is the Cabibbo-favored one and its amplitudes
are proportional to the Cabibbo-Kobayashi-Maskawa (CKM) matrix
PhysRevLett.10.531 ; PTP.49.652  element ${|}V_{cs}{|}$ ${\sim}$ ${\cal
O}(1)$. This decay would have a relatively large branching ratio among the
$D_{s}^{\ast}$ meson weak decays, and hence should have a high priority to be
studied. In addition, the charm quark is somewhat massive and can be regarded
as one bridge between the perturbative and nonperturbative regimes. The charm
quark decays offer a laboratory to test various phenomenological models and
study the behaviors of the strong interactions near the scale of ${\cal
O}(m_{c})$.

Experimentally, the curved tracks of the charged pion and kaon plunged into
magnetic field will be unambiguously detectable by the highly sensitive
detectors. So, the final states are easily identified for the $D_{s}^{\ast}$
${\to}$ ${\phi}{\pi}$ decays, where ${\phi}$ and ${\pi}$ mesons with a
definite momentum are back-to-back in the center-of-mass frame of the
$D_{s}^{\ast}$ meson, and the ${\phi}$ meson can be well reconstructed from
the kaon pairs. It is expected to have a higher signal-to-background ratio and
a better identification efficiency, and have a big competitive advantage over
both the pure leptonic decays $D_{s}^{\ast}$ ${\to}$ ${\ell}\bar{\nu}$ and
semileptonic decays $D_{s}^{\ast}$ ${\to}$ ${\phi}{\ell}\bar{\nu}$ which
suffer from the additional complications caused by the final neutrinos.

In this paper, we will study the $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decay
within SM by using the phenomenological factorization approach zpc.34.103 ,
and estimate the branching ratio in order to provide a ready reference for
future experimental analysis. This paper is organized as follows. The
amplitudes for the $D_{s}^{\ast}$ decay in question using the factorization
approximation is given in Sec. II. Branching ratio and event numbers of the
$D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decay are listed in Sec. III. Section IV
devotes to a summary.

##  II The theoretical framework

At the quark level, the effective Hamiltonian responsible for the nonleptonic
decay $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ can be written as
RevModPhys.68.1125 ,

${\cal H}_{\rm
eff}\,=\,\frac{G_{F}}{\sqrt{2}}\,V_{cs}^{\ast}\,V_{ud}\,\big{\\{}C_{1}\,O_{1}+C_{2}\,O_{1}\big{\\}}+{\rm
h.c.},$ (1)

where the Fermi constant $G_{F}$ is the weak interaction coupling coefficient,
$G_{F}$ ${\approx}$ $1.166{\times}10^{-5}$ ${\rm GeV}^{-2}$ pdg2020 .
$V_{cs}^{\ast}\,V_{ud}$ is the product of CKM matrix elements, which has been
determined precisely by experiments, ${|}V_{ud}{|}$ $=$ $0.97370(14)$ and
${|}V_{cs}{|}$ $=$ $0.987(11)$ pdg2020 . The Wilson coefficients $\vec{C}$ $=$
$\\{C_{1},C_{2}\\}$ can be obtained with the renormalization group equation,

$\vec{C}({\mu}_{c})\,=\,U_{4}({\mu}_{c},m_{b})\,M(m_{b})\,U_{5}(m_{b},m_{W})\,\vec{C}(m_{W}),$
(2)

where ${\mu}_{c}$ ${\sim}$ ${\cal O}(m_{c})$ is the scale for the charm quark
decays. $m_{b}$ and $m_{W}$ are the mass of the bottom quark and the charged
$W$ gauge boson, respectively. $U_{f}({\mu}_{f},{\mu}_{i})$ and $M(m_{b})$ are
the evolution matrix and threshold matching matrix, respectively. The
expressions of $\vec{C}(m_{W})$, $U_{f}({\mu}_{f},{\mu}_{i})$ and $M(m_{b})$
can be found in Ref. RevModPhys.68.1125 . The effective operators are defined
as follows.

$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle\big{[}\bar{s}_{\alpha}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,c_{\alpha}\big{]}\,\big{[}\bar{u}_{\beta}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,d_{\beta}\big{]},$
(3) $\displaystyle O_{2}$ $\displaystyle=$
$\displaystyle\big{[}\bar{s}_{\alpha}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,c_{\beta}\big{]}\,\big{[}\bar{u}_{\beta}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,d_{\alpha}\big{]},$
(4)

where ${\alpha}$ and ${\beta}$ are the color indices. Because the
$D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decay is an external $W$ emission
process, there are only two current-current operator $O_{1,2}$ and without the
penguin operators, and the contributions from new physics beyond SM to this
decay are negligible.

The initial and final states are hadrons, while the operators are the specific
combinations of four quarks. The influence of the long-distance strong
interactions on the transitions between quarks and hadrons makes the
predictions of nonleptonic decays notoriously difficult. To obtain the decay
amplitudes for the $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decay, the remaining
work is to evaluate the hadronic matrix elements (HMEs)
${\langle}{\phi}{\pi}{|}O_{i}{|}D_{s}^{\ast}{\rangle}$.

Phenomenologically, one of the most frequently used methods to deal with HME
is the naive factorization (NF) approach zpc.34.103 . The NF approach is based
on the color transparency hypothesis npbps.11.325  that a nearly collinear and
relativistic light quark-antiquark pair originating from the heavy quark
decays might be approximated as a color singlet before its hadronization and
complete separation from the interaction points. According to the color
transparency hypothesis, it is possible to replace the product of the quark
currents in the effective Hamiltonian of Eq.(1) by product of the
corresponding hadron currents, and express the color singlet quark currents in
terms of the participating hadron fields Stech.1985 . The outgoing light
hadrons of two-body decays are back-to-back and energetic in the heavy quark
limit, and fly away far from each other before the interference with the soft
gluons. It may be a good approximation to neglect the final state interactions
for the moment. In addition, the asymptotic freedom property of the strong
interactions implies that the creation of quark pairs of high energy from the
vacuum by hard virtual gluon is highly suppressed npb.133.315 , i.e., it is
believed that the $W$-annihilation amplitudes for the nonleptonic heavy-
flavored hadron decays might be much smaller than the $W$-emission amplitudes.

Under the assumption of factorization, the decay amplitudes are written as,

$\displaystyle{\cal
A}(D_{s}^{\ast}{\to}{\phi}{\pi})\,=\,{\langle}{\phi}\,{\pi}{|}{\cal H}_{\rm
eff}{|}D_{s}^{\ast}{\rangle}$ (5) $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\,V_{cs}^{\ast}\,V_{ud}\,a_{1}\,{\langle}{\phi}\,{\pi}{|}(\bar{s}\,c)_{H}\,(\bar{u}\,d)_{H}{|}D_{s}^{\ast}{\rangle}$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\,V_{cs}^{\ast}\,V_{ud}\,a_{1}\,{\langle}{\pi}{|}(\bar{u}\,d)_{H}{|}0{\rangle}\,{\langle}{\phi}{|}(\bar{s}\,c)_{H}{|}D_{s}^{\ast}{\rangle},$

where $(\bar{s}\,c)_{H}$ and $(\bar{u}\,d)_{H}$ are the color singlet $V$-$A$
hadron currents, and the subscript $H$ is introduced to indicate the change to
hadron currents and distinguish with quark currents of Eq.(3) and Eq.(4). The
effects from the color exchanges are embodied into the coefficient $a_{1}$ $=$
$C_{1}$ $+$ ${\xi}\,C_{2}$. It is expected ${\xi}$ $=$ $1/N_{c}$ $=$ $1/3$
from color matching. ${\xi}$ or $a_{1}$ sometimes is regarded as a parameter
for different factorization approaches, because of the uncertain contributions
of color octet current product and nonfactorizable contributions. The
approximation of $a_{1}$ ${\approx}$ $1.1$ is frequently used in many
phenomenological studies of nonleptonic decays for charmed hadron mesons, such
as Refs. Stech.1985 ; npb.133.315 ; cpc.26.665 ; cpc.27.759 ; epjc.42.391 ;
jpg.34.637 ; PhysRevD.81.074021 ; PhysRevD.84.074019 ; PhysRevD.86.014014 ;
PhysRevD.86.036012 ; ijmpa.30.1550094 ; PhysRevD.100.093002 .

Using the parameterization for amplitude in Eq.(5), the decay widths can be
given in terms of measurable physical HMEs. The HMEs of hadron currents in
Eq.(5) are related to the decay constants and hadron transition form factors.
The one-body HMEs are relevant to decay constants of hadrons,

$\displaystyle{\langle}0{|}\bar{d}\,{\gamma}_{\mu}\,u{|}{\pi}^{+}(p){\rangle}$
$\displaystyle=$ $\displaystyle 0,$ (6)
$\displaystyle{\langle}0{|}\bar{d}\,{\gamma}_{\mu}\,{\gamma}_{5}\,u{|}{\pi}^{+}(p){\rangle}$
$\displaystyle=$ $\displaystyle i\,f_{\pi}\,p_{\mu}.$ (7)

The charged pion decay constant has been well determined from numerical
lattice QCD simulations, $f_{\pi}$ $=$ $130.2{\pm}1.2$ MeV (See Ref. pdg2020
for a summary review). With the conventions of Refs. jhep.1912.102 , the form
factors are defined as,

$\displaystyle{\langle}{\phi}({\epsilon}_{2},p_{2}){|}\,\bar{s}\,{\gamma}_{\mu}\,c\,{|}D_{s}^{\ast}({\epsilon}_{1},p_{1}){\rangle}$
(8) $\displaystyle=$
$\displaystyle-({\epsilon}_{1}{\cdot}{\epsilon}_{2}^{\ast})\,\big{\\{}P_{\mu}\,V_{1}(q^{2})-q_{\mu}\,V_{2}(q^{2})\big{\\}}-({\epsilon}_{1}{\cdot}q)\,{\epsilon}_{2,{\mu}}^{\ast}\,V_{5}(q^{2})+({\epsilon}_{2}^{\ast}{\cdot}q)\,{\epsilon}_{1,{\mu}}\,V_{6}(q^{2})$
$\displaystyle+\frac{({\epsilon}_{1}{\cdot}q)\,({\epsilon}_{2}^{\ast}{\cdot}q)}{m_{D_{s}^{\ast}}^{2}-m_{{\phi}}^{2}}\,\big{\\{}\big{[}P_{\mu}-\frac{m_{D_{s}^{\ast}}^{2}-m_{{\phi}}^{2}}{q^{2}}\,q_{\mu}\big{]}\,V_{3}(q^{2})+\frac{m_{D_{s}^{\ast}}^{2}-m_{{\phi}}^{2}}{q^{2}}\,q_{\mu}\,V_{4}(q^{2})\big{\\}},$
$\displaystyle{\langle}{\phi}({\epsilon}_{2},p_{2}){|}\,\bar{s}\,{\gamma}_{\mu}\,{\gamma}_{5}\,c\,{|}D_{s}^{\ast}({\epsilon}_{1},p_{1}){\rangle}$
(9) $\displaystyle=$
$\displaystyle-i\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{1}^{\alpha}\,{\epsilon}_{2}^{{\ast}{\beta}}\,\big{\\{}\big{[}P^{\nu}-\frac{m_{D_{s}^{\ast}}^{2}-m_{{\phi}}^{2}}{q^{2}}\,q^{\nu}\big{]}\,A_{1}(q^{2})+\frac{m_{D_{s}^{\ast}}^{2}-m_{{\phi}}^{2}}{q^{2}}\,q^{\nu}\,A_{2}(q^{2})\big{\\}}$
$\displaystyle-\frac{i\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,P^{\alpha}\,q^{\beta}}{m_{D_{s}^{\ast}}^{2}-m_{{\phi}}^{2}}\,\big{\\{}({\epsilon}_{2}^{\ast}{\cdot}q)\,{\epsilon}_{1}^{\nu}\,A_{3}(q^{2})-({\epsilon}_{1}{\cdot}q)\,{\epsilon}_{2}^{{\ast},{\nu}}\,A_{4}(q^{2})\big{\\}},$

where ${\epsilon}_{i}$ denotes the polarization vector of the vector mesons.
The momentum $P$ $=$ $p_{1}$ $+$ $p_{2}$ and $q$ $=$ $p_{1}$ $-$ $p_{2}$. At
the pole $q^{2}$ $=$ $0$, there is,

