# Coherent reaction between molecular and atomic Bose-Einstein condensates:
integrable model

Rajesh K. Malla  Theoretical Division, and the Center for Nonlinear Studies,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  Vladimir Y.
Chernyak  Department of Chemistry, Wayne State University, 5101 Cass Ave,
Detroit, Michigan 48202, USA  Department of Mathematics, Wayne State
University, 656 W. Kirby, Detroit, Michigan 48202, USA  Chen Sun  School of
Physics and Electronics, Hunan University, Changsha 410082, China  Nikolai A.
Sinitsyn  Theoretical Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA<EMAIL_ADDRESS>
###### Abstract

We solve a model that describes a stimulated conversion between ultracold
bosonic atoms and molecules. The reaction is triggered by a linearly time-
dependent transition throughout the Feshbach resonance. Our solution predicts
a nonexponential dependence, with a dynamic phase transition, of the reaction
efficiency on the transition rate. We find that the emerging phase can have a
thermalized energy distribution with the temperature defined by the rate of
the transition. This phase, however, has strong purely quantum correlations.

Recently, a coherent conversion between Cs atomic Bose-Einstein condensate
(BEC) and the condensate of molecules Cs2 was demonstrated by changing a
magnetic field in time that pushed the ultracold atoms across the Feshbach
resonance Zhang  _et al._ (2021). To suppress irreversible scatterings, the
experimentalists in Zhang  _et al._ (2021) confined their rotating BECs in a
quasi-2D trap. In the created state, possibly up to 50% of molecules formed a
condensate, and in the reversed process, about 40% of the molecules coherently
dissociated into atoms. Further improvements of the conversion efficiency are
expected. This will enable the chemistry of coherent BECs for flexible
engineering of macroscopic correlated quantum states, with applications in
sensing and information-processing.

Every established branch of physics has benefited from solving certain models
without any approximations. For example, the solution for the hydrogen atom
explains the basic observations in atomic physics, for which it serves as a
starting point for numerous approximate techniques.

In this Letter, we solve a model that can play a similar role in the chemistry
of BECs. Its Hamiltonian describes a stimulated conversion between bosonic
atoms and molecules during a sweep of a linearly time-dependent magnetic field
across a narrow Feshbach resonance. Without approximations, the model captures
the main features of the process: many-body interactions, energy dispersion of
the atomic states, different initial populations of these states, and an
arbitrary passage rate:

$\displaystyle H(t)=-\beta
t\hat{\Psi}^{\dagger}\hat{\Psi}+\sum_{k}\Big{\\{}\varepsilon^{a}_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k}+\varepsilon^{b}_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}$
$\displaystyle+g\left(\hat{\Psi}^{\dagger}\hat{a}_{k}\hat{b}_{k}+\hat{\Psi}\hat{a}^{\dagger}_{k}\hat{b}^{\dagger}_{k}\right)\Big{\\}}.$
(1)

Here, $\hat{a}_{k}$, $\hat{b}_{k}$ and $\hat{\Psi}$ are the boson annihilation
operators; $\varepsilon^{a,b}_{k}$ are the energies of the free atoms; $\beta$
is proportional to the ramp of the magnetic field that sweeps the system
across the Feshbach resonance, and $g$ is the coupling for the conversion of
the atomic pairs into the molecules; $\hat{\Psi}$ is the molecular field, and
$\hat{a}_{k}$ and $\hat{b}_{k}$ describe the modes of the ultracold atoms.
They are generally different due to momentum and spin conservation Kayali and
Sinitsyn (2003). However, we allow for any reaction channel $k$ to identify
$\hat{b}_{k}$ with $\hat{a}_{k}$.

We assume that as $t\rightarrow-\infty$ the system is close to its ground
state. Hence, (Coherent reaction between molecular and atomic Bose-Einstein
condensates: integrable model) for $\beta>0$ describes the driven transition
from atoms as $t\rightarrow-\infty$ into an initially empty molecular mode
[Fig. 1(a)]. We will call this a forward process. The reverse process, of the
molecular condensate dissociation in Fig. 1(b), corresponds to $\beta<0$, so
that the ground state with a molecular condensate is at $t=-\infty$, and it is
coherently converted into atomic pairs. Our goal is to find the final state as
$t\rightarrow+\infty$.

The creation of ultracold molecules by sweeping the magnetic field across the
resonance has been studied theoretically Yurovsky  _et al._ (2002); Altland
_et al._ (2009); Sun and Sinitsyn (2016); Itin and Törmä (2009), and used
experimentally Herbig  _et al._ (2003); Xu  _et al._ (2003); Bohn  _et al._
(2017); Matyjaśkiewicz  _et al._ (2008); Duda  _et al._ (2021) for two
decades. However, a considerable fraction of the molecular condensate, until
the experiment Zhang  _et al._ (2021), was found only for the reactions of
fermionic atoms, which had fewer than bosonic possibilities to create
detrimental excitations. Therefore, the purely bosonic reactions are yet to be
understood.

Figure 1: The reaction between the atoms and the molecular condensate in (a)
the forward and (b) the reverse process. The E-axis is energy; violet arrows
show the direction of the transition throughout the resonance. In the
potential trap, molecular and atomic modes are shown as pink and green
surfaces, respectively. (c) and (d) characterize the average number of
molecules $\langle n\rangle$ and of atomic pairs $\langle m\rangle=N-\langle
n\rangle$ that are found for different values of the inverse sweep rate,
$1/\beta$, at $g=1$, after the transitions starting from the ground state in
the forward and the backward processes in (6), respectively. The discrete
points are the exact predictions for the initial number of particles
$N=10^{4}$ sup , and the solid curves in (c) and (d) are the large-$N$
predictions of, respectively, Eq. (10) and Eq. (13) from sup .

The Hamiltonian $H(t)$ is explicitly time-dependent and has cubic
interactions. There are no restrictions on all of its parameters, so (Coherent
reaction between molecular and atomic Bose-Einstein condensates: integrable
model) can describe both slow (small $\beta$) and fast (large $\beta$)
transitions. The dependence on $\beta$ has been accessible both in Zhang  _et
al._ (2021) and previous experiments with fermion-molecular reactions. Hence,
here we consider the questions: how the molecular-atomic conversion efficiency
depends on $\beta$ and how by tuning $\beta$ one can control properties of the
emerging state after the reaction?

To solve the model, we note that
$Q_{k}=\hat{a}_{k}^{\dagger}\hat{a}_{k}-\hat{b}^{\dagger}_{k}\hat{b}_{k}$
commutes with $H(t)$, and introduce operators
$K_{k}^{+}\equiv\hat{a}_{k}^{\dagger}\hat{b}_{k}^{\dagger}$ and
$K_{k}^{-}\equiv\hat{a}_{k}\hat{b}_{k}$, which satisfy

$[\hat{n}_{k},K^{\pm}_{k}]=\pm
2K_{k}^{\pm},\quad[K_{k}^{-},K_{k}^{+}]=(\hat{n}_{k}+1),$ (2)

where
$\hat{n}_{k}\equiv\hat{a}^{\dagger}_{k}\hat{a}_{k}+\hat{b}^{\dagger}_{k}\hat{b}_{k}.$
We also introduce parameters
$\varepsilon_{k}=(\varepsilon_{k}^{a}+\varepsilon_{k}^{b})$, and $\tau$ to
rewrite $H(t)$ as

$H(t)=-\beta
t\hat{\Psi}^{\dagger}\hat{\Psi}+\sum_{k}\big{\\{}\tau\varepsilon_{k}\hat{n}_{k}/2+g\left[\hat{\Psi}^{\dagger}K^{-}_{k}+\hat{\Psi}K^{+}_{k}\right]\big{\\}}.$
(3)

At $\tau=1$, we reproduce (Coherent reaction between molecular and atomic
Bose-Einstein condensates: integrable model) up to $Q_{k}$-dependent terms
that do not change the dynamics. Our main observation is that $H(t)$ in (3)
commutes with

$\displaystyle H^{\prime}(t)$ $\displaystyle=$
$\displaystyle\sum_{k}\big{\\{}\varepsilon_{k}(t+\frac{\tau\varepsilon_{k}}{\beta})\frac{\hat{n}_{k}}{2}+\frac{g\varepsilon_{k}}{\beta}\left[\hat{\Psi}^{\dagger}K^{-}_{k}+\hat{\Psi}K^{+}_{k}\right]\big{\\}}+$
(4) $\displaystyle+$ $\displaystyle\frac{g^{2}}{\beta\tau}\sum_{i,j;\,i\neq
j}\left(K_{i}^{+}K_{j}^{-}-(\hat{n}_{i}+1)(\hat{n}_{j}+1)/4\right),$

and these two operators satisfy the relation

$\partial H/\partial\tau=\partial H^{\prime}/\partial t.$ (5)

According to Sinitsyn  _et al._ (2018); Chernyak  _et al._ (2020), this makes
our model an integrable multistate Landau-Zener model, for which any rescaling
by $\tau$ does not change the scattering probabilities between any eigenstates
of $H(t)$.