$V_{3}(0)\,=\,V_{4}(0),$ (10) $A_{1}(0)\,=\,A_{2}(0).$ (11)

The values of formfactors for the $D_{s}^{\ast}$ ${\to}$ ${\phi}$ transition
have been obtained with the light front approach jhep.1912.102 , for example,
$A_{1}(0)$ $=$ $0.65$, $V_{1}(0)$ $=$ $0.71$, $V_{4}(0)$ $=$ $0.28$,
$V_{5}(0)$ $=$ $1.54$, and $V_{6}(0)$ $=$ $0.86$.

Finally, the decay amplitude can be expressed by three invariant amplitudes.
They are defined by the decomposition,

$\displaystyle{\cal A}(D_{s}^{\ast}{\to}{\phi}{\pi})$ (12) $\displaystyle=$
$\displaystyle
a\,({\epsilon}_{D_{s}^{\ast}}{\cdot}{\epsilon}_{\phi}^{\ast})+\frac{b}{m_{D_{s}^{\ast}}\,m_{\phi}}\,({\epsilon}_{D_{s}^{\ast}}{\cdot}p_{\pi})\,({\epsilon}_{\phi}^{\ast}{\cdot}p_{\pi})+\frac{c}{m_{D_{s}^{\ast}}\,m_{\phi}}\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{D_{s}^{\ast}}^{\alpha}\,{\epsilon}_{\phi}^{{\ast}{\beta}}\,p_{\pi}^{\mu}\,(p_{D_{s}^{\ast}}+p_{\phi})^{\nu}$
$\displaystyle=$
$\displaystyle{\epsilon}_{D_{s}^{\ast}}^{\alpha}\,{\epsilon}_{\phi}^{{\ast}{\beta}}\,\big{\\{}a\,g_{{\alpha}{\beta}}+\frac{b}{m_{D_{s}^{\ast}}\,m_{\phi}}\,p_{{\pi},{\alpha}}\,p_{{\pi},{\beta}}\,+\frac{c}{m_{D_{s}^{\ast}}\,m_{\phi}}\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,p_{\pi}^{\mu}\,(p_{D_{s}^{\ast}}+p_{\phi})^{\nu}\big{\\}},$

and the invariant amplitudes $a$, $b$, and $c$ describe the $s$-, $d$-, and
$p$-wave contributions.

$\displaystyle a$ $\displaystyle=$
$\displaystyle-i\,\frac{G_{F}}{\sqrt{2}}\,V_{cs}^{\ast}\,V_{ud}\,f_{\pi}\,(m_{D_{s}^{\ast}}^{2}-m_{{\phi}}^{2})\,a_{1}\,V_{1}(0),$
(13) $\displaystyle b$ $\displaystyle=$
$\displaystyle-i\,\frac{G_{F}}{\sqrt{2}}\,V_{cs}^{\ast}\,V_{ud}\,f_{\pi}\,m_{D_{s}^{\ast}}\,m_{\phi}\,a_{1}\,\big{\\{}V_{5}(0)-V_{6}(0)-V_{4}(0)\big{\\}},$
(14) $\displaystyle c$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\,V_{cs}^{\ast}\,V_{ud}\,f_{\pi}\,m_{D_{s}^{\ast}}\,m_{\phi}\,a_{1}\,A_{1}(0).$
(15)

In the rest frame of the $D_{s}^{\ast}$ meson, branching ratio is defined as,

$\displaystyle{\cal B}(D_{s}^{\ast}{\to}{\phi}{\pi})$ $\displaystyle=$
$\displaystyle\frac{1}{24\,{\pi}}\,\frac{p_{\rm
c.m.}}{m_{D_{s}^{\ast}}^{2}\,{\Gamma}_{D_{s}^{\ast}}}{|}{\cal
A}(D_{s}^{\ast}{\to}{\phi}{\pi}){|}^{2}$ (16) $\displaystyle=$
$\displaystyle\frac{1}{24\,{\pi}}\,\frac{p_{\rm
c.m.}}{m_{D_{s}^{\ast}}^{2}\,{\Gamma}_{D_{s}^{\ast}}}\big{\\{}{|}a{|}^{2}\,(2+x^{2})+{|}b{|}^{2}\,(x^{2}-1)^{2}$
$\displaystyle+{|}2\,c{|}^{2}\,2\,(x^{2}-1)-2\,{\rm
R}e(a\,b^{\ast})\,x\,(x^{2}-1)\big{\\}},$

where the center-of-mass momentum of final states is of magnitude,

$p_{\rm
c.m.}\,=\,\displaystyle\frac{{\lambda}^{\frac{1}{2}}(m_{D_{s}^{\ast}}^{2},m_{\phi}^{2},m_{\pi}^{2})}{2\,m_{D_{s}^{\ast}}},$
(17)

the parameter $x$ is defined as,

$x\,=\,\frac{p_{D_{s}^{\ast}}{\cdot}p_{\phi}}{m_{D_{s}^{\ast}}\,m_{\phi}}\,=\,\frac{E_{\phi}}{m_{\phi}}\,=\,\frac{m_{D_{s}^{\ast}}^{2}+m_{\phi}^{2}-m_{\pi}^{2}}{2\,m_{D_{s}^{\ast}}\,m_{\phi}},$
(18) ${\lambda}(x,y,z)\,=\,x^{2}+y^{2}+z^{2}-2\,x\,y-2\,y\,z-2\,z\,x,$ (19)
$p_{\rm c.m.}^{2}\,=\,m_{\phi}^{2}\,(x^{2}-1).$ (20)

##  III numerical results and discussion

The total decay width ${\Gamma}_{D_{s}^{\ast}}$ $<$ $1.9$ MeV was set at the
90% confidence level by the CLEO collaboration in 1995 PhysRevLett.75.3232 . A
quantitative and concrete result currently comes from theoretical estimations.
Because of the lion’s share ${\cal B}(D_{s}^{\ast}{\to}{\gamma}D_{s})$ $=$
$(93.5{\pm}0.7)\%$ pdg2020 , an approximation ${\Gamma}_{D_{s}^{\ast}}$
${\approx}$ ${\Gamma}(D_{s}^{\ast}{\to}{\gamma}D_{s})$ is often used in
theoretical calculation. The decay width for the magnetic dipole transition is
epjc.81.1110 ,

${\Gamma}(D_{s}^{\ast}{\to}D_{s}{\gamma})\,=\,\frac{4}{3}\,{\alpha}_{\rm
em}\,k_{\gamma}^{3}\,{\mu}_{D_{s}^{\ast}D_{s}}^{2}\,\,{\approx}\,0.36\,\text{keV},$
(21)

where ${\mu}_{D_{s}^{\ast}D_{s}}$ is the magnetic dipole moment and
$k_{\gamma}$ is the momentum of photon.

Using Eq.(16), we can obtain branching ratio,

${\cal
B}(D_{s}^{\ast}{\to}{\phi}{\pi})\,{\approx}\,2.4{\times}\,\frac{0.36\,\text{keV}}{{\Gamma}_{D_{s}^{\ast}}}\,{\times}\,10^{-7},$
(22)

and the corresponding partial decay width,
${\Gamma}(D_{s}^{\ast}{\to}{\phi}{\pi})$ ${\approx}$
$0.86\,{\times}\,10^{-13}$ GeV, is more than twice as large as the recent
estimate using the QCD light cone sum rules in Ref. cheng2203  where a
relatively smaller coefficient $a_{1}$ ${\approx}$ $1.0$ is used.

We will make two comments on branching ratio. (1) There are many factors which
influence the numerical results, such as the final state interactions. It is
foreseeable that there will very large theoretical uncertainties. For example,
using a much smaller decay width ${\Gamma}_{D_{s}^{\ast}}$ ${\approx}$ $0.07$
keV from the lattice QCD simulations PhysRevLett.112.212002 , branching ratio
will be increased five times. Our focus is whether there is feasible to
explore the $D_{s}^{\ast}$ meson via the ${\phi}{\pi}$ final states at the
future experiments. A rough estimate rather than precise calculation on
branching ratio is enough. (2) For the tree-dominated and color-favored
nonleptonic heavy flavored meson decays arising from the external $W$ emission
weak interaction, there is a consensus that NF approximation does hold and can
give a reasonable and correct magnitude order estimation on branching ratio.
In this sence, ${\cal B}(D_{s}^{\ast}{\to}{\phi}{\pi})$ ${\sim}$ ${\cal
O}(10^{-7})$ seems credible.

Based on the above analysis, it can be conclude that the $D_{s}^{\ast}$
${\to}$ ${\phi}{\pi}$ decay should be measurable in the future experiments,
such as STCF, SuperKEKB, CEPC, FCC-ee and LHCb. The potential event numbers of
the $D_{s}^{\ast}$ mesons and the $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decays
are listed in Table 1. It is clearly seen from Table 1 that the natural
properties of the $D_{s}^{\ast}$ meson can be investigated via the
$D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ weak decays, particularly in the future
FCC-ee and LHCb experiments.