Hence, without affecting the final result, we can set $\tau\rightarrow\infty$,
which renders all atomic states well-separated by times of their individual
resonant interaction with the molecular mode. Near each such resonance, we can
safely disregard all other reaction channels and then treat their effects
sequentially in their chronological order.

The Hamiltonian restricted to a single reaction channel,
$\hat{\Psi}\leftrightarrow\hat{a}\hat{b}$, is given by

$\displaystyle H(t)=-\beta
t\hat{\Psi}^{\dagger}\hat{\Psi}+g\left(\hat{\Psi}^{\dagger}\hat{a}\hat{b}+\hat{\Psi}\hat{a}^{\dagger}\hat{b}^{\dagger}\right),$
(6)

where we shifted the timescale to set the resonance at $t=0$. The dynamics
with (6) conserves two quantities:

$N=\hat{\Psi}^{\dagger}\hat{\Psi}+\hat{b}^{\dagger}\hat{b},\quad
Q=\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b}.$ (7)

This allows us to express the microstates only via the the number of molecules
$n$: $|n\rangle\equiv|n;N-n+Q,N-n\rangle,$ where $N-n+Q$ and $N-n$ are the
numbers of atoms in $a$ and $b$ modes, respectively. Note also that at $Q=1$
(6) has the same matrix elements as the Hamiltonian with a single atomic mode:
${\hat{a}}\equiv{\hat{b}}$.

Our second observation is that (6) can be mapped to the driven Tavis-Cummings
model (DTCM) Sinitsyn and Li (2016); Sun and Sinitsyn (2016):

$H_{TC}=\beta
t(\hat{\psi}^{\dagger}\hat{\psi}-N)+g(\hat{\psi}^{\dagger}\hat{S}^{-}+\hat{\psi}\hat{S}^{+}),$
(8)

where $\hat{\psi}$ is a boson annihilation operator, and $\hat{S}^{\pm}$ are
raising/lowering operators of a spin with size $S=(N+Q)/2$. The DTCM conserves
$N=\hat{\psi}^{\dagger}\hat{\psi}+(S+\hat{S}_{z}).$ Hence, we can mark its
states as $|n\rangle$ where $n=S+S_{z},$ Then,

$\langle n+1|H_{TC}|n\rangle=\sqrt{(N-n)(N+Q-n)(n+1)}.$ (9)

Comparing with (6), we find

$\langle n|H|n^{\prime}\rangle=\langle n|H_{TC}|n^{\prime}\rangle,\quad\forall
n,n^{\prime}.$

This map is not intuitive in the sense that the number of molecules $n$ in (6)
is not the same as the number of bosons in the DTCM. Instead, $\langle
n|\hat{\psi}^{\dagger}\hat{\psi}|n\rangle=N-n$. Therefore, the dynamics that
was called the forward process in DTCM Sun  _et al._ (2019) is mapped to the
reverse process in (6), and vice versa. Thus, we predict strongly different
efficiencies in purely bosonic versus fermion-bosonic reactions.

Reaction efficiency. The DTCM has been solved previously Sinitsyn and Li
(2016). Its transition probabilities between any microstates can be found in
Eqs. (23-24) in Sun and Sinitsyn (2016), where the limits of large $N$ are
also described. Using our map to this model, we find for the forward sweep sup
that if all atoms are initially in the ground state with some number of pairs
$N\gg 1$ and $Q=O(1)$ then the probability distribution of finding
$n$-molecules, for $\langle n\rangle\gg 1$, at the end is sharply peaked near
the average value

$\langle
n\rangle\equiv\langle\hat{\Psi}^{\dagger}\hat{\Psi}\rangle_{t\rightarrow\infty}=N+\frac{\log{\left(2-x^{N+Q}\right)}}{\log{x}},\quad
x=e^{-\frac{2\pi g^{2}}{{\beta}}}.$ (10)

For the reverse sweep, starting with $N$ molecules and no atoms, after the
passage through one resonance, the probability to produce $m$ atomic pairs is
given by sup

$P_{m}=x^{N-m}(x^{N-m+1},x)_{m},\quad x=e^{-2\pi g^{2}/\beta},$ (11)

where

$(a,x)_{m}\equiv\prod_{k=0}^{m-1}(1-ax^{k})=(1-a)(1-ax)\cdots(1-ax^{m-1})$
(12)

is the q-Pochhammer symbol. According to Sun  _et al._ (2019), this
distribution predicts a dynamic phase transition. Namely, if the number of
molecules $N$ is initially macroscopically large, then the fraction of
molecules converted to atoms, $\langle m\rangle/N=1-\langle n\rangle/N,$
behaves discontinuously as a function of, $g^{2}/\beta$. Let

$f=\frac{2\pi g^{2}}{\beta}\frac{N}{{\rm log_{e}}N},$ (13)

then the distribution (11) has the property

$\displaystyle\langle m\rangle/N$ $\displaystyle=$ $\displaystyle 0\quad{\rm
for}\quad f<1,\quad N\rightarrow\infty,$ $\displaystyle\langle m\rangle/N$
$\displaystyle=$ $\displaystyle\frac{f-1}{f}\quad{\rm for}\quad f\geq 1,\quad
N\rightarrow\infty.$ (14)

In Figs. 1(c,d) we confirm the predictions (10), (14) using numerically exact
transition probabilities sup  for $N=10^{4}$, and also verify robustness of
such predictions against an initial asymmetry in atomic population due to
nonzero $Q$. Here we note similarity of Fig. 1(d) with the experimentally
obtained Fig.4(a) in Zhang  _et al._ (2021). Both figures describe the number
of produced atoms from molecules: in our case as a function of $1/\beta$ but
as a function of time of the resonant interaction in Zhang  _et al._ (2021).
We attribute this to the fact that $g/\beta$ in our model characterizes the
effective time of the transition through the resonance, so both figures
describe essentially the same physics: it takes initially certain critical
time for the process to evolve without visible effects, after which a
macroscopic number of molecules dissociates quickly due to superradiance Sun
_et al._ (2019), which can occur before entering stationary equilibrium phases
Radzihovsky (2004); Romans  _et al._ (2004); Santos  _et al._ (2006); Duncan
_et al._ (2007); Santos  _et al._ (2010).

Coherent thermalization. Consider now the effect of the dispersion
$\varepsilon_{k}$ of atomic modes in the general model (Coherent reaction
between molecular and atomic Bose-Einstein condensates: integrable model). For
the forward-sweep, the integrability means that only higher energy atomic
pairs can influence the lower energy ones. Hence, if the initial state is the
atomic ground state then the higher energy states remain empty at the end, and
Eq. (10) applies to this case as well.

For the reverse process, if the initial state has $N$ molecules, then Eq. (11)
applies to the first encountered resonance. It also applies to the following
resonances but the number of entering molecules must be reduced by the amount
that has already dissociated. If there are many resonances, all molecules will
dissociate.

The final multi-mode population distribution has a very simple structure. To
show this, we write the joint transition probability to produce $m_{1}$ atomic
pairs in the first and $m_{2}$ pairs in the second resonance:

$P_{m_{1},m_{2}}=P_{m_{1}}x^{N-{m_{1}}-m_{2}}(x^{N-m_{1}-m_{2}+1},x)_{m_{2}},$
(15)

and compare two probabilities of the populations that are different by moving
one atomic pair from the lower energy mode to the higher energy one. Taking
the ratio of such probabilities, we find

$\frac{P_{m_{1}-1,m_{2}+1}}{P_{m_{1},m_{2}}}=e^{-2\pi g^{2}/\beta},$ (16)

which does not depend on $m_{1}$ and $m_{2}$. The same is true for any pair of
the nearest energy atomic modes. This means that all probabilities satisfy the
detailed balance conditions that are found in the Gibbs distribution:

$P_{\\{m_{k}\\}}=\frac{1}{Z}e^{-\frac{1}{k_{B}T}\sum_{k=1}^{\infty}km_{k}}\delta\left(N-\sum_{k}m_{k}\right),$
(17)

where $\\{m_{k}\\}$ is the vector of the final atomic $a_{k}b_{k}$-mode
populations, assuming that the mode indices grow according to the ordering of
energies $\varepsilon_{k}$ in (3). Here, $Z$ is a normalization factor, the
delta function follows from the particle conservation, and

$k_{B}T\equiv\beta/(2\pi g^{2}).$ (18)

The experiment Zhang  _et al._ (2021), indeed, found that after the reverse
sweep the atomic energy population was thermalized at elevated temperatures.
Physically, however, (17) would be thermal only for a linear energy
dispersion, $\varepsilon_{k}\sim k$. We attribute the agreement with Zhang
_et al._ (2021) to the fact that in its 2D trap geometry the atomic energy
dispersion is expected to be linear because
$A\iint_{0}^{\infty}\frac{dk_{x}dk_{y}}{(2\pi)^{2}}\,\delta(E-k^{2}/(2m_{a}))=A\int_{0}^{\infty}\frac{kdk}{(2\pi)}\delta(E-k^{2}/(2m_{a}))=Am_{a}/(2\pi),$
where $m_{a}$ is the atomic mass, and $A$ is the area of the trap. Hence, we
predict that the physical temperature is $k_{B}T=A\beta m_{a}/(2\pi g)^{2}$.
The physical coupling $g$ decays with $A$, so only the linear dependence on
$\beta$ is our testable prediction. The dynamic phase transition in Fig. 1(d)
now has a new interpretation: the final atomic distribution coincides with the
equilibrium one for free bosons, which form a BEC below a critical
temperature.

Figure 2: (a) The ratio $|\langle A|\hat{a}^{2}|A\rangle|/N$ oscillates for
changing $\beta$ and $N$. (b) The overlap of $|A\rangle$ with Glauber states
$|\alpha\rangle$, $\alpha=\alpha_{x}+i\alpha_{y}$, reveals two peaks. Here,
the parameters for $|A\rangle$ are: $\phi_{M}=0$,
$|\alpha_{m}|=g^{2}/\beta=5$.

Phase coherence. The Gibbs distribution (17) does not mean that the system
after the reaction is truly thermalized because it is still described by a
coherent state vector. This can be revealed by measuring the relative phases
of the final states, which can also be found exactly. In sup , we show how
they can be derived but we postpone the detailed analysis to a follow-up work,
and consider here only the adiabatic limit. Although it is experimentally
unreachable, our goal here is to show that the coherent behavior in the model
is, indeed, rich.

Let the molecular condensate be in a coherent state:

$|\alpha_{m}\rangle=e^{-|\alpha_{m}|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha_{m}^{n}}{\sqrt{n!}}|n\rangle.$
(19)

The condensate phase $\phi_{M}$ is then defined from
$\langle\alpha_{m}|\hat{\Psi}|\alpha_{m}\rangle\sim\sqrt{N}e^{i{\rm
arg}(\alpha_{m})}$, where $N=|\alpha|^{2}$, so $\phi_{M}={\rm
arg}(\alpha_{m})$. Consider the adiabatic dissociation of this condensate into
a single atomic mode: $H=-\beta
t\hat{\Psi}^{\dagger}\hat{\Psi}+g[\hat{\Psi}^{\dagger}\hat{a}^{2}+\hat{\Psi}(\hat{a}^{\dagger})^{2}]$.
Each $|n\rangle$ in (19) is then converted to $e^{i\phi_{n}}|2n\rangle_{a}$,
where $a$ marks the atomic states. In sup  we derive an exact formula for the
scattering phase of the complete dissociation amplitude:

$\phi_{n}=3n\pi/4-\sum_{k=1}^{n}arg\Gamma[i(g^{2}/\beta)(k+1)].$ (20)

For $g^{2}/\beta\gg 1$ it is simplified because, for $s\gg 1$,
$\pi/4+arg\Gamma[is]\approx s({\rm log}_{e}s-1)$. As $t\rightarrow+\infty$,
the atomic state becomes

$|A\rangle=e^{-|\alpha_{m}|^{2}/2}\sum_{n=0}^{\infty}\frac{(e^{3i\pi/4}\alpha_{m})^{n}}{\sqrt{n!}}e^{i\phi_{n}}|2n\rangle_{a}.$
(21)

This state contains information about the original molecular state. Thus,
imagine that the molecular condensate has a vortex, such that $\phi_{M}$ winds
by $2\pi$ around some spatial point. This topological property is preserved
after the reaction because the circulation

$-i\int_{0}^{2\pi}\langle A|\frac{d}{d\phi_{M}}|A\rangle\,d\phi_{M}=2\pi N$
(22)

remains the same as it would be for the initial state $|\alpha_{m}\rangle$.
Hence, the stimulated reaction should preserve vortex-like spatial
distributions of particles and currents.

The state $|A\rangle$ is far from a coherent state because $\langle
A|\hat{a}|A\rangle=0$. To test whether it can be close to a squeezed state, in
Fig. 2(a) we plot a numerically calculated ratio $|\langle
A|\hat{a}^{2}|A\rangle|/N$ as a function of $g^{2}/\beta$. Its values close to
$1$ at large $N$ indicate the emergence of the squeezed state. We find that
this ratio is generally small except at narrow resonant values of width $\sim
1/\sqrt{N}$ near $g^{2}{\rm log}_{e}(g^{2}N/\beta)/\beta=2\pi n$,
$n=1,2\ldots$, where it exceeds $0.5$. Figure 2(b) shows the overlap of a
typical $|A\rangle$ with coherent states $|\alpha\rangle$, revealing two peaks
with opposite signs of the corresponding $\alpha$; this makes $|A\rangle$ akin
to a macroscopic cat-state. As $\phi_{M}$ changes from $0$ to $2\pi$, the axis
connecting the two peaks rotates so that the peaks exchange their positions at
the end. This is a topological consequence of the geometric phase (22).

In conclusion, our solution provides a nonperturbative description of the
reaction efficiency and quantum correlations for stimulated BEC reactions. It
predicts a dynamic phase transition and thermalization of the atomic energy
distribution, which were observed experimentally in Zhang  _et al._ (2021).
The emerging state after the reaction has considerable nonclassical
correlations with properties that should be of interest for quantum sensing
and topological quantum computing.

## Acknowledgements

The authors thank Avadh Saxena and Junyu Lin for useful discussions. This work
was supported (R.K.M, V.Y.C, N.A.S.) by the U.S. Department of Energy, Office
of Science, Basic Energy Sciences, Materials Sciences and Engineering
Division, Condensed Matter Theory Program. The completion of this work was
also supported (V.Y.C and N.A.S.) by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences, under Award Number DE-SC0022134. R.K.M. was
partly supported by LANL Center for Nonlinear Studies. C.S. contributed to the
discussion that lead to the idea of this article. He was supported by NSFC
(No. 12105094) and by the Fundamental Research Funds for the Central
Universities from China.

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# Mixing and CP-violation studies in charm decays at LHCb

On behalf of the LHCb collaboration
The University of Oxford
E-mail

###### Abstract:

Studies of charm physics with the 2010 LHCb data sample are presented. Time-
integrated searches for CP violation in $D^{+}\to K^{-}K^{+}\pi^{+}$ and
$D^{0}\to K^{-}K^{+},~{}\pi^{-}\pi^{+}$ are discussed.

##  1 Introduction

The charm sector is a promising place to probe for new physics effects. Mixing
is now well-established [1] at a level which is consistent with but at the
upper end of Standard Model (SM) expectations [2]. Three types of $C\\!P$
violation (CPV) are possible: in the decay amplitudes, in the mixing between
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, and in the
interference between mixing and decay. The first is referred to as direct CPV,
and the second and third as indirect CPV. Only direct CPV is possible in
$D^{+}$ decays, due to the absence of mixing. In the SM indirect $C\\!P$
violation is expected to be small and direct $C\\!P$ violation in singly-
Cabibbo-suppressed modes such as those discussed below is naively expected to
be $\mathcal{O}(10^{-3})$ or less [3], though larger values cannot be excluded
from first principles [4]. In the presence of new physics the rate of $C\\!P$
violation could plausibly be enhanced to $\mathcal{O}(10^{-2})$. At the time
of the conference no evidence for CPV in charm had yet been found, though
first indications have since emerged in the 2011 LHCb data [5].

##  2 Search for CPV in $D^{+}\to K^{-}K^{+}\pi^{+}$

Direct $C\\!P$ violation arises when two different amplitudes with non-zero
relative weak and strong phases contribute to decays to the same final state.
In two-body decays this must imply contributions from different Feynmann
diagrams, such as from tree and penguin processes. In multi-body decays the
same mechanism exists, but in addition a rich variety of intermediate resonant
states can contribute to the decay, each naturally producing a different
strong phase with well-defined variation across the Dalitz plane. Thus, the
interference between these amplitudes can give rise to observable asymmetries
which change across the Dalitz plane.

We search for such asymmetries at LHCb [6] by comparing the Dalitz plot
distributions of $D^{+}\to K^{-}K^{+}\pi^{+}$ and its conjugate process
$D^{-}\to K^{+}K^{-}\pi^{-}$ (Fig. 1), applying a model-independent technique
of comparing the binned, normalized distributions. Normalizing the two Dalitz
plots to the same total number of events cancels any production asymmetry and
suppresses many systematic effects that are mainly expressed as an overall
efficiency asymmetry. The statistical technique used to test for consistency
between the $D^{+}$ and $D^{-}$ Dalitz plots, and to localize the asymmetry if
one is found, is based on the Miranda approach (see Ref. [7] and also Ref.
[8]). A variety of different binnings are used in order to test for different
manifestations of $CP$ violation.

Figure 1:  Mass spectra and Dalitz plot. The mass spectra after selection for
(a) $K^{-}\pi^{+}\pi^{+}$ and (b) $K^{-}K^{+}\pi^{+}$ are shown, with the
signal and sideband mass windows indicated. For those candidates in the
$D^{+}\to K^{-}K^{+}\pi^{+}$ signal window, the Dalitz plot is shown on the
right.