Table 1: The potential event numbers of the $D_{s}^{\ast}$ meson available and the $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decays in the future experiments, with the branching ratio ${\cal B}(Z^{0}{\to}c\bar{c})$ ${\approx}$ $12\%$ pdg2020  and ${\cal B}(D_{s}^{\ast}{\to}{\phi}{\pi})$ ${\approx}$ $3\,{\times}\,10^{-7}$, the fragmentation fraction $f(c{\to}D_{s}^{\ast})$ ${\approx}$ $5.5\%$ epjc.76.397  and the identification efficiency ${\epsilon}$ ${\sim}$ $20\%$. experiment | $N_{D_{s}^{\ast}}$ | $N_{D_{s}^{\ast}{\to}{\phi}{\pi}}$ | ${\epsilon}{\times}N_{D_{s}^{\ast}{\to}{\phi}{\pi}}$ | remarks
---|---|---|---|---
STCF STCF ; SCTF  |  $10^{10}$ epjc.81.1110  | $3000$ | $600$ | with $10\,ab^{-1}$ data
SuperKEKB PTEP.2019.123C01  | $5.5{\times}10^{9}$ | $1600$ | $300$ | with $5{\times}10^{10}$ charm quark pairs
CEPC cepc  | $1.3{\times}10^{10}$ | $4000$ | $800$ | from $10^{12}$ $Z^{0}$ boson decays
FCC-ee fcc  | $6.6{\times}10^{10}$ | $2{\times}10^{4}$ | $4000$ | from $5{\times}10^{12}$ $Z^{0}$ boson decays
LHCb@HL-LHC epjst.228.1109  | $4{\times}10^{13}$ | $10^{7}$ | $2{\times}10^{6}$ | with $300\,fb^{-1}$ data

##  IV Summary

Inspired by the inadequate understanding of the properties of $D_{s}^{\ast}$
meson, and the promisingly experimental prospects of investigating the
$D_{s}^{\ast}$ meson in the future high-luminosity experiments, the
$D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ decay was studied by using the NF
approach within SM. The nonleptonic $D_{s}^{\ast}$ ${\to}$ ${\phi}{\pi}$ weak
decay offers a fresh arena and a tempting opportunity to explore the wanted
$D_{s}^{\ast}$ meson, although with a very tiny occurrence probability of
${\sim}$ ${\cal O}(10^{-7})$. The final states of the $D_{s}^{\ast}$ ${\to}$
${\phi}{\pi}$ decay have the relatively larger momenta than those of the
predominant electromagnetic decays $D_{s}^{\ast}$ ${\to}$ $D_{s}{\gamma}$ and
${\to}$ $D_{s}{\pi}$, and can be more easily identified by the sensitive
detectors. It is found that several thousands of events for the $D_{s}^{\ast}$
${\to}$ ${\phi}{\pi}$ decay are expected to be accessible at the STCF,
SuperKEKB, CEPC and FCC-ee experiments, several millions of events at LHCb@HL-
LHC experiments. It is practicable to experimentally study the $D_{s}^{\ast}$
${\to}$ ${\phi}{\pi}$ weak decay in the future.

## Acknowledgments

The work is supported by the National Natural Science Foundation of China
(Grant Nos. 11705047, U1632109, 11875122) and Natural Science Foundation of
Henan Province (Grant No. 222300420479), the Excellent Youth Foundation of
Henan Province (Grant No. 212300410010).

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# Set-theoretical solutions of the pentagon equation
on Clifford semigroups111This work was partially supported by the Dipartimento
di Matematica e Fisica “Ennio De Giorgi” - Università del Salento and the
Departament de Matemàtiques - Universitat de València. The first and the third
authors are members of GNSAGA (INdAM) and of the non-profit association ADV-
AGTA.

Marzia MAZZOTTA<EMAIL_ADDRESS>Vicent PÉREZ-CALABUIG
<EMAIL_ADDRESS>Paola STEFANELLI<EMAIL_ADDRESS>Dipartimento di Matematica e Fisica “Ennio De Giorgi”,
Università del Salento,
Via Provinciale Lecce-Arnesano,
73100 Lecce (Italy)
Departament de Matemàtiques de València,
Dr. Moliner, 50,
46100 Burjassot, València (Spain)

###### Abstract

Given a set-theoretical solution of the pentagon equation $s:S\times S\to
S\times S$ on a set $S$ and writing $s(a,b)=(a\cdot b,\,\theta_{a}(b))$, with
$\cdot$ a binary operation on $S$ and $\theta_{a}$ a map from $S$ into itself,
for every $a\in S$, one naturally obtains that $\left(S,\,\cdot\right)$ is a
semigroup.
In this paper, we focus on solutions on Clifford semigroups
$\left(S,\,\cdot\right)$ satisfying special properties on the set of the
idempotents $\operatorname{E}(S)$. Into the specific, we provide a complete
description of _idempotent-invariant solutions_ , namely, those solutions for
which $\theta_{a}$ remains invariant in $\operatorname{E}(S)$, for every $a\in
S$. Moreover, considering $(S,\,\cdot)$ as a disjoint union of groups, we
construct a family of _idempotent-fixed solutions_ , i.e., those solutions for
which $\theta_{a}$ fixes every element in $\operatorname{E}(S)$, for every
$a\in S$, starting from a solution on each group.

###### keywords:

pentagon equation , set-theoretical solution , inverse semigroups, Clifford
semigroups

###### MSC:

[2022] 16T25, 81R50, 20M18

## Introduction

If $V$ is a vector space over a field $F$, a linear map $\mathcal{S}:V\otimes
V\to V\otimes V$ is said to be a _solution of the pentagon equation_ on $V$ if
it satisfies the relation

$\displaystyle\mathcal{S}_{12}\mathcal{S}_{13}\mathcal{S}_{23}=\mathcal{S}_{23}\mathcal{S}_{12},$
(1)

where $\mathcal{S}_{12}=\mathcal{S}\otimes\operatorname{id}_{V}$,
$\mathcal{S}_{23}=\operatorname{id}_{V}\otimes\,\mathcal{S}$,
$\mathcal{S}_{13}=(\operatorname{id}_{V}\otimes\,\Sigma)\,\mathcal{S}_{12}\;(\operatorname{id}_{V}\otimes\,\Sigma)$,
with $\Sigma$ the flip operator on $V\otimes V$, i.e., $\Sigma(u\otimes
v)=v\otimes u$, for all $u,v\in V$. The pentagon equation arose at first at
the beginning of ’80 in [5] as the Biedenharn-Elliott identity for Wigner
$6j-$symbols and Racah coefficients in the representation theory for the
rotation group. Maillet [21] showed that solutions of the pentagon equation
lead to solutions of the tetrahedron equation [31], a generalization of the
well-known quantum Yang-Baxter equation [29, 4]. Moreover, in [25, Theorem
3.2], Militaru showed that bijective solutions on finite vector spaces are
equivalent to finite Hopf algebras, and so the classification of the latter is
reduced to the classification of solutions. In the subsequent years, the
pentagon equation appeared in literature in several forms with different
terminologies according to the specific research areas. We highlight some
interesting works as [11, 30, 22, 27, 16, 28, 2, 26, 17, 3, 25, 15, 13], just
to name a few. For a fuller treatment of some applications in which the
pentagon equation appears, we suggest the recent paper by Dimakis and Müller-
Hoissen [10] (along with the references therein), where the authors dealt with
an infinite family of equations named _polygon equations_.

As well as Drinfel’d in [12] translated the study of solutions of the Yang-
Baxter equation into set-theoretical terms, Kashaev and Sergeev in [19] began
the study of the pentagon equation with a set-theoretical approach. Namely, if
$S$ is a set, a map $s:S\times S\to S\times S$ satisfying the following
“reversed” relation

$s_{23}s_{13}s_{12}=s_{12}s_{23},$ (2)

where $s_{12}=s\times\operatorname{id}_{S}$,
$s_{23}=\operatorname{id}_{S}\times s$,
$s_{13}=(\operatorname{id}_{S}\times\tau)\,s_{12}\,(\operatorname{id}_{S}\times\tau)$,
and $\tau(a,b)=(b,a)$, for all $a,b\in S$, is said to be a _set-theoretical
solution of the pentagon equation_ , or briefly _solution_ , on $S$. If, in
particular, $s$ is a solution on a finite set $S$, then the linear map
$\mathcal{S}:F^{S\times S}\to F^{S\times S}$ defined by
$\mathcal{S}(f)(a,b)=f(s(a,b))$, for all $a,b\in S$, is a solution of (1) on
the vector space $F^{S}$ of all functions from $S$ to $F$.
For their purposes, the authors in [19] investigated only bijective maps. This
class of solutions was also studied by Kashaev and Reshetikhin in [18], where
it is shown that each symmetrically factorizable Lie group is related to a
bijective solution. Among these solutions, a description of all those that are
involutive, i.e., $s^{2}=\operatorname{id}_{S\times S}$, has been recently
given by Colazzo, Jespers, and Kubat in [9].
As one can see in [6, Proposition 8], any arbitrary solution $s$ on a set $S$
can be written as $s(a,b)=(a\cdot b,\,\theta_{a}(b))$, with $\cdot$ a binary
operation on $S$ and $\theta_{a}$ a map from $S$ into itself, for every $a\in
S$. In this way, $S$ is inherently endowed with a structure of a semigroup
$\left(S,\,\cdot\right)$ and it appears natural the study of solutions on
specific classes of semigroups. For brevity, we will denote the multiplication
in $S$ as a concatenation. In this vein, in [6, Theorem 15] the authors
provide a description of all solutions on a group, by showing that they are
determined by its normal subgroups. Moreover, in [7], we can find several
constructions of solutions on semigroups, such as on the matched product of
two semigroups, that is a semigroup including the classical Zappa-Szép
product. In the same paper, the authors investigate maps that are both
solutions of the pentagon and the Yang-Baxter equations [12]. Furthermore, in
[23], the first author study the idempotent solutions, namely, maps satisfying
the property $s^{2}=s$, and describes this kind of solutions on monoids having
central idempotents.

In this paper, we begin the study of solutions on Clifford semigroups, namely,
inverse semigroups whose idempotent elements are central. Recalling that a
semigroup $S$ is inverse if, for each $a\in S$, there exists a unique
$a^{-1}\in S$ satisfying $aa^{-1}a=a$ and $a^{-1}aa^{-1}=a^{-1}$, it is clear
that the behaviour of Clifford semigroups is very close to that of groups. In
light of this fact and the description of solutions on groups in [6], it is
natural to wonder if a description of solutions can be obtained also on this
class of semigroups. However, such an aim appears challenging and some
considerations on the set of solutions must be considered. It is easy to check
that every solution on a group $G$ satisfies that $\theta_{a}(1)=1$, for every
$a\in G$. Therefore, it motivates us to consider both classes of solutions on
a Clifford semigroup $S$ such that $\theta_{a}$, respectively, fixes every
idempotent or remains invariant on every idempotent, for every $a\in S$. We
call them, respectively, _idempotent-fixed_ and _idempotent-invariant_
solutions.
The main results of this paper are the following. Firstly, we provide a
complete description of the first class of solutions on a Clifford semigroup
$S$, which includes that made in the context of groups. To this aim, we
introduce the _kernel_ of an arbitrary solution on $S$, which turns out to be
a normal subsemigroup, that is a subsemigroup containing the idempotents and
closed by conjugation. Secondly, for the second class, considering that any
Clifford semigroup is a union of a family of pairwise disjoint groups
$\\{G_{e}\\}_{e\in\operatorname{E}(S)}$, we give a construction of solutions
obtained starting from a solution on each group $G_{e}$.

##  1 Preliminaries

The aim of this section is to briefly introduce some basics of set-theoretical
solutions of the pentagon equation. Initially, we recall some notions related
to Clifford semigroups useful for our purposes. For a fuller treatment of this
topic, we refer the reader to [8] and [20].

###  1.1 Basics on Clifford semigroups

Recall that $S$ is an _inverse semigroup_ if for each $a\in S$ there exists a
unique $a^{-1}\in S$ such that $a=aa^{-1}a$ and $a^{-1}=a^{-1}aa^{-1}$. They
hold $(ab)^{-1}=b^{-1}a^{-1}$ and $(a^{-1})^{-1}=a$, for all $a,b\in S$.
Moreover, $\operatorname{E}(S)=\\{\,aa^{-1}\ |\ a\in S\,\\}=\\{\,a^{-1}a\ |\
a\in S\,\\}$ and one can consider the following natural partial order relation

$\displaystyle\forall\ e,f\in\operatorname{E}(S)\qquad e\leq f\
\Longleftrightarrow\ e=ef=fe.$

An inverse semigroup $S$ is _Clifford_ if $aa^{-1}=a^{-1}a$, for any $a\in S$,
or, equivalently, the idempotents are central in the sense that commute with
every element in $S$.

Given a Clifford semigroup $S$, we introduce the following relations and the
properties involved themselves. They are an easy consequence of the fact that
all Green’s relations coincide in $S$ and they characterize the structure of
$S$ itself. If $a,b\in S$, we define

  1. 1.

$a\leq b$ if, and only if, $aa^{-1}\leq bb^{-1}$, which is an extension of the
natural partial order in $S$;

  2. 2.

$a\,\mathcal{R}\,b$ if, and only if, $a\leq b$ and $b\leq a$.

It follows that $\leq$ is a preorder on $S$ and $\mathcal{R}$ is an
equivalence relation on $S$ such that

$G_{aa^{-1}}:=[a]_{\mathcal{R}}=\\{b\in S\,\mid\,bb^{-1}=aa^{-1}\\}$

is a group with identity $aa^{-1}$, for every $a\in S$. On the other hand, for
all $a,b\in S$,

$\displaystyle a\leq b\,\Longleftrightarrow\,\exists\,u\in S\quad
a=ub\,\,\vee\,\,a=bu.$ (3)

Moreover, $\leq$ induces an order relation on the equivalence classes of
$\mathcal{R}$, namely, for all $e,f\in\operatorname{E}(S)$, $G_{e}\leq G_{f}$
if, and only if, $e\leq f$. The following theorem describes Clifford
semigroups.

######  Theorem 1.

Let $S$ be a Clifford semigroup. Then,

  1. 1.

$S$ is a union of a family of pairwise disjoint groups
$\\{G_{e}\\}_{e\in\operatorname{E}(S)}$;

  2. 2.

the map $\varphi_{f,e}\colon G_{f}\rightarrow G_{e}$ given by
$\varphi_{f,e}(b)=eb$, for every $b\in G_{f}$, is a group homomorphism, for
all $e,f\in\operatorname{E}(S)$ such that $e\leq f$;

  3. 3.

for all $e,f,g\in\operatorname{E}(S)$ such that $e\leq f\leq g$, then
$\varphi_{g,e}=\varphi_{f,e}\varphi_{g,f}$.

As a consequence of the previous theorem, the product in Clifford semigroups
can be written by means of the group homomorphisms $\varphi_{e,f}$, namely,

$\displaystyle
ab=(ae)(fb)=(efa)(efb)=\varphi_{e,ef}\left(a\right)\varphi_{f,ef}\left(b\right)\in
G_{ef},$

for all $a\in G_{e}$, $b\in G_{f}$. In particular, for all $a\in S$,
$e\in\operatorname{E}(S)$ such that $a\leq e$, then $ae=ea=a$.

For the sake of completeness, the converse of 1 is also true.

###  1.2 Basics on solutions

Kashaev and Sergeev [19] first dealt with solutions from an algebraic point of
view. Recently, the study of these solutions has been recovered in [6, 7, 9,
23]. Following the notation introduced in these works, given a set $S$ and a
map $s$ from $S\times S$ into itself, we will write

$s(a,b):=(ab,\,\theta_{a}(b))$,

for all $a,b\in S$, where $\theta_{a}$ is a map from $S$ into itself, for
every $a\in S$. Then, $s$ is briefly a _solution_ on $S$ if, and only if, the
following conditions hold

$\displaystyle(ab)c$ $\displaystyle=a(bc)$
$\displaystyle\theta_{a}(b)\theta_{ab}(c)$ $\displaystyle=\theta_{a}(bc)$ (P1)
$\displaystyle\ \theta_{\theta_{a}(b)}\theta_{ab}$ $\displaystyle=\theta_{b}$
(P2)

for all $a,b,c\in S$. Thus, the first identity naturally gives rise to a
semigroup structure on $S$, which leads the study of solutions to focus on
specific classes of semigroups. When describing solutions, it serves to
distinguish those solutions that are not isomorphic.

######  Definition 2.

Let $S,T$ be two semigroups and $s(a,b)=(ab,\theta_{a}(b))$,
$t(u,v)=(uv,\eta_{u}(v))$ two solutions on $S$ and $T$, respectively. Then,
$s$ and $t$ are _isomorphic_ if there exists an isomorphism $\psi:S\to T$ such
that

$\displaystyle\psi\theta_{a}(b)=\eta_{f\left(a\right)}\psi(b),$ (4)

for all $a,b\in S$, or, equivalently, $(\psi\times\psi)s=t(\psi\times\psi)$.

The following are easy examples of solutions used throughout this paper.

######  Examples 1.

  1. 1.

Let $S$ be a set and $f,g:S\to S$ idempotent maps such that $fg=gf$. Then,
$s(a,b)=\left(f\left(a\right),\,g\left(b\right)\right)$ is a solution on $S$
(cf. [24]).

  2. 2.

Let $S$ be a semigroup and $\gamma\in\operatorname{End}(S)$ such that
$\gamma^{2}=\gamma$. Then, the map $s$ given by
$s(a,b)=\left(ab,\gamma\left(b\right)\right),$ for all $a,b\in S$, is a
solution on $S$ (see [6, Examples 2-2.]).

Let us observe that every Clifford semigroup $S$ gives rise to the following
solutions

$\displaystyle\mathcal{I}(a,b)=(ab,b),\qquad\mathcal{F}(a,b)=\left(ab,bb^{-1}\right),\qquad\mathcal{E}(a,b)=(ab,e),$
(5)

where $e\in\operatorname{E}(S)$ is a fixed idempotent of $S$, belonging to the
class of solutions in $2.$ of 1.

In [1], solutions of (1) are defined on Hilbert spaces in terms of commutative
and cocommutative multiplicative unitary operators (see [1, Definition 2.1]).
These operators motivate the following classes of solutions in the set-
theoretical case.

######  Definition 3.

A solution $s:S\times S\to S\times S$ is said to be _commutative_ if
$s_{12}s_{13}=s_{13}s_{12}$ and _cocommutative_ if
$s_{13}s_{23}=s_{23}s_{13}$.

Solutions in 1-$1.$ are both commutative and cocommutative. In [9, Corollary
3.4], it is proved that if $s$ is an involutive solution, i.e.,
$s^{2}=\operatorname{id}_{S\times S}$, then $s$ is both commutative and
cocommutative.

Convention: In the sequel, we assume that $S$ is a Clifford semigroup and
simply write that $s$ is a solution on $S$ instead of
$s(a,b)=(ab,\theta_{a}(b))$, for all $a,b\in S$.

##  2 Properties of solutions on Clifford semigroups

In this section, we show the existence of a normal subsemigroup associated to
any solution $s$ on $S$. We point out that the properties we proved are
consistent with those given in the context of groups [6].

######  Proposition 4.

Let $s$ be a solution on $S$. Then, the following statements hold:

  1. 1.

$\theta_{a}\left(a^{-1}\right)=\theta_{aa^{-1}}\left(a\right)^{-1}$,

  2. 2.

$\theta_{a}\left(a^{-1}a\right)=\theta_{a}\left(a^{-1}\right)\theta_{a}\left(a^{-1}\right)^{-1}\in\operatorname{E}(S)$,

  3. 3.

$\theta_{aa^{-1}}=\theta_{\theta_{a^{-1}}\left(aa^{-1}\right)}\theta_{a^{-1}}$,

for every $a\in S$.

###### Proof.

Let $a\in S$. Then, by (P1), we have

$\displaystyle\theta_{a}\left(a^{-1}\right)\theta_{aa^{-1}}\left(a\right)\theta_{a}\left(a^{-1}\right)$
$\displaystyle=\theta_{a}\left(a^{-1}a\right)\theta_{a}\left(a^{-1}\right)=\theta_{a}\left(a^{-1}a\right)\theta_{aa^{-1}a}\left(a^{-1}\right)$
$\displaystyle=\theta_{a}\left(a^{-1}aa^{-1}\right)=\theta_{a}\left(a^{-1}\right)$

and
$\theta_{aa^{-1}}\left(a\right)\theta_{a}\left(a^{-1}\right)\theta_{aa^{-1}}\left(a\right)=\theta_{aa^{-1}}\left(aa^{-1}\right)\theta_{aa^{-1}}\left(a\right)=\theta_{aa^{-1}}\left(aa^{-1}a\right)=\theta_{aa^{-1}}\left(a\right),$
hence $\theta_{a}\left(a^{-1}\right)=\theta_{aa^{-1}}\left(a\right)^{-1}$, so
$1.$ is satisfied.
Moreover, by $1.$, we get
$\theta_{a}\left(a^{-1}a\right)=\theta_{a}\left(a^{-1}\right)\theta_{aa^{-1}}\left(a\right)=\theta_{a}\left(a^{-1}\right)\theta_{a}\left(a^{-1}\right)^{-1}$,
thus $\theta_{a}\left(a^{-1}a\right)$ is an idempotent of $S$.
Finally, by (P2),
$\theta_{aa^{-1}}=\theta_{\theta_{a^{-1}}\left(aa^{-1}\right)}\theta_{a^{-1}aa^{-1}}=\theta_{\theta_{a^{-1}}\left(aa^{-1}\right)}\theta_{a^{-1}}$,
which is our claim. ∎

Note that the previous result also holds in any inverse semigroup that is not
necessarily Clifford.

Now, let us introduce a crucial object in studying solutions on Clifford
semigroups.

######  Definition 5.

If $s$ is a solution on $S$, the following set

$\displaystyle K=\\{a\in S\,\mid\,\forall\ e\in\operatorname{E}(S),\,\,e\leq
a\quad\theta_{e}(a)\in\operatorname{E}(S)\\}$

is called the _kernel_ of $s$.

Consistently with [6, Lemma 13], our aim is to show that $K$ is a _normal
subsemigroup_ of the Clifford $S$, namely, $\operatorname{E}(S)\subseteq K$
and $a^{-1}Ka\subseteq K$, for every $a\in S$. To this end, we first provide a
preliminary result.

######  Lemma 6.

Let $s$ be a solution on $S$ and $K$ the kernel of $s$. Then, they hold:

  1. 1.

$\theta_{a}(e)\in\operatorname{E}(S)$, for all $a\in S$ and
$e\in\operatorname{E}(S)$ such that $a\leq e$;

  2. 2.

$\theta_{ea}(k)\in\operatorname{E}(S)$, for all $a\in S$, $k\in K$, and
$e\in\operatorname{E}(S)$ such that $e\leq a$, $e\leq k$.