Control modes are analysed to validate the method. The main tool is the
Cabibbo-favoured $D_{s}^{+}\to K^{-}K^{+}\pi^{+}$ control mode, which has the
same final state as the signal as well as similar kinematics and Dalitz plot
structure. As expected, no evidence of any asymmetry is found in this mode
(e.g. p-value of 34% for 25-bin adaptive binning), nor in the sidebands around
the $D^{+}$ mass window. In addition, the analysis is repeated for the
Cabibbo-favoured $D^{+}\to K^{-}\pi^{+}\pi^{+}$ mode. This is more sensitive
to systematic effects, since (a) the yield is ten times larger than that of
the signal mode, and (b) the kaon imbalance can induce momentum-dependent
detector efficiency asymmetries which would not be present in the signal mode.
Nonetheless, only weak indications of asymmetries are seen (e.g. p-value of
12% for 25-bin adaptive binning). Thus, systematic effects in the more robust
$D^{+}\to K^{-}K^{+}\pi^{+}$ signal mode are negligible. The final, unblinded
results are shown in Table 1: no evidence of $CP$ violation is found in the
2010 data. For further details, see Ref. [9, 5].

Binning | Bins | $\chi^{2}/{\rm ndf}$ |  $p$-value (%)
---|---|---|---
Adaptive I | 25 | 32.0/24 | 12.7
Adaptive II | 106 | 123.4/105 | 10.6
Uniform I | 199 | 191.3/198 | 82.1
Uniform II | 530 | 519.5/529 | 60.5
Table 1: $\chi^{2}/{\rm ndf}$ and $p$-values for consistency with no CPV for
the $D^{+}\to K^{-}K^{+}\pi^{+}$ decay mode with four different binnings.

##  3 Search for CPV in $D^{0}\to K^{-}K^{+},~{}\pi^{-}\pi^{+}$

As discussed in Section 1, both direct and indirect CPV can contributed to the
time-integrated $CP$ asymmetry in these singly Cabibbo suppressed decays to
$CP$-even final states. The indirect $CP$ asymmetry is universal to a very
good approximation [10], although the measured value is affected by the
$D^{0}$ decay time acceptance of the experiment [11]. However, the direct $CP$
asymmetry in general varies between final states, and in the limit of U-spin
symmetry is equal and opposite between $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ [3].
Thus, the difference in time-integrated asymmetry between the two final
states, $\Delta A_{CP}$, is sensitive to direct CPV but has limited
sensitivity to indirect CPV:

$\Delta
A_{CP}=a^{\mathrm{dir}}_{C\\!P}(K^{-}K^{+})-a^{\mathrm{dir}}_{C\\!P}(\pi^{-}\pi^{+})+\frac{\Delta\langle
t\rangle}{\tau}\,a^{\mathrm{ind}}_{C\\!P},$

where $\frac{\Delta\langle t\rangle}{\tau}=0.10\pm 0.01$ is the difference in
normalized time acceptance for the two final states at LHCb,
$a^{\mathrm{dir}}_{C\\!P}(f)$ is the direct $CP$ asymmetry for final state
$f$, and $a^{\mathrm{ind}}_{C\\!P}$ is the indirect $CP$ asymmetry.

The observable $\Delta A_{CP}$ also has the advantage of being highly robust
against systematic effects. The measured (raw) asymmetry between $D^{0}\to f$
and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\to\bar{f}$, where the
initial flavour of the $D$ is established with a $D^{*+}\to
D^{0}\pi_{\mathrm{s}}$ tag, can be written at first order as:

$A_{\mathrm{RAW}}(f)\approx
A_{C\\!P}(f)\,+\,A_{\mathrm{D}}(f)\,+\,A_{\mathrm{D}}(\pi_{\mathrm{s}})\,+\,A_{\mathrm{P}}(D^{*+}),$
(1)

where $A_{CP}$, $A_{\mathrm{D}}$, and $A_{\mathrm{P}}$ are the relevant
physics, detector efficiency, and production asymmetries, respectively. Within
a local kinematic region, $A_{\mathrm{D}}(\pi_{\mathrm{s}})$ and
$A_{\mathrm{P}}(D^{*+})$ are independent of the $D^{0}$ decay mode and thus
cancel in the difference $\Delta A_{CP}$. Further,
$A_{\mathrm{D}}(K^{-}K^{+})$ and $A_{\mathrm{D}}(\pi^{-}\pi^{+})$ are zero by
construction, since the final state is spinless and self-conjugate. Thus, all
detector and production effects cancel in $\Delta A_{CP}$ at first order. To
ensure good behaviour at second order, the data are divided into 12 disjoint
kinematic bins, as well as being partitioned according to trigger conditions
and magnetic field polarity. Taking the weighted average of the individual
measurements, we obtain $\Delta A_{CP}=(-0.28\pm 0.70\pm 0.25)\%$, where the
first uncertainty is statistical and the second is systematic (taking into
account modeling of the lineshapes [0.06%], the $D^{0}$ mass window [0.20%],
multiple candidates [0.13%], and the kinematic binning [0.01%]). For further
details, see Ref. [12].

##  4 Conclusions and prospects

LHCb’s charm physics programme is off to a strong start. Several proof-of-
concept measurements have been made on the 2010 data sample of 38
$\mbox{\,pb}^{-1}$, and the first results on the much larger 2011 and 2012
data sets are now forthcoming.

## References

  * [1] Heavy Flavor Averaging Group, D. Asner et al., Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton Properties, arXiv:1010.1589.
  * [2] A. F. Falk, Y. Grossman, Z. Ligeti, Y. Nir and A. A. Petrov, The $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mass difference from a dispersion relation, Phys. Rev. D 69 (2004) 114021, [hep-ph/0402204].
  * [3] See e.g. S. Bianco, F. L. Fabbri, D. Benson and I. Bigi, A Cicerone for the physics of charm, Riv. Nuovo Cim. 26N7 (2003) 1 [hep-ex/0309021]; M. Bobrowski, A. Lenz, J. Riedl and J. Rohrwild, How large can the SM contribution to $CP$ violation in $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing be?, JHEP 1003 (2010) 009 [arXiv:1002.4794]; Y. Grossman, A. L. Kagan and Y. Nir, New physics and $CP$ violation in singly Cabibbo suppressed $D$ decays, Phys. Rev. D 75 (2007) 036008 [hep-ph/0609178].
  * [4] J. Brod, A. L. Kagan and J. Zupan, On the size of direct $CP$ violation in singly Cabibbo-suppressed $D$ decays, arXiv:1111.5000.
  * [5] LHCb Collaboration, R. Aaij et al., Evidence for $CP$ violation in time-integrated $D^{0}\to h^{-}h^{+}$ decay rates, arXiv:1112.0938 (submitted to Phys. Rev. Lett.).
  * [6] LHCb Collaboration, A. A. Alves, Jr. et al., The LHCb Detector at the LHC, JINST 3 (2008) S08005.
  * [7] I. Bediaga, I. I. Bigi, A. Gomes, G. Guerrer, J. Miranda and A. C. d. Reis, On a $CP$ anisotropy measurement in the Dalitz plot, Phys. Rev. D 80 (2009) 096006 [arXiv:0905.4233].
  * [8] BABAR Collaboration, B. Aubert et al., Search for $CP$ Violation in Neutral $D$ Meson Cabibbo-suppressed Three-body Decays, Phys. Rev. D 78 (2008) 051102 [arXiv:0802.4035].
  * [9] LHCb Collaboration, R. Aaij et al., Search for CP violation in $D^{+}\to K^{-}K^{+}\pi^{+}$ decays, arXiv:1110.3970 (accepted by Phys. Rev. D).
  * [10] A. L. Kagan and M. D. Sokoloff, On Indirect $CP$ Violation and Implications for $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $B^{0}_{s}$-$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing, Phys. Rev. D 80 (2009) 076008 [arXiv:0907.3917].
  * [11] CDF Collaboration, T. Aaltonen et al., Measurement of $CP$-violating asymmetries in $D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to K^{+}K^{-}$ decays at CDF, arXiv:1111.5023.
  * [12] LHCb Collaboration, A search for time-integrated $CP$ violation in $D^{0}\rightarrow h^{+}h^{-}$ decays and a measurement of the $D^{0}$ production asymmetry, LHCb-CONF-2011-023.


††thanks: The first author’s research was supported by the Warsaw University
of Technology under grant number 504/04259/1120.

# Mal’tsev products of varieties, I

T. Penza  Faculty of Mathematics and Information Science
Warsaw University of Technology
00-661 Warsaw, Poland<EMAIL_ADDRESS>A.B. Romanowska  Faculty of
Mathematics and Information Science
Warsaw University of Technology
00-661 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract.

We investigate the Mal’tsev product $\mathcal{V}\circ\mathcal{W}$ of two
varieties $\mathcal{V}$ and $\mathcal{W}$ of the same similarity type. Such a
product is usually a quasivariety but not necessarily a variety. We give an
equational base for the variety generated by $\mathcal{V}\circ\mathcal{W}$ in
terms of identities satisfied in $\mathcal{V}$ and $\mathcal{W}$. Then the
main result provides a new sufficient condition for
$\mathcal{V}\circ\mathcal{W}$ to be a variety: If $\mathcal{W}$ is an
idempotent variety and there are terms $f(x,y)$ and $g(x,y)$ such that
$\mathcal{W}$ satisfies the identity $f(x,y)=g(x,y)$ and $\mathcal{V}$
satisfies the identities $f(x,y)=x$ and $g(x,y)=y$, then
$\mathcal{V}\circ\mathcal{W}$ is a variety. We also provide a number of
examples and applications of this result.

###### keywords:

Mal’tsev product of varieties. Identities true in Mal’tsev products.

###### 1991 Mathematics Subject Classification:

08B05, 08C15, 08A30

##  1\. Introduction

In his 1967 paper [14], Mal’tsev introduced a product of classes of algebras
that is now known as the Mal’tsev product. Let
$\tau\colon\Omega\rightarrow\mathbb{N}$ be a similarity type of
$\Omega$-algebras. Let $\mathcal{B}$ and $\mathcal{C}$ be subclasses of a
class $\mathcal{D}$ of $\Omega$-algebras. Then the _Mal’tsev product
$\mathcal{B}\circ_{\mathcal{D}}\mathcal{C}$ of $\mathcal{B}$ and $\mathcal{C}$
relative to $\mathcal{D}$_ consists of all algebras $A$ in $\mathcal{D}$, with
a congruence $\theta$, such that $A/\theta$ belongs to $\mathcal{C}$ and every
congruence class of $\theta$ that is a subalgebra of $A$ belongs to
$\mathcal{B}$. If $\mathcal{D}$ is the variety $\mathcal{T}$ of all
$\Omega$-algebras, then the Mal’tsev product
$\mathcal{B}\circ_{\mathcal{T}}\mathcal{C}$ is called simply the _Mal’tsev
product_ of $\mathcal{B}$ and $\mathcal{C}$, and is denoted by
$\mathcal{B}\circ\mathcal{C}$.

In the same paper, Mal’tsev proved that, if $\mathcal{Q}$ and $\mathcal{R}$
are subquasivarieties of a quasivariety $\mathcal{K}$ of $\Omega$-algebras,
then the Mal’tsev product $\mathcal{Q}\circ_{\mathcal{K}}\mathcal{R}$ is also
a quasivariety, albeit under the assumption of a finite similarity type for
$\mathcal{K}$. If one assumes that the second quasivariety $\mathcal{R}$ is
idempotent, then there is no restriction on the type. This “closure” property
makes the class of subquasivarieties of a given quasivariety a natural domain
for consideration of Mal’tsev products. Moreover, the congruence $\theta$ in
the definition of Mal’tsev product of two quasivarieties may be taken to be
the $\mathcal{R}$-replica congruence $\varrho$ of $A$, the smallest congruence
of $A$ whose induced quotient falls into the quasivariety $\mathcal{R}$. (See
e.g. [20, Ch. 3].) So in this case, the Mal’tsev product may be described
formally as follows:

$\mathcal{Q}\circ_{\mathcal{K}}\mathcal{R}=\\{A\in\mathcal{K}\mid(\forall{a\in
A})\,\,(a/\varrho\leq A\,\Rightarrow\,a/\varrho\in\mathcal{Q})\\}.$ (1.1)

Idempotent elements of $\mathcal{R}$-algebras (i.e. algebras in $\mathcal{R}$)
play a significant role in the theory of Mal’tsev products
$\mathcal{Q}\circ_{\mathcal{K}}\mathcal{R}$ of quasivarieties $\mathcal{Q}$
and $\mathcal{R}$. Recall that an element $a$ of an $\Omega$-algebra $A$ is
_idempotent_ if $\\{a\\}$ is a subalgebra of $A$. An algebra $A$ is idempotent
if every element is idempotent. A class of $\Omega$-algebras is idempotent if
every member algebra is idempotent.

The following easy lemma shows a general relation between subalgebras of an
algebra and idempotents of its quotients.

######  Lemma 1.1.

Let $A$ be an $\Omega$-algebra with congruence $\theta$, and let $a\in A$.
Then $a/\theta$ is a subalgebra of $A$ if and only if $a/\theta$ is an
idempotent element of $A/\theta$.

###### Proof.

If $a/\theta$ is a subalgebra of $A$, then for $a_{1},\dots,a_{n}\in a/\theta$
and each ($n$-ary) $\omega\in\Omega$, we have $a_{1}\dots
a_{n}\omega\,\in\,a/\theta$. Hence $a/\theta\dots a/\theta\,\omega=a\dots
a\omega/\theta=a/\theta$.

Now let $a/\theta$ be an idempotent element of $A/\theta$. This means that
$a/\theta\dots a/\theta\,\omega=a/\theta$. Hence for $a_{1},\dots,a_{n}\in
a/\theta$ and each $n$-ary $\omega\in\Omega$, we have $a_{1}\dots
a_{n}\omega/\theta=a_{1}/\theta\dots a_{n}/\theta\,\omega=a/\theta\dots
a/\theta\,\omega=a/\theta$. Consequently $a/\theta$ is a subalgebra of $A$. ∎

Let us note that if no non-trivial $\mathcal{R}$-algebra has idempotent
elements, then $\mathcal{Q}\circ_{\mathcal{K}}\mathcal{R}$ contains all
$\mathcal{K}$-algebras, since each of them has the $\mathcal{R}$-replica
congruence and no corresponding congruence classes are subalgebras. So in this
case $\mathcal{Q}\circ_{\mathcal{K}}\mathcal{R}$ coincides with the class
$\mathcal{K}$.

Another extreme case concerns the situation when $\mathcal{R}$ is an
idempotent quasivariety. In this case, each $\varrho$-class of an algebra $A$
in $\mathcal{Q}\circ_{\mathcal{K}}\mathcal{R}$ is a subalgebra of $A$, and
then

$\mathcal{Q}\circ_{\mathcal{K}}\mathcal{R}=\\{A\in\mathcal{K}\mid(\forall{a\in
A})\,\,(a/\varrho\in\mathcal{Q})\\}.$ (1.2)

Finally it may happen that, though $\mathcal{R}$ is not an idempotent
quasivariety, all $\mathcal{R}$-algebras have some idempotent elements.
(Examples are provided by inverse semigroups.)

We usually assume that $\Omega$-algebras considered in this paper have no
nullary basic operations. Note however that this assumption is not essential.
If $\Omega$ contains a symbol $c$ of a nullary operation, one can replace it
by a symbol $c(x)$ of a unary operation as follows. If $\mathcal{Q}$ is a
quasivariety of $\Omega$-algebras of a type $\tau$ containing the symbol $c$,
then instead of $\mathcal{Q}$ one can consider the equivalent quasivariety
$\mathcal{Q}^{\prime}$ of the type obtained from $\tau$ by taking the symbol
$c(x)$ in place of $c$, satisfying the quasi-identities true in $\mathcal{Q}$
and additionally the identity $c(x)=c(y)$. Hence, in particular, the unary
operation is constant on all algebras of $\mathcal{Q}^{\prime}$. Sometimes we
have to consider varieties of algebras with at least one non-unary operation
and with no nullary basic operations. Such a similarity type is called
_plural_ [20].

An $\Omega$-term $t$ such that $\mathcal{V}$ satisfies the identities

$t\dots t\,\omega=t,$ (1.3)

for each $\omega\in\Omega$, will be called a _term idempotent_ of
$\mathcal{V}$.

For varieties of algebras the following version of [5, Thm. 9] holds.

######  Lemma 1.2.

Let $\mathcal{V}$ be a variety of $\Omega$-algebras of a type without nullary
operation symbols. Then each algebra in $\mathcal{V}$ has idempotents if and
only if the variety $\mathcal{V}$ has a term idempotent. Moreover, if
$\mathcal{V}$ is of a plural type and has a term idempotent, then it has term
idempotents of any arity $n\geq 1$.

###### Proof.

If all algebras in $\mathcal{V}$ have idempotents, then in particular also the
free algebra $X\Omega$ on one-element set $X$ has an idempotent, which is some
unary term $t$. And as an idempotent, this term satisfies (1.3).

On the other hand, if there is an $n$-ary $\Omega$-term $t$ such that
$\mathcal{V}$ satisfies (1.3), then for any algebra $A$ in $\mathcal{V}$ and
any $a\in A$, one has

$(a\dots at)\dots(a\dots at)\omega=a\dots at$

for each $\omega\in\Omega$, whence $a\dots at$ is an idempotent of $A$.

Finally note that if $x_{1}\dots x_{n}t$ is an $\Omega$-term satisfying (1.3)
for each $\omega\in\Omega$, then the unary $x\dots xt$ also satisfies it. And
moreover for arbitrary $\Omega$-term $u$ of any arity, we also have

$(u\dots ut)\dots(u\dots ut)\,\omega=u\dots ut$

for each $\omega\in\Omega$. ∎

A special case concerns the situation of so-called polarized varieties. A
variety $\mathcal{V}$ of $\Omega$-algebras is _polarized_ if there is a unary
term idempotent of $\mathcal{V}$ that is constant on every member of
$\mathcal{V}$. This constant is called the _pole_ of the algebra, and the
algebra is called _polarized_. The pole of an algebra is unique (if it
exists), and a congruence class of a polarized algebra is a subalgebra if and
only if it is the congruence class of the pole. A (quasi)variety of
$\Omega$-algebras is _polarized_ if all its members are polarized algebras.
(See Mal’tsev [14].)