###### Proof.

Let $a\in S$ and $e\in\operatorname{E}(S)$. If $a\leq e$, by (P1), we obtain
$\theta_{a}(e)=\theta_{a}(e)\theta_{ae}(e)=\theta_{a}(e)^{2}$, hence $1.$
follows. Now, if $k\in K$ and $e\leq a$, $e\leq k$, then
$\theta_{e}(k)\in\operatorname{E}(S)$ and, by (P2),

$\displaystyle\theta_{ea}(k)=\theta_{\theta_{a^{-1}}\left(ea\right)}\theta_{a^{-1}ea}(k)=\theta_{\theta_{a^{-1}}\left(ea\right)}\theta_{e}(k).$

If we prove that $\theta_{a^{-1}}\left(ea\right)\leq\theta_{e}(k)$, by $1.$,
we obtain that $\theta_{ea}(k)\in\operatorname{E}(S)$. We get

$\displaystyle\theta_{a^{-1}}\left(ea\right)$
$\displaystyle=\theta_{a^{-1}}\left(eakk^{-1}\right)=\theta_{a^{-1}}\left(ea\right)\theta_{a^{-1}ea}\left(kk^{-1}\right)=\theta_{a^{-1}}\left(ea\right)\theta_{e}\left(kk^{-1}\right)$
$\displaystyle=\theta_{a^{-1}}\left(ea\right)\theta_{e}\left(k\right)\theta_{ek}\left(k^{-1}\right).$

Hence, by (3), $\theta_{a^{-1}}\left(ea\right)\leq\theta_{e}\left(k\right)$.
Therefore, the claim follows. ∎

######  Corollary 7.

Let $s$ be a solution on $S$. If $a,b\in S$ are such that $a\leq b$, then
$\theta_{a}(b)\in G_{\theta_{a}\left(bb^{-1}\right)}$. Moreover, they hold
$\theta_{a}(bb^{-1})=\theta_{a}(b)\theta_{a}(b)^{-1}$ and
$\theta_{a}(b)^{-1}=\theta_{ab}(b^{-1})$.

###### Proof.

If $a,b\in S$ are such that $a\leq b$, then $a\leq bb^{-1}$ and by Lemma
6-$1.$, $\theta_{a}(bb^{-1})\in\operatorname{E}(S)$. Now,

$\theta_{a}(b)=\theta_{a}\left(bb^{-1}b\right)=\theta_{a}\left(bb^{-1}\right)\theta_{abb^{-1}}(b)=\theta_{a}\left(bb^{-1}\right)\theta_{a}(b)$

and
$\theta_{a}\left(bb^{-1}\right)=\theta_{a}(b)\theta_{ab}\left(b^{-1}\right)$.
Thus, by (3), $\theta_{a}(b)\leq\theta_{a}(bb^{-1})$ and
$\theta_{a}(bb^{-1})\leq\theta_{a}(b)$, i.e. $\theta_{a}(b)\in
G_{\theta_{a}\left(bb^{-1}\right)}$. In addition, by the equality
$\theta_{a}\left(bb^{-1}\right)=\theta_{a}\left(b^{-1}b\right)=\theta_{a}\left(b^{-1}\right)\theta_{ab^{-1}}\left(b\right)$
and the previous paragraph, it follows that $\theta_{a}(b)$,
$\theta_{a}(b^{-1})$, and $\theta_{a}(bb^{-1})$ are in the same group with
identity $\theta_{a}(bb^{-1})$. Moreover,
$\theta_{a}(b)^{-1}=\theta_{ab}\left(b^{-1}\right)$, which completes the
proof. ∎

######  Theorem 8.

Let $s$ be a solution on $S$. Then, the kernel $K$ of $s$ is a normal
subsemigroup of $S$.

###### Proof.

Initially, by Lemma 6-$1.$, $\operatorname{E}(S)\subseteq K$. Now, if $k,h\in
K$ and $e\in\operatorname{E}(S)$ are such that $e\leq kh$, then $e\leq k$ and
$e\leq h$ and thus, $\theta_{e}(k),\theta_{e}(h)\in\operatorname{E}(S)$. By
Lemma 6-$2.$, we obtain that
$\theta_{ek}\left(h\right)\in\operatorname{E}(S)$, and so that
$\theta_{e}\left(kh\right)=\theta_{e}\left(k\right)\theta_{ek}\left(h\right)\in\operatorname{E}(S)$.
Now, if $a\in S$, $k\in K$, and $e\in\operatorname{E}(S)$ are such that $e\leq
a^{-1}ka$, then $e\leq a$, $e\leq a^{-1}$, and $e\leq k$. Then,
$\theta_{e}(k)\in\operatorname{E}(S)$. Besides,

$\displaystyle\theta_{e}\left(a^{-1}ka\right)=\theta_{e}\left(a^{-1}\right)\theta_{ea^{-1}}(k)\theta_{ea^{-1}k}(a).$

By Lemma 6-$1.$, $\theta_{e}\left(a^{-1}\right)\in\operatorname{E}(S)$ and, by
Lemma 6-$2.$, $\theta_{ea^{-1}}(k)\in\operatorname{E}(S)$. Furthermore, also
$\theta_{ea^{-1}k}(a)\in\operatorname{E}(S)$. In fact, by (P2),

$\displaystyle\theta_{ea^{-1}k}\left(a\right)=\theta_{\theta_{k^{-1}a}\left(ea^{-1}k\right)}\theta_{k^{-1}aea^{-1}k}\left(a\right)=\theta_{\theta_{k^{-1}a}\left(ea^{-1}k\right)}\theta_{e}\left(a\right)$

and, since

$\displaystyle\theta_{k^{-1}a}\left(ea^{-1}k\right)$
$\displaystyle=\theta_{k^{-1}a}\left(ea^{-1}kaa^{-1}\right)\theta_{k^{-1}a}\left(ea^{-1}k\right)\theta_{k^{-1}aea^{-1}k}\left(aa^{-1}\right)$
$\displaystyle=\theta_{k^{-1}a}\left(ea^{-1}k\right)\theta_{e}\left(a\right)\theta_{ea}\left(a^{-1}\right),$

we obtain that, by (3),
$\theta_{k^{-1}a}\left(ea^{-1}k\right)\leq\theta_{e}\left(a\right)$. So, as
before, by Lemma 6-$1.$, we obtain $\theta_{ea^{-1}k}\left(a\right)\in
E\left(S\right)$. Therefore, the claim follows. ∎

We conclude the section by describing the commutative and cocommutative
solutions on Clifford semigroups. It is easy to check that a solution
$s(a,b)=(ab,\theta_{a}(b))$ is commutative if, and only if,

$\displaystyle acb=abc$ (C1) $\displaystyle\theta_{a}=\theta_{ab}$ (C2)

and $s$ is cocommutative if, and only if,

$\displaystyle a\theta_{b}(c)=ac$ (CC1)
$\displaystyle\theta_{a}\theta_{b}=\theta_{b}\theta_{a}$ (CC2)

for all $a,b,c\in S$.

######  Proposition 9.

Let $s$ be a solution on $S$. Then,

  1. 1.

$s$ is commutative if, and only if, $S$ is a commutative Clifford semigroup
and $\theta_{a}=\gamma$, for every $a\in S$, with
$\gamma\in\operatorname{End}(S)$ and $\gamma^{2}=\gamma$.

  2. 2.

$s$ is cocommutative if, and only if, $\theta_{a}(b)=b$, for all $a,b\in S$,
i.e., $s=\mathcal{I}$.

###### Proof.

At first, we suppose that $s(a,b)=(ab,\theta_{a}(b))$ is a commutative
solution. Then, by (C1), taking $a=cc^{-1}$, we obtain that $S$ is
commutative. Moreover, by (C2), we get
$\theta_{a}=\theta_{ab}=\theta_{ba}=\theta_{b}$. Hence, $\theta_{a}=\gamma$,
for every $a\in S$, and by the definition of solution we obtain the rest of
the claim. The converse trivially follows by $2.$ in 1.
Now, assume that $s(a,b)=(ab,\theta_{a}(b))$ is a cocommutative solution.
Then, by (CC1), taking $a=cc^{-1}$, we obtain

$cc^{-1}\theta_{b}(c)=c,\quad\text{for all $b,c\in S$.}$

Set $e_{0}:=\theta_{b}(c)\theta_{b}(c)^{-1}$, it follows that $cc^{-1}\leq
e_{0}$. On the other hand, again by (CC1), $e\theta_{b}(c)=ec$, for every
$e\in\operatorname{E}(S)$. In particular,
$\theta_{b}(c)=e_{0}\theta_{b}(c)=e_{0}c$. Thus, $e_{0}\leq cc^{-1}$ and so
$e_{0}=cc^{-1}$. Therefore, we get $\theta_{b}(c)=c$, that is our claim. ∎

##  3 A description of idempotent-invariant solutions

In this section, we provide a description of a specific class of solutions on
a Clifford semigroup, the idempotent-invariant ones, which includes the result
contained in [6, Theorem 15].

######  Definition 10.

A solution $s$ on $S$ is said to be _idempotent-invariant_ or
_$E\left(S\right)$ -invariant_ if it holds the identity

$\displaystyle\theta_{a}(e)=\theta_{a}(f),$ (6)

for all $a\in S$ and $e,f\in\operatorname{E}(S)$.

An easy example of $\operatorname{E}(S)$-invariant solution is
$\mathcal{E}(a,b)=(ab,e)$ in (5), with $e\in\operatorname{E}(S)$.

######  Example 2.

Let us consider the commutative Clifford monoid $S=\\{1,\,a,\,b\\}$ with
identity $1$ and such that $a^{2}=a$, $b^{2}=a$, and $ab=b$. Then, other than
the map $\mathcal{E}$ in (5), there exists the idempotent-invariant solution
$s(a,b)=(ab,\gamma(b))$ with $\gamma:S\to S$ the map given by
$\gamma(1)=\gamma(a)=a$ and $\gamma(b)=b$, which belongs to the class of
solutions in $2.$ of 1.

Next, we show how to construct an idempotent-invariant solution on $S$
starting from a specific congruence on $S$. Recall that the restriction of a
congruence $\rho$ in a Clifford semigroup $S$ to $\operatorname{E}(S)$ is also
a congruence on $\operatorname{E}(S)$, called the _trace_ of $\rho$ and
usually denoted by $\tau=\operatorname{tr}\rho$ (for more details, see [14,
Section 5.3]).

######  Proposition 11.

Let $S$ be a Clifford semigroup, $\rho$ a congruence on $S$ such that $S/\rho$
is a group, and $\mathcal{R}$ a system of representatives of $S/\rho$. If
$\mu:S\to\mathcal{R}$ is a map such that

$\mu\left(ab\right)=\mu\left(a\right)\mu\left(a\right)^{-1}\mu\left(ab\right),$
(7)

for all $a,b\in S$, and $\mu(a)\in[a]_{\rho}$, for every $a\in S$, then the
map $s:S\times S\to S\times S$ given by

$s(a,b)=\left(ab,\mu\left(a\right)^{-1}\mu\left(ab\right)\right),$

for all $a,b\in S$, is an $\operatorname{E}(S)$-invariant solution on $S$.