######  Corollary 1.3.

Each algebra in $\mathcal{V}$ has precisely one idempotent if and only if the
variety $\mathcal{V}$ is polarized.

In this paper we are interested in Mal’tsev products
$\mathcal{V}\circ\mathcal{W}$ of two varieties $\mathcal{V}$ and $\mathcal{W}$
of the same type $\tau\colon\Omega\rightarrow\mathbb{N}$. Such a product is
usually a quasivariety, however need not be a variety. For some results
providing sufficient conditions for the Mal’tsev product of two varieties to
be a variety see e.g. [7, S. 57, 63], [17], [14], [10] and [11]. They usually
concern Mal’tsev products relative to some special varieties. The best known
may be the following theorem of Mal’tsev.

######  Theorem 1.4.

[14] If $\mathcal{V}$ and $\mathcal{W}$ are subvarieties of a congruence
permutable polarized variety $\mathcal{K}$, then
$\mathcal{V}\circ_{\mathcal{K}}\mathcal{W}$ is a variety.

Such products were considered earlier in the case of groups (see Hanna
Neumann’s book [17] on varieties of groups). The Mal’tsev product of any two
varieties of groups relative to the variety of all groups is a variety.
Sufficient conditions of a kind similar to Theorem 1.4 were then obtained by
A.A. Iskander in [10] and [11], and some versions of Theorem 1.4 by C. Bergman
in [1] and [2].

In particular, almost the same proof as the proof of Mal’tsev’s Theorem 1.4
may be used to show the following.

######  Theorem 1.5.

[2] If $\mathcal{V}$ and $\mathcal{W}$ are idempotent subvarieties of a
congruence permutable variety $\mathcal{K}$, then
$\mathcal{V}\circ_{\mathcal{K}}\mathcal{W}$ is a variety.

In fact, it is sufficient to assume that only the variety $\mathcal{W}$ is
idempotent.

Another special case concerns the Mal’tsev product of a strongly irregular
variety $\mathcal{V}$ of any plural type $\tau$ and the variety of
semilattices considered as the variety of the same type $\tau$. Recall that a
strongly irregular variety is a variety $\mathcal{V}_{t}$ defined by a set of
regular identities and a (strongly irregular) identity $t(x,y)=x$, where
$t(x,y)$ is a binary term containing both variables $x$ and $y$. The variety
$\mathcal{S}$ of semilattices may be considered as the variety
$\mathcal{S}_{\tau}$ of any plural type $\tau$ (definitionally) equivalent to
$\mathcal{S}$. (See [4] for details.)

######  Theorem 1.6.

[4] If $\mathcal{V}_{t}$ is a strongly irregular variety of a plural type
$\tau$ and $\mathcal{S}_{\tau}$ is the variety of the same type $\tau$
equivalent to the variety of semilattices, then
$\mathcal{V}_{t}\circ\mathcal{S}_{\tau}$ is a variety.

Also in [4], it was shown how to derive an equational base for the product
$\mathcal{V}_{t}\circ\mathcal{S}_{\tau}$ from equational bases for
$\mathcal{V}_{t}$ and $\mathcal{S}_{\tau}$, and even more generally, how to
derive all identities true in the product
$\mathcal{V}\circ\mathcal{S}_{\tau}$, for any variety $\mathcal{V}$ of
$\Omega$-algebras of a plural type $\tau$.

The aim of this paper is to describe the identities true in the Mal’tsev
product $\mathcal{V}\circ\mathcal{W}$ of two varieties of a type $\tau$
without symbols of nullary operations in terms of the identities satisfied by
its components, and then to extend Theorem 1.6.

The identities satisfied in $\mathcal{V}\circ\mathcal{W}$ are described in
Section 2. They are obtained from the identities true in $\mathcal{V}$ by
substituting for their variables $\mathcal{W}$-equivalent term idempotents of
$\mathcal{W}$. In the case when the Mal’tsev product
$\mathcal{V}\circ\mathcal{W}$ is a variety, one then obtains its equational
base.

Note that if $\mathcal{U}$ is a variety of type $\tau$ and
$\mathcal{V}\circ\mathcal{W}$ is a variety, then
$\mathcal{V}\circ_{\mathcal{U}}\mathcal{W}=(\mathcal{V}\circ\mathcal{W})\cap\mathcal{U}$
is also a variety. However, it is possible that
$\mathcal{V}\circ_{\mathcal{U}}\mathcal{W}$ is a variety but
$\mathcal{V}\circ\mathcal{W}$ is not. It was shown in [4], that the Mal’tsev
product $\mathcal{S}\circ\mathcal{S}$, where $\mathcal{S}$ is the variety of
semilattices, is not a variety. But since
$\mathcal{S}\subseteq\mathcal{S}\circ\mathcal{S}$, it follows that
$\mathcal{S}\circ_{\mathcal{S}}\mathcal{S}$ is the variety $\mathcal{S}$.

Theorem 3.3 of Section 3 provides a new sufficient condition for
$\mathcal{V}\circ\mathcal{W}$ to be a variety. We show that if $\mathcal{W}$
is idempotent and there are terms $f(x,y)$ and $g(x,y)$ such that
$\mathcal{V}$ satisfies the identities $f(x,y)=x$ and $g(x,y)=y$, and
$\mathcal{W}$ satisfies the identity $f(x,y)=g(x,y)$, then
$\mathcal{V}\circ\mathcal{W}$ is a variety. Some consequences of this theorem
are derived. Section 4 discusses the case when the variety $\mathcal{W}$ is
equivalent to a variety of bands.

We use notation and conventions similar to those of [4], [19, 20]. For details
and further information concerning quasivarieties and Mal’tsev product of
quasivarieties we refer the reader to [14] and [15], and then also to [2] and
[20, Ch. 2]; for universal algebra, see [3] and [19], [20].

##  2\. Identities true in Mal’tsev products

In the whole paper we assume that $\mathcal{V}$ and $\mathcal{W}$ are
varieties of the same type $\tau\colon\Omega\rightarrow\mathbb{N}$ without
symbols of nullary operations. Let $\mathrm{Id}(\mathcal{V})$ and
$\mathrm{Id}(\mathcal{W})$ denote the sets of identities true in $\mathcal{V}$
and $\mathcal{W}$, respectively. Members of the absolutely free
$\Omega$-algebra $X\Omega$ over a countably infinite set $X$, i.e.
$\Omega$-terms, are denoted $x_{1}\dots x_{k}u,x_{1}\dots x_{k}v$, etc. or
briefly $\mathbf{x}u,\mathbf{x}v$, etc. or just $u,v$, etc. Note that
$\mathbf{x}u$ may not actually contain a particular variable among
$x_{1},\dots,x_{k}$. We use a similar notation for elements of an
$\Omega$-algebra $A$. If $a_{1},\dots,a_{k}$ are members of $A$, we write
$\mathbf{a}u$, when $u$ is applied to elements $a_{1},\dots,a_{k}$, even if a
particular variable $x_{i}$ does not appear in $u$. If the identity $u=v$
holds in a variety $\mathcal{W}$, then the terms $u$ and $v$ will sometimes be
called $\mathcal{W}$-_equivalent_.

######  Definition 2.1.

Let

$x_{1}\dots x_{k}u=x_{1}\dots x_{k}v$ ($\sigma$)

be an identity true in $\mathcal{V}$. Define

$\displaystyle\sigma^{p}:=$ $\displaystyle\\{r_{1}\dots r_{k}u=r_{1}\dots
r_{k}v\,\mid$ $\displaystyle r_{1},\dots,r_{k}\in X\Omega,$
$\displaystyle\forall\,1\leq i,j\leq k,\,\mathcal{W}\models r_{i}=r_{j},$
$\displaystyle\forall\,\omega\in\Omega,\,\mathcal{W}\models r_{1}\dots
r_{1}\omega=r_{1}\\}.$

For a subset $\Sigma$ of $\mathrm{Id}(\mathcal{V})$, define

$\Sigma^{p}:=\bigcup_{\sigma\in\Sigma}\sigma^{p}.$

Note that the last condition of the definition of $\sigma^{p}$ implies that
$\mathcal{W}\models r_{i}\dots r_{i}\omega=r_{i}$ for all $1\leq i\leq k$ and
$\omega\in\Omega$. If $\mathcal{W}$ is an idempotent variety, then this
condition is always satisfied.

In other words, we may say that the identities $\sigma^{p}$ are obtained from
$\sigma$ by substituting for variables of $\sigma$, pairwise
$\mathcal{W}$-equivalent term idempotents of $\mathcal{W}$. However, by Lemma
1.2, such terms can exist only if all $\mathcal{W}$-algebras contain
idempotent elements. Otherwise the set $\Sigma^{p}$ is empty.

######  Lemma 2.2.

Let $\Sigma$ be any set of identities true in $\mathcal{V}$. Then the Mal’tsev
product $\mathcal{V}\circ\mathcal{W}$ satisfies the identities $\Sigma^{p}$.

###### Proof.

First note that if $\mathcal{W}$ has no term idempotents, that means there are
no terms $r$ such that $\mathcal{W}$ satisfies $r\dots r\omega=r$ for all
$\omega\in\Omega$, then the set $\Sigma^{p}$ is empty, and
$\mathcal{V}\circ\mathcal{W}$ satisfies $\Sigma^{p}$ vacuously.

Now we assume that there exist such terms. Let $A$ be in
$\mathcal{V}\circ\mathcal{W}$ with the $\mathcal{W}$-replica congruence
$\varrho$. Let $x_{1}\dots x_{k}u=x_{1}\dots x_{k}v$ be an identity of
$\Sigma$, and let $y_{1}\dots y_{n}r_{1},\,\dots,\,y_{1}\dots y_{n}r_{k}$ be
pairwise $\mathcal{W}$-equivalent term idempotents of $\mathcal{W}$. Since
$A/\varrho\in\mathcal{W}$, it follows that for any $\mathbf{a}\in A^{n}$, we
have

$(\mathbf{a}r_{i})/\varrho=(\mathbf{a}r_{j})/\varrho$ (2.1)

for every $1\leq i<j\leq k$. Thus all of the elements $\mathbf{a}r_{i}$, for
$i=1,\dots,k$, lie in the same congruence class of $\varrho$.

Since all $r_{i}$ are term idempotents of $\mathcal{W}$, it follows that

$(\mathbf{a}r_{1})/\varrho\dots(\mathbf{a}r_{1})/\varrho\,\,\omega=(\mathbf{a}r_{1})/\varrho$

for each $\omega\in\Omega$, that means $(\mathbf{a}r_{1})/\varrho$ is an
idempotent of $A/\varrho$. Then by Lemma 1.1, $(\mathbf{a}r_{1})/\varrho$ is a
subalgebra of $A$. Consequently, since $A\in\mathcal{V}\circ\mathcal{W}$, we
have $(\mathbf{a}r_{1})/\varrho\in\mathcal{V}$. Since
$\mathbf{a}r_{1},\dots,\mathbf{a}r_{k}\in(\mathbf{a}r_{1})/\varrho$, it
follows that
$\mathbf{a}r_{1}\dots\mathbf{a}r_{k}u=\mathbf{a}r_{1}\dots\mathbf{a}r_{k}v$.
Thus the identity

$r_{1}\dots r_{k}u=r_{1}\dots r_{k}v$

is satisfied in $A$. ∎

Let us note that Lemma 2.2 holds for any equational base $\Sigma$ of
$\mathcal{V}$. In particular, it holds for $\Sigma=\mathrm{Id}(\mathcal{V})$.

######  Corollary 2.3.

For any equational base $\Sigma$ of $\mathcal{V}$, the Mal’tsev product
$\mathcal{V}\circ\mathcal{W}$ is contained in the variety $\mathcal{U}$
defined by $\Sigma^{p}$:

$\mathcal{V}\circ\mathcal{W}\subseteq\mathcal{U}.$ (2.2)

The following simple examples illustrate how the set $\Sigma^{p}$ is built
from the set $\Sigma$.

######  Example 2.4.

Let $\mathcal{V}$ be the variety $\mathcal{LZ}$ of left-zero semigroups
defined by the identity $x\cdot y=x$, and let $\mathcal{W}$ be the variety
$\mathcal{S}$ of semilattices. It is known that $r=s$ is an identity true in
$\mathcal{S}$ precisely if the terms $r$ and $s$ have the same sets of
variables. As the set $\Sigma$ we take the one-element set $\\{\sigma\\}$,
where $\sigma$ is the single identity defining $\mathcal{LZ}$. Since
semilattices are idempotent, the set $\Sigma^{p}$ consists of all identities
of the form

$x_{1}\dots x_{m}\,r\cdot x_{1}\dots x_{m}\,s=x_{1}\dots x_{m}\,r,$

where both $r$ and $s$ contain precisely the same variables
$x_{1},\dots,x_{m}$ and $m=1,2,\dots$. Note that $\Sigma^{p}$ contains the
idempotent law $x\cdot x=x$, but neither the commutative law nor the
associative law are contained in $\Sigma^{p}$. By Lemma 2.2, the Mal’tsev
product $\mathcal{LZ}\circ\mathcal{S}$ satisfies the identities $\Sigma^{p}$.
It follows by results of [4] that in fact these identities define
$\mathcal{LZ}\circ\mathcal{S}$.

######  Example 2.5.

Let $\mathcal{V}$ be the variety $\mathcal{LZ}$ and let $\mathcal{W}$ be the
variety $\mathcal{CS}$ of constant semigroups defined by the identity $x\cdot
y=z\cdot t$. It is easy to see that $r=s$ is a (non-trivial) identity true in
$\mathcal{CS}$ precisely if none of the terms $r$ and $s$ is a variable, or in
different words, if each is the product of two groupoid terms, i.e.

$r=r_{1}\cdot r_{2}\,\,\mbox{and}\,\,s=s_{1}\cdot s_{2}.$

As the set $\Sigma$ we take the one-element set $\\{\sigma\\}$, where $\sigma$
is the identity defining $\mathcal{LZ}$. To describe $\Sigma^{p}$, note that
for $r=r_{1}\cdot r_{2}$,

$r\cdot r=r_{1}\cdot r_{2}=r.$

Consequently the last condition of the definition of $\sigma^{p}$ is
satisfied. It follows that the set $\Sigma^{p}$ consists of the identities of
the form

$(r_{1}\cdot r_{2})\cdot(s_{1}\cdot s_{2})=r_{1}\cdot r_{2},$

where $r_{1},r_{2},s_{1},s_{2}$ are any groupoid terms. If we define
$c:=x\cdot x$, then $\Sigma^{p}$ may be written as follows.

$\Sigma^{p}=\\{r\cdot s=r\mid\mathcal{CS}\models r=s=c\\}.$

Note that $\Sigma^{p}$ contains neither the idempotent nor the associative,
nor the commutative law. By Lemma 2.2, the Mal’tsev product
$\mathcal{LZ}\circ\mathcal{CS}$ satisfies the identities $\Sigma^{p}$.

######  Example 2.6.

In this example $\mathcal{V}$ will be again the variety $\mathcal{LZ}$ with
$\Sigma$ and $\sigma$ defined as in the previous examples. And $\mathcal{W}$
will be the variety $\mathcal{GP}$ of groups, considered as algebras
$(G,\cdot,^{-1})$ with one binary and one unary operation, where the group
identity $e$ is defined as $x\cdot x^{-1}$. (See e.g. [4].) Left-zero bands
are considered as algebras of the same type, with $x^{-1}:=x$. Let $r$ and $s$
be groupoid terms. Then $r=s$ is an identity satisfied in $\mathcal{GP}$ if
$r$ and $s$ have the same reduced (or canonical) form. (See e.g. [9].) Recall
that each group has only one idempotent, namely the identity $e$. Hence
$\sigma^{p}$ consists of the identities of the form

$r\cdot s=r,$

where $r=s$ are the identities satisfied in $\mathcal{GP}$ such that $r\cdot
r=r$ and $r^{-1}=r$. However, the last two conditions are satisfied precisely
if $r=e$. It follows that

$\Sigma^{p}=\\{r\cdot s=r\mid\mathcal{GP}\models r=s=e\\}.$

By Lemma 2.2, the Mal’tsev product $\mathcal{LZ}\circ\mathcal{GP}$ satisfies
the identities $\Sigma^{p}$.

In the next example, algebras in $\mathcal{W}$ have more than one idempotent.

######  Example 2.7.

In this example $\mathcal{V}$, $\sigma$ and $\Sigma$ are defined as in Example
2.6. The variety $\mathcal{W}$ is the regularisation
$\widetilde{\mathcal{GP}}$ of the variety $\mathcal{GP}$ of groups, that means
it is defined by the regular identities true in $\mathcal{GP}$. (See e.g. [20,
ch. 4] and [4].) This class consists precisely of the Płonka sums of groups
(in other words: strong semilattices of groups or Clifford semigroups), and
forms a subvariety of the variety of inverse semigroups. (We refer the reader
to [18] and [20, Ch. 4] for necessary properties of Płonka sums and to [13]
for properties of inverse semigroups.) It is clear that
$\widetilde{\mathcal{GP}}$ is contained in $\mathcal{GP}\circ\mathcal{S}$. Let
$A=\sum_{i\in I}A_{i}$ be the Płonka sum of groups $A_{i}$. Then each $A_{i}$
contains one idempotent, the identity $e_{i}$ of $A_{i}$. We will show that
two elements $a,b\in A$ belong to one summand of $A$ precisely if
$aa^{-1}=bb^{-1}$ and $a^{-1}a=b^{-1}b$. Note however that in each Clifford
semigroup $x^{-1}x=xx^{-1}$. This product is the identity of the summand. To
show that $a^{-1}a=b^{-1}b$ for elements $a$ and $b$ in one summand of $A$,
recall that $a,b\in A_{i}$ for some $i\in I$ precisely if $ab^{-1}b=a$ and
$ba^{-1}a=b$. If these conditions hold, then
$a^{-1}a=a^{-1}ab^{-1}b=b^{-1}ba^{-1}a=b^{-1}b$, since Clifford semigroups
satisfy the identity $x^{-1}xy^{-1}y=y^{-1}yx^{-1}x$. On the other hand, if
$a^{-1}a=b^{-1}b$, then $ab^{-1}b=aa^{-1}a=a$ and $ba^{-1}a=bb^{-1}b=b$, since
Clifford semigroups satisfy $xx^{-1}x=x$. Hence $a$ and $b$ belong to the same
summand.