###### Proof.

Let $a,b,c\in S$. Set
$\theta_{a}(b):=\mu\left(a\right)^{-1}\mu\left(ab\right)$, by (7), we obtain

$\displaystyle\theta_{a}(b)\theta_{ab}(c)=\mu\left(a\right)^{-1}\mu\left(ab\right)\mu\left(ab\right)^{-1}\mu\left(abc\right)=\mu\left(a\right)^{-1}\mu\left(abc\right)=\theta_{a}(bc).$

Now, if we compare

$\displaystyle\theta_{\theta_{a}(b)}\theta_{ab}(c):$
$\displaystyle=\mu\left(\mu\left(a\right)^{-1}\mu\left(ab\right)\right)^{-1}\mu\left(\mu\left(a\right)^{-1}\mu\left(ab\right)\mu\left(ab\right)^{-1}\mu\left(abc\right)\right)$
$\displaystyle=\mu\left(\mu\left(a\right)^{-1}\mu\left(ab\right)\right)^{-1}\mu\left(\mu\left(a\right)^{-1}\mu\left(abc\right)\right)$
by (7)

and $\theta_{b}(c):=\mu(b)^{-1}\mu(bc)$, to get the claim it is enough to show
that

$\displaystyle\mu(x)^{-1}\mu(xy)=\mu(y),$

for all $x,y\in S$. Indeed, by [14, Proposition 5.3.1],
$\operatorname{tr}\rho=\operatorname{E}(S)\times\operatorname{E}(S)$, and so

$\displaystyle\mu(x)^{-1}\mu(xy)\ \rho\ x^{-1}xy\ \rho\ y^{-1}yy\ \rho\ y\
\rho\ \mu(y).$

Finally, if $a\in S$ and $e,f\in\operatorname{E}(S)$, we obtain that

$\displaystyle\mu(ae)\ \rho\ ae\ \rho\ af\rho\ \mu(af),$

hence $\mu\left(ae\right)=\mu\left(af\right)$. Thus,
$\theta_{a}(e)=\mu(a)^{-1}\mu\left(ae\right)=\mu(a)^{-1}\mu\left(af\right)=\theta_{a}(f)$.
Therefore, the claim follows. ∎

Our aim is to show that all idempotent invariant solutions can be constructed
exactly as in 11. Firstly, let us collect some useful properties of these
maps.

######  Lemma 12.

Let $s$ be an $E\left(S\right)$-invariant solution on $S$. Then, the following
hold:

  1. 1.

$\theta_{e}=\theta_{f}$,

  2. 2.

$\theta_{ae}=\theta_{a}$,

  3. 3.

$\theta_{a}\left(e\right)\in\operatorname{E}\left(S\right)$,

  4. 4.

$\theta_{e}\theta_{a}=\theta_{e}$,

  5. 5.

$\theta_{a}\left(b\right)=\theta_{a}\left(eb\right)$,

  6. 6.

$\theta_{e}(a)^{-1}=\theta_{ea}\left(a^{-1}\right)$,

for all $e,f\in\operatorname{E}\left(S\right)$ and $a,b\in S$.

###### Proof.

Let $e,f\in\operatorname{E}(S)$ and $a,b\in S$.
$1.$ Since
$\theta_{e}=\theta_{\theta_{f}(e)}\theta_{fe}=\theta_{\theta_{f}(fe)}\theta_{ffe}=\theta_{fe}$
and, similarly $\theta_{f}=\theta_{ef}$, it yields that
$\theta_{f}=\theta_{e}$.
$2.$ We have that

$\displaystyle\theta_{ae}$
$\displaystyle=\theta_{\theta_{a^{-1}}(ae)}\theta_{aa^{-1}e}$
$\displaystyle=\theta_{\theta_{a^{-1}}(a)\theta_{a^{-1}a}(e)}\theta_{aa^{-1}}$
$aa^{-1}e\in\operatorname{E}\left(S\right)$
$\displaystyle=\theta_{\theta_{a^{-1}}(a)\theta_{a^{-1}a}\left(a^{-1}a\right)}\theta_{aa^{-1}}$
by (6)
$\displaystyle=\theta_{\theta_{a^{-1}}\left(a\right)}\theta_{aa^{-1}}=\theta_{a}.$

$3.$ According to $2.$, it follows that
$\theta_{a}\left(e\right)=\theta_{a}\left(ee\right)=\theta_{a}\left(e\right)\theta_{ae}\left(e\right)=\theta_{a}\left(e\right)\theta_{a}\left(e\right)$,
i.e., $\theta_{a}\left(e\right)\in\operatorname{E}\left(S\right)$.
$4.$ According to $2.$, we obtain that
$\theta_{e}=\theta_{\theta_{a}\left(e\right)}\theta_{ae}=\theta_{e}\theta_{ae}=\theta_{e}\theta_{a}$.
$5.$ Note that, by $2.$,
$\theta_{a}\left(b\right)=\theta_{a}\left(bb^{-1}b\right)=\theta_{a}\left(bb^{-1}\right)\theta_{abb^{-1}}\left(b\right)=\theta_{a}\left(e\right)\theta_{ae}\left(b\right)=\theta_{a}\left(eb\right)$.
$6.$ Applying $1.$, we get
$\theta_{e}\left(a\right)\theta_{ea}\left(a^{-1}\right)\theta_{e}(a)=\theta_{e}\left(aa^{-1}\right)\theta_{eaa^{-1}}(a)=\theta_{e}(a)$
and, on the other hand,

$\theta_{ea}\left(a^{-1}\right)\theta_{e}(a)\theta_{ea}\left(a^{-1}\right)=\theta_{ea}\left(a^{-1}\right)\theta_{e}\left(aa^{-1}\right)=\theta_{ea}\left(a^{-1}\right)\theta_{eaa^{-1}}\left(aa^{-1}\right)=\theta_{e}\left(a^{-1}\right).$

Therefore, the claim follows. ∎

To prove the converse of 11, we need to recall the notion of the congruence
pair of inverse semigroups that are Clifford (see [14, p. 155]). Given a
Clifford semigroup $S$, a congruence $\tau$ on $\operatorname{E}(S)$ is said
to be _normal_ if

$\displaystyle\forall\ e,f\in\operatorname{E}(S)\quad e\ \tau\ f\
\Longrightarrow\ \forall\ a\in S\quad a^{-1}ea\ \tau\ a^{-1}fa.$

If $K$ is a normal subsemigroup of $S$, the pair $(K,\tau)$ is named a
_congruence pair_ of $S$ if

$\displaystyle\forall\ a\in S,\ e\in\operatorname{E}(S)\quad ae\in K\ \
\text{and}\ \ (e,a^{-1}a)\in\tau\ \Longrightarrow\ a\in K.$

Given a congruence $\rho$, denoted by $\operatorname{Ker}\rho$ the union of
all the idempotent $\rho$-classes, its properties can be described entirely in
terms of $\operatorname{Ker}\rho$ and $\operatorname{tr}\rho$.

######  Theorem 13 (cf. Theorem 5.3.3 in [14]).

Let $S$ be an inverse semigroup. If $\rho$ is a congruence on $S$, then
$(\operatorname{Ker}\rho,\operatorname{tr}\rho)$ is a congruence pair.
Conversely, if $(K,\tau)$ is a congruence pair, then

$\displaystyle\rho_{(K,\tau)}=\\{(a,b)\in S\times
S\,\mid\,\left(a^{-1}a,b^{-1}b\right)\in\tau,\,ab^{-1}\in K\\}$

is a congruence on $S$. Moreover, $\operatorname{Ker}\rho_{(K,\tau)}=K$,
$\operatorname{tr}\rho_{(K,\tau)}=\tau$, and
$\rho_{(\operatorname{Ker}\rho,\operatorname{tr}\rho)}=\rho$.

######  Lemma 14.

Let $s$ be an $\operatorname{E}\left(S\right)$-invariant solution on $S$,
$\tau=\operatorname{E}(S)\times\operatorname{E}(S)$, and $K$ the kernel of
$s$. Then, $\left(K,\tau\right)$ is a congruence pair of $S$.

###### Proof.

At first, let us observe that the kernel $K$ of $s$ can be written as

$\displaystyle K=\\{a\in S\,\mid\,\forall\
e\in\operatorname{E}(S)\quad\theta_{e}(a)\in\operatorname{E}(S)\\}.$

Now, let $a\in S$ and $e\in\operatorname{E}(S)$ such that $ae\in K$. To get
the claim it is enough to show that if $f\in\operatorname{E}(S)$, then
$\theta_{f}\left(a\right)\in\operatorname{E}\left(S\right)$, i.e., $a\in K$.
By $1.$ and $5.$ in Lemma 12, we obtain that

$\displaystyle\theta_{f}\left(a\right)=\theta_{ef}\left(a\right)=\theta_{ef}\left(ae\right)\in\operatorname{E}(S),$

which is our claim. ∎

The following result completely describes idempotent-invariant solutions.

######  Theorem 15.

Let $s$ be an $\operatorname{E}\left(S\right)$-invariant solution on $S$.
Then, the map $\theta_{e}$ satisfies (7), for every $e\in\operatorname{E}(S)$,
and

$\displaystyle\theta_{a}(b)=\theta_{e}(a)^{-1}\theta_{e}(ab),$

for all $a,b\in S$ and $e\in\operatorname{E}\left(S\right)$. Moreover, there
exists the congruence pair $\left(K,\tau\right)$, with $K$ the kernel of $S$
and $\tau=\operatorname{E}(S)\times\operatorname{E}(S)$, such that
$\theta_{e}\left(S\right)$ is a system of representatives of the group
$S/\rho_{\left(K,\tau\right)}$ and
$\left(\theta_{e}\left(a\right),a\right)\in\rho_{\left(K,\tau\right)}$, for
all $e\in\operatorname{E}(S)$ and $a\in S$.

###### Proof.

Initially, (7) is satisfied since

$\displaystyle\theta_{e}(a)^{-1}\theta_{e}(a)\theta_{e}(ab)=\theta_{e}(a)^{-1}\theta_{e}(a)\theta_{e}(a)\theta_{ea}(b)=\theta_{e}(a)\theta_{ea}(b)=\theta_{e}(ab),$

for all $a,b\in S$ and $e\in\operatorname{E}\left(S\right)$. Besides,

$\displaystyle\theta_{a}(b)$ $\displaystyle=\theta_{a}\left(a^{-1}ab\right)$
by Lemma 12-$5.$
$\displaystyle=\theta_{a}\left(a^{-1}\right)\theta_{aa^{-1}}(ab)$
$\displaystyle=\theta_{aa^{-1}}(a)^{-1}\theta_{aa^{-1}}(ab),$ by 4-$1.$
$\displaystyle=\theta_{e}(a)^{-1}\theta_{e}(ab)$ by Lemma 12-$1.$

for all $a,b\in S$ and $e\in\operatorname{E}\left(S\right)$. Moreover, by
Lemma 14, $\left(K,\tau\right)$ is a congruence pair and so, by 13,
$\rho_{\left(K,\tau\right)}$ is a congruence such that
$\tau=\operatorname{tr}\rho_{\left(K,\tau\right)}$. Besides, by [14,
Proposition 5.3.1], since
$\operatorname{tr}\rho_{\left(K,\tau\right)}=\operatorname{E}(S)\times\operatorname{E}(S)$,
$S/\rho_{\left(K,\tau\right)}$ is a group. Now, let $a\in S$ and
$e\in\operatorname{E}\left(S\right)$ and let us check that
$\left(\theta_{e}\left(a\right),a\right)\in\rho_{\left(K,\tau\right)}$ by
proving that $a^{-1}\theta_{e}\left(a\right)\in K$, i.e.,
$\theta_{e}\left(a^{-1}\theta_{e}\left(a\right)\right)\in\operatorname{E}\left(S\right)$.
To this end, note that

$\displaystyle\theta_{e}\left(a^{-1}\theta_{e}\left(a\right)\right)$
$\displaystyle=\theta_{e}\theta_{a}\left(a^{-1}\theta_{e}\left(a\right)\right)$
by Lemma 12-$4.$
$\displaystyle=\theta_{e}\left(\theta_{a}\left(a^{-1}\right)\theta_{aa^{-1}}\theta_{e}\left(a\right)\right)$
$\displaystyle=\theta_{e}\left(\theta_{a}\left(a^{-1}\right)\theta_{aa^{-1}}\left(a\right)\right)$
by Lemma 12-$4.$
$\displaystyle=\theta_{e}\left(\theta_{a}\left(a^{-1}\right)\theta_{a}\left(a^{-1}\right)^{-1}\right),$
by 4-$1.$

hence, by Lemma 12-$3.$,
$\theta_{e}\left(a^{-1}\theta_{e}\left(a\right)\right)\in\operatorname{E}\left(S\right)$.
Now, let us verify that $\theta_{e}\left(S\right)$ is a system of
representatives of $S/\rho_{\left(K,\tau\right)}$. Clearly,
$\theta_{e}\left(S\right)\neq\emptyset$ since
$\theta_{e}\left(e\right)\in\operatorname{E}\left(S\right)$. Besides, if
$\left(\theta_{e}\left(b\right),a\right)\in\rho_{\left(K,\tau\right)}$ we have
that $a\,\rho_{\left(K,\tau\right)}\,b$, since
$\left(\theta_{e}\left(a\right),a\right)\in\rho_{\left(K,\tau\right)}$. Thus,
$ab^{-1}\in K$ and so
$\theta_{e}\left(ab^{-1}\right)\in\operatorname{E}\left(S\right)$. This
implies that

$\displaystyle\theta_{e}\left(b\right)$
$\displaystyle=\theta_{e}\left(bb^{-1}\right)\theta_{ebb^{-1}}\left(b\right)$
$\displaystyle=\theta_{e}\left(bb^{-1}\right)\theta_{\theta_{e}\left(ab^{-1}\right)}\left(b\right)$
by Lemma 12-$1.$
$\displaystyle=\theta_{e}\theta_{e}\left(ab^{-1}\right)\theta_{\theta_{e}\left(ab^{-1}\right)}\theta_{eab^{-1}}\left(b\right)$
by (6) and Lemma 12-$4.$
$\displaystyle=\theta_{e}\left(ab^{-1}\right)\theta_{ab^{-1}}\left(b\right)$
by Lemma 12-$4.$
$\displaystyle=\theta_{e}\left(ab^{-1}\right)\theta_{eab^{-1}}\left(b\right)$
by Lemma 12-$2.$ and (P2) $\displaystyle=\theta_{e}\left(ab^{-1}b\right)$
$\displaystyle=\theta_{e}\left(a\right).$ by Lemma 12-$5.$

Therefore, the claim follows. ∎

######  Proposition 16.

Let $s(a,b)=(ab,\theta_{a}(b))$ and $t(u,v)=(uv,\theta_{u}(v))$ be two
$\operatorname{E}(S)$-invariant solutions on $S$. Then, $s$ and $t$ are
isomorphic if, and only if, there exists an isomorphism $\psi$ of $S$ such
that $\psi\theta_{e}=\eta_{e}\psi$, i.e., $\psi$ sends the system of
representatives $\theta_{e}(S)$ into the other one
$\eta_{e}\left(\psi(S)\right)$, for every $e\in\operatorname{E}(S)$.

###### Proof.

Indeed, making explicit the condition (4), we obtain

$\displaystyle\psi\left(\theta_{e}(a)^{-1}\theta_{e}(ab)\right)=\eta_{e}\left(\psi(a)\right)^{-1}\eta_{e}\left(\psi(ab)\right),$

for all $a,b\in S$ and $e\in\operatorname{E}(S)$. Applying Lemma 12-$6.$ and
taking $b=a^{-1}$, we get
$\psi\left(\theta_{e}(a)^{-1}\right)=\eta_{e}\left(\psi(a)\right)^{-1}$. Thus,
the claim follows. ∎

##  4 A construction of idempotent-fixed solutions

In this section, we deal with a class of solutions different from the
idempotent-invariant ones, what we call idempotent-fixed solutions. Bearing in
mind that a Clifford semigroup can be seen as a union of groups satisfying
certain properties, it is natural to contemplate whether it is possible or not
to construct a global solution in a Clifford semigroup from solutions obtained
in each of its groups. In this regard, in the case of idempotent-fixed
solutions, we manage to construct a family of solutions obtained by starting
from given solutions on each group.

######  Definition 17.

Let $s$ be a solution on $S$. Then, $s$ is _idempotent-fixed_ or
_$\operatorname{E}(S)$ -fixed_ if

$\displaystyle\theta_{a}(e)=e,$ (8)

for all $a\in S$ and $e\in\operatorname{E}(S)$.

The maps $\mathcal{I}(a,b)=(ab,b)$ and
$\mathcal{F}(a,b)=\left(ab,bb^{-1}\right)$ in (5) are idempotent-fixed
solutions on $S$. Clearly, if $S$ is a Clifford that is not a group, i.e.,
$|\operatorname{E}(S)|>1$, then a solution on $S$ can not be both idempotent-
fixed and idempotent-invariant.

The next results contained several properties of idempotent-fixed solutions.

######  Proposition 18.

Let $s$ be an idempotent-fixed solution on $S$. Then,
$\theta_{e}=\theta_{e}\theta_{ae}$, for all $a\in S$ and
$e\in\operatorname{E}(S)$. In particular, $\theta_{e}$ is an idempotent map.

###### Proof.

It follows by
$\theta_{e}=\theta_{\theta_{a}(e)}\theta_{ae}=\theta_{e}\theta_{ae}$, for all
$a\in S$ and $e\in\operatorname{E}(S)$. Taking $a=e$, we obtain that the map
$\theta_{e}$ is idempotent. ∎

######  Proposition 19.

Let $s$ be an idempotent-fixed solution on $S$. Then, the following hold:

  1. 1.

$\theta_{a}(b)=bb^{-1}\theta_{a}(b)$,

  2. 2.

$\theta_{a}\left(b\right)\theta_{a}\left(b\right)^{-1}=bb^{-1}$,

  3. 3.

$\theta_{a}(b)=\theta_{abb^{-1}}(b)$,

for all $a,b\in S$.

###### Proof.

Let $a,b\in S$. Then,
$\theta_{a}\left(b\right)=\theta_{a}\left(b\right)\theta_{ab}\left(b^{-1}b\right)=\theta_{a}\left(b\right)bb^{-1}$.
Moreover, we have that
$\theta_{a}\left(b\right)^{-1}=\theta_{ab}\left(b^{-1}\right)$ since

$\displaystyle\theta_{a}\left(b\right)\theta_{ab}\left(b^{-1}\right)\theta_{a}\left(b\right)=\theta_{a}\left(bb^{-1}\right)\theta_{a}\left(b\right)=bb^{-1}\theta_{a}\left(b\right)=\theta_{a}\left(b\right)$

and

$\displaystyle\theta_{ab}\left(b^{-1}\right)\theta_{a}\left(b\right)\theta_{ab}\left(b^{-1}\right)$
$\displaystyle=\theta_{ab}\left(b^{-1}\right)\theta_{a}\left(bb^{-1}\right)=b^{-1}b\,\theta_{ab}\left(b^{-1}\right)=\theta_{ab}\left(bb^{-1}\right)\theta_{abb^{-1}b}\left(b^{-1}\right)$
$\displaystyle=\theta_{ab}\left(b^{-1}\right).$

It follows that
$\theta_{a}\left(b\right)\theta_{a}\left(b\right)^{-1}=\theta_{a}\left(b\right)\theta_{ab}\left(b^{-1}\right)=\theta_{a}\left(bb^{-1}\right)=bb^{-1}.$
Finally, by $1.$, we have that

$\displaystyle\theta_{abb^{-1}}\left(b\right)=bb^{-1}\theta_{abb^{-1}}\left(b\right)=\theta_{a}\left(bb^{-1}\right)\theta_{abb^{-1}}\left(b\right)=\theta_{a}\left(bb^{-1}b\right)=\theta_{a}\left(b\right)$

that completes the proof. ∎

As a consequence of 19-$1.$, if $s$ is an idempotent-fixed solution on the
Clifford $S$, it follows that every group in $S$ remains invariant by
$\theta_{a}$, for all $a\in S$. Thus, motivated by the fact that solutions on
groups are well-described, it makes sense to provide a method to construct
this type of solutions from solutions on each group in $S$. To this end, the
inner structure of a Clifford semigroup makes clear that conditions relating
to different solutions on the groups of $S$ must be considered. For instance,
19-$3.$ shows that
$\theta_{a}\left(b\right)=\theta_{\varphi_{e,f}\left(a\right)}\left(b\right)$,
for all $e,f\in\operatorname{E}(S)$, with $e\geq f$, and all $a\in G_{e}$,
$b\in G_{f}$. In light of these observations, we provide the following family
of idempotent-fixed solutions.

######  Theorem 20.

Let $s^{[e]}(a,b)=\left(ab,\,\theta^{[e]}_{a}\left(b\right)\right)$ be a
solution on $G_{e}$, for every $e\in\operatorname{E}(S)$. Moreover, for all
$e,f\in\operatorname{E}(S)$, let $\epsilon_{e,f}:G_{e}\to G_{f}$ be maps such
that $\epsilon_{e,f}=\varphi_{e,f}$ if $e\geq f$. If the following conditions
are satisfied

$\displaystyle\theta^{[h]}_{\epsilon_{ef,h}(ab)}$
$\displaystyle=\theta^{[h]}_{\epsilon_{e,h}(a)\epsilon_{f,h}(b)},$ (9)
$\displaystyle\epsilon_{f,h}\theta^{[f]}_{\epsilon_{e,f}(a)}(b)$
$\displaystyle=\theta^{[h]}_{\epsilon_{e,h}(a)}\epsilon_{f,h}(b),$ (10)

for all $e,f,h\in\operatorname{E}(S)$ and $a\in G_{e}$ and $b\in G_{f}$, set

$\theta_{a}(b):=\theta^{[f]}_{\epsilon_{e,f}(a)}(b),$

for all $a\in G_{e}$ and $b\in G_{f}$. Then, the map $s:S\times S\to S\times
S$ given by $s(a,b)=(ab,\theta_{a}(b))$ is an idempotent-fixed solution on
$S$.

###### Proof.

Let $e,f,h\in\operatorname{E}(S)$, $a\in G_{e}$, $b\in G_{f}$, and $c\in
G_{h}$. Then, since $s^{[fh]}$ is a solution on $G_{fh}$, we obtain

$\displaystyle\theta_{a}\left(bc\right)$
$\displaystyle=\theta_{a}\left(\varphi_{f,fh}\left(b\right)\varphi_{h,fh}\left(c\right)\right)=\theta^{[fh]}_{\epsilon_{e,fh}\left(a\right)}\left(\varphi_{f,fh}\left(b\right)\varphi_{h,fh}\left(c\right)\right)$
$\displaystyle=\theta^{[fh]}_{\epsilon_{e,fh}\left(a\right)}\varphi_{f,fh}\left(b\right)\theta^{[fh]}_{\epsilon_{e,fh}\left(a\right)\varphi_{f,fh}\left(b\right)}\varphi_{f,fh}\left(c\right).$

Besides, we have that

$\displaystyle\theta_{a}\left(b\right)\theta_{ab}\left(c\right)$
$\displaystyle=\theta^{[f]}_{\epsilon_{e,f}\left(a\right)}\left(b\right)\theta^{[h]}_{\epsilon_{ef,h}\left(ab\right)}\left(c\right)=\varphi_{f,fh}\theta^{[f]}_{\epsilon_{e,f}\left(a\right)}\left(b\right)\varphi_{h,fh}\theta^{[h]}_{\epsilon_{ef,h}\left(ab\right)}\left(c\right).$

Hence, noting that, by (9),

$\displaystyle\theta^{[fh]}_{\epsilon_{e,fh}\left(a\right)}\varphi_{f,fh}\left(b\right)=\theta^{[fh]}_{\epsilon_{e,fh}\left(a\right)}\epsilon_{f,fh}\left(b\right)=\epsilon_{f,fh}\theta^{[f]}_{\epsilon_{e,f}\left(a\right)}\left(b\right)=\varphi_{f,fh}\theta^{[f]}_{\epsilon_{e,f}\left(a\right)}\left(b\right)$

and

$\displaystyle\theta^{[fh]}_{\epsilon_{e,fh}\left(a\right)\varphi_{f,fh}\left(b\right)}\varphi_{f,fh}\left(c\right)$
$\displaystyle=\theta^{[fh]}_{\epsilon_{e,fh}\left(a\right)\epsilon_{f,fh}\left(b\right)}\epsilon_{f,fh}\left(c\right)$
$\displaystyle=\theta^{[fh]}_{\epsilon_{ef,fh}\left(ab\right)}\epsilon_{f,fh}\left(c\right)$
by (9)
$\displaystyle=\epsilon_{h,fh}\theta^{[h]}_{\epsilon_{ef,h}\left(ab\right)}\left(c\right)$
by (10)
$\displaystyle=\varphi_{h,fh}\theta^{[h]}_{\epsilon_{ef,h}\left(ab\right)}\left(c\right),$

it follows that (P1) is satisfied. In addition,

$\displaystyle\theta_{\theta_{a}\left(b\right)}\theta_{ab}\left(c\right)$
$\displaystyle=\theta_{\theta^{[f]}_{\epsilon_{e,f}\left(a\right)}\left(b\right)}\theta^{[h]}_{\epsilon_{ef,h}\left(ab\right)}\left(c\right)$
$\displaystyle=\theta^{[h]}_{\epsilon_{f,h}\theta^{[f]}_{\epsilon_{e,f}\left(a\right)}\left(b\right)}\theta^{[h]}_{\epsilon_{ef,h}\left(ab\right)}\left(c\right)$
$\displaystyle=\theta^{[h]}_{\theta^{[h]}_{\epsilon_{e,h}\left(a\right)}\epsilon_{f,h}\left(b\right)}\theta^{[h]}_{\epsilon_{e,h}\left(a\right)\epsilon_{f,h}\left(b\right)}\left(c\right)$
by (10) and (9)
$\displaystyle=\theta^{[h]}_{\epsilon_{f,h}\left(b\right)}\left(c\right)$
$s^{[h]}$ is a solution on $G_{h}$ $\displaystyle=\theta_{b}\left(c\right),$

thus (P2) holds. Finally, by [6, Lemma 11-$1.$],
$\theta_{a}(f)=\theta^{[f]}_{\epsilon_{e,f}(a)}(f)=f$ and so $s$ is
idempotent-fixed. ∎

The following is a class of idempotent-fixed solutions on $S$ that can be
constructed through 20 and includes the solutions $\mathcal{I}(a,b)=(ab,b)$
and $\mathcal{F}(a,b)=\left(ab,bb^{-1}\right)$ in (5).

######  Example 3.

Let $s^{[e]}\left(a,b\right)=\left(ab,\gamma^{[e]}\left(b\right)\right)$ be
the solution on $G_{e}$ as in $2.$ of 1 with $\gamma^{[e]}$ an idempotent
endomorphism of $G_{e}$, for every $e\in\operatorname{E}(S)$. Then, by
choosing maps $\epsilon_{e,f}:G_{e}\to G_{f}$, for all
$e,f\in\operatorname{E}(S)$ such that
$\varphi_{e,f}\gamma^{[e]}=\gamma^{[f]}\varphi_{e,f}$ if $e\geq f$ and
$\epsilon_{e,f}\left(x\right):=f$ otherwise, then conditions (9) and (10) are
satisfied. Hence, the map

$\displaystyle s(a,b)=\left(ab,\gamma^{[f]}(b)\right),$

for all $a\in G_{e}$ and $b\in G_{f}$, is a solution on $S$.

As a consequence of 20, the following construction provides a subclass of
idempotent-fixed solutions in Clifford semigroups in which each group $G_{f}$
is an epimorphic image of $G_{e}$, whenever $f\leq e$, for all
$e,f\in\operatorname{E}(S)$.

######  Corollary 21.

Let $S$ be a Clifford semigroup such that $\varphi_{e,f}$ is an epimorphism,
for all $e,f\in\operatorname{E}(S)$ with $f\leq e$. Let
$s^{[e]}(a,b)=\left(ab,\theta_{a}^{[e]}(b)\right)$ be a solution on $G_{e}$
and set $N_{e}:=\prod\limits_{f\leq e}\ker\varphi_{e,f}$, for every
$e\in\operatorname{E}(S)$. Suppose that

  1. 1.

$\theta_{a}^{[e]}=\theta_{b}^{[e]}$, for all $e\in\operatorname{E}(S)$ and all
$a,b\in G_{e}$ with $aN_{e}=bN_{e}$,

  2. 2.

$\varphi_{e,f}\theta_{a}^{[e]}(b)=\theta_{\varphi_{e,f}(a)}^{[f]}\varphi_{e,f}(b)$,
for all $e,f\in\operatorname{E}(S)$ with $f\leq e$, and all $a,b\in G_{e}$.

Set $\theta_{a}(b):=\theta_{b^{\prime}}^{[f]}(b)$, with $b^{\prime}\in G_{f}$
such that $\varphi_{f,ef}(b)=\varphi_{e,ef}(a)$, for all
$e,f\in\operatorname{E}(S)$, and all $a\in G_{e}$, $b\in G_{f}$. Then, the map
$s\colon S\times S\rightarrow S\times S$ given by $s(a,b)=(ab,\theta_{a}(b))$
is an idempotent-fixed solution on $S$.

###### Proof.

Initially, by $1.$, note that $\theta_{a}$ is well-defined, for every $a\in
S$. Now, let $e,f\in\operatorname{E}(S)$ and consider $T_{e,f}$ a system of
representatives of $\ker\varphi_{f,ef}$ in $G_{f}$. Since $\varphi_{f,ef}$ is
an epimorphism, for every $a\in G_{e}$, we can define a map
$\epsilon_{e,f}(a):=x\in T_{e,f}$, with $\varphi_{e,ef}(a)=\varphi_{f,ef}(x)$.
Specifically, in the case that $f\leq e$, it follows that
$\epsilon_{e,f}=\varphi_{e,f}$. Therefore, for all $e,f\in\operatorname{E}(S)$
and all $a\in G_{e}$, $b\in G_{f}$, it holds
$\theta_{a}(b)=\theta_{\epsilon_{e,f}(a)}^{[f]}(b)$. Note that, by $1.$, the
last equality is independent of the choice of $T_{e,f}$. Moreover, applying
properties in 1 of homomorphisms $\varphi_{e,f}$, for all
$e,f\in\operatorname{E}(S)$ with $f\leq e$, and the assumptions, it is a
routine computation to check that conditions (9) and (10) of 20 are satisfied.

∎

Let us observe that the kernel of an idempotent-fixed solution $s$ can be
rewritten as

$\displaystyle K=\\{a\in S\,\mid\,\forall\,e\in\operatorname{E}(S),\,e\leq a,\
\theta_{e}(a)=aa^{-1}\\}.$

Denoted by $K_{e}$ the kernel of each solution $s^{[e]}$ on $G_{e}$, i.e., the
normal subgroup

$\displaystyle K_{e}=\\{a\in G_{e}\,\mid\,\theta^{[e]}_{e}(a)=e\\}.$

of $G_{e}$, we have the following result that clarifies the previous
construction in 20 is not a description.

######  Proposition 22.

Let $s$ be an idempotent-fixed solution on $S$ constructed as in 20 and
suppose that $\epsilon_{e,f}(e)=f$, for all $e,f\in\operatorname{E}(S)$ with
$e\leq f$. Assume that each $G_{e}$ admits a solution $s^{[e]}$ and let
$K_{e}$ be the kernel of such a map $s^{[e]}$, for every
$e\in\operatorname{E}(S)$. Then,
$K=\displaystyle\bigcup_{e\in\operatorname{E}(S)}\,K_{e}$.

###### Proof.

Indeed, let $a\in K\cap G_{e}$. Then, we get
$e=aa^{-1}=\theta_{e}(a)=\theta^{[e]}_{e}(a)$. Thus, $a\in K_{e}$. On the
other hand, if $a\in K_{e}$ and $f\in\operatorname{E}(S)$ is such that $f\leq
a$, then, since $\epsilon_{e,f}(e)=f$, we obtain
$\theta_{f}(a)=\theta^{[e]}_{\epsilon_{f,e}(f)}(a)=\theta^{[e]}_{e}(a)=e$,
i.e., $a\in K$. ∎

######  Question 1.

Complete a description of all the idempotent-fixed solutions.

To conclude, we observe that not every solution on $S$ lies in the class of
idempotent invariant or idempotent-fixed solutions. Indeed, even in Clifford
semigroups of low order, it is possible to construct such an example.

######  Example 4.

Let $S=\\{1,\,a,\,b\\}$ be the Clifford monoid in 2. Then, the maps

$\displaystyle\theta_{1}(x)=a,\quad\text{for every $x$}\in S,$
$\displaystyle\theta_{a}=\theta_{b}:S\to S,\quad\text{given
by}\,\,\theta_{a}(1)=1,\,\,\theta_{a}(a)=\theta_{a}(b)=a$

give rise to a solution on $S$ that is neither idempotent invariant, nor
idempotent fixed.

######  Question 2.

Study other classes of solutions on Clifford semigroups, including, for
instance, the map in 4.

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