The elements of the form $aa^{-1}$ form all idempotents of the semigroup $A$.
Note also the following identities true in all Clifford semigroups

$(xx^{-1})\cdot(xx^{-1})=xx^{-1}\,\,\mbox{and}\,\,(xx^{-1})^{-1}=xx^{-1}.$

It follows that $\sigma^{p}$ consists of the identities of the form

$r\cdot s=r,$

where $r=s$ are identities satisfied in $\widetilde{\mathcal{GP}}$ such that
$r\cdot r=r$ and $r^{-1}=r$. Identities satisfied in
$\widetilde{\mathcal{GP}}$ are described by Płonka’s Theorem (see [20, Ch. 4]
and [4]), whereas the last two conditions hold precisely if
$r=rr^{-1}=ss^{-1}=s$. Hence

$\Sigma^{p}=\\{r\cdot s=r\mid\widetilde{\mathcal{GP}}\models
r=rr^{-1}=ss^{-1}=s\\}.$

By Lemma 2.2, the Mal’tsev product $\mathcal{LZ}\circ\widetilde{\mathcal{GP}}$
satisfies the identities $\Sigma^{p}$.

######  Example 2.8.

Consider the variety $\mathcal{MU}$ of monounary algebras $(A,u)$. Recall that
all terms of monounary type have the form $u^{m}(x)$ for $m\in\mathbb{N}$.
There are two types of proper subvarieties of $\mathcal{MU}$: the varieties
$\mathcal{U}_{k}$, for $k\in\mathbb{N}$, each defined by the identity
$u^{k}(x)=u^{k}(y)$, and the varieties $\mathcal{U}_{n,k}$, each defined by
the identity $u^{n+k}(x)=u^{k}(x)$, for $k\in\mathbb{N}$ and
$n\in\mathbb{Z}^{+}$. (See e.g. [12, S. 9.1].) Note that an idempotent unary
operation of an algebra $A$ is just the identity mapping on $A$. The only
idempotent subvariety of $\mathcal{MU}$ is the variety $\mathcal{U}_{1,0}$
defined by the identity $u(x)=x$. A term idempotent $u^{m}(x)$ of a subvariety
of $\mathcal{MU}$ must satisfy $u^{1+m}(x)=u^{m}(x)$. The only subvarieties of
$\mathcal{MU}$ having term idempotents are the varieties $\mathcal{U}_{k}$ and
$\mathcal{U}_{1,k}$ for $k\in\mathbb{N}$. Term idempotents have the form
$u^{k+l}(x)$ for $k,l\in\mathbb{N}$. (See [12] again.) Now let $v=w$ be the
unique identity defining a proper subvariety $\mathcal{V}$ of $\mathcal{MU}$.
As the set $\Sigma$ we take the one element set consisting of this identity.
Let $\mathcal{W}$ be any of the varieties $\mathcal{U}_{k}$ or
$\mathcal{U}_{1,k}$. Then the identities of $\Sigma^{p}$ are obtained from
$v=w$ by substituting for variables in $v=w$, $\mathcal{W}$-equivalent term
idempotents of $\mathcal{W}$. Finally note that the varieties
$\mathcal{U}_{n,k}$ for $n>1$ have no term idempotents, so algebras of these
varieties have no idempotent elements, hence the set $\Sigma^{p}$ is empty. In
this case the variety $\mathsf{H}(\mathcal{V}\circ\mathcal{U}_{n,k})$
generated by the Mal’tsev product $\mathcal{V}\circ\mathcal{U}_{n,k}$
coincides with $\mathcal{MU}$.

If $\mathcal{U}$ is the variety defined by $\Sigma^{p}$ for any equational
base $\Sigma$ of $\mathcal{V}$, then

$\mathcal{W}\subseteq\mathcal{V}\circ\mathcal{W}\subseteq\mathcal{U}.$

By basic properties of free algebras in quasivarieties (see e.g. [20, §3.3])
it is known that, if $\varrho$ is the $\mathcal{W}$-replica congruence of the
free $\mathcal{U}$-algebra $X\mathcal{U}$ over a set $X$, then

$X\mathcal{U}/\varrho\,\cong\,X\mathcal{W},$

where $X\mathcal{W}$ is the free $\mathcal{W}$-algebra over $X$.

######  Proposition 2.9.

Let $\mathcal{U}$ be the variety defined by $\Sigma^{p}$, where $\Sigma$ is
any equational base of $\mathcal{V}$. Let $X\mathcal{U}$ be the free
$\mathcal{U}$-algebra over a set $X$. Then

$X\mathcal{U}\in\mathcal{V}\circ\mathcal{W}.$

###### Proof.

We already know that the $\mathcal{W}$-replica $\varrho$ of $F:=X\mathcal{U}$
is isomorphic to $X\mathcal{W}$. Let $a\in F$ be such that the congruence
class $B:=a/\varrho$ is a subalgebra of $F$. We will show that $B$ belongs to
$\mathcal{V}$.

Let $\sigma$ be an identity $x_{1}\dots x_{k}u=x_{1}\dots x_{k}v$ from
$\Sigma$ and let $r_{1},\dots,r_{k}\in B$. Since $B$ is a single
$\varrho$-class, it follows that $r_{i}/\varrho=r_{j}/\varrho$. As $F/\varrho$
is free in $\mathcal{W}$, we get that $\mathcal{W}$ satisfies the identities
$r_{i}=r_{j}$. Similarly, since $B$ is a subalgebra of $F$, we conclude that
$(r_{1}\dots r_{1}\omega)/\varrho=r_{1}/\varrho$ for each $\omega\in\Omega$,
whence $\mathcal{W}$ satisfies the identities $r_{1}\dots r_{1}\omega=r_{1}$.
Thus the identity $r_{1}\dots r_{k}u=r_{1}\dots r_{k}v$ belongs to
$\sigma^{p}\subseteq\Sigma^{p}$. So it is satisfied in $\mathcal{U}$. In
particular in the free $\mathcal{U}$-algebra $F$ we have the equality

$r_{1}\dots r_{k}u=r_{1}\dots r_{k}v.$

This shows that $B$ satisfies the identities of $\Sigma$, so it belongs to
$\mathcal{V}$. ∎

######  Theorem 2.10.

Let $\mathcal{V}$ and $\mathcal{W}$ be varieties of the same similarity type
and let $\Sigma$ be an equational base for $\mathcal{V}$. The variety
$\mathsf{H}(\mathcal{V}\circ\mathcal{W})$ generated by the Mal’tsev product
$\mathcal{V}\circ\mathcal{W}$ is defined by the identities $\Sigma^{p}$.

###### Proof.

Let $\mathcal{U}$ be the variety defined by $\Sigma^{p}$. By Corollary 2.3, we
know that $\mathsf{H}(\mathcal{V}\circ\mathcal{W})\subseteq\mathcal{U}$, and
by Proposition 2.9, that
$X\mathcal{U}\in\mathcal{V}\circ\mathcal{W}\subseteq\mathsf{H}(\mathcal{V}\circ\mathcal{W})$.
Thus the varieties $\mathsf{H}(\mathcal{V}\circ\mathcal{W})$ and $\mathcal{U}$
share the same free algebras. Since free algebras are uniquely defined and
determine the variety, it follows that
$\mathsf{H}(\mathcal{V}\circ\mathcal{W})=\mathcal{U}$. ∎

######  Corollary 2.11.

If not all $\mathcal{W}$-algebras contain idempotent elements, then
$\mathsf{H}(\mathcal{V}\circ\mathcal{W})$ does not depend on $\mathcal{V}$,
and coincides with the variety $\mathcal{T}$ of all $\Omega$-algebras.

######  Corollary 2.12.

If $\Sigma$ is an equational base for $\mathcal{V}$ and the Mal’tsev product
$\mathcal{V}\circ\mathcal{W}$ is a variety, then $\Sigma^{p}$ is an equational
base for $\mathcal{V}\circ\mathcal{W}$.

Let us recall that if $\mathcal{W}$ is an idempotent variety, then all blocks
of congruences of algebras in $\mathcal{V}\circ\mathcal{W}$ are subalgebras,
and the last condition in the definition of $\sigma^{p}$ is redundant. Note,
as well, that if $\mathcal{W}$ is idempotent, then each algebra $A$ in
$\mathcal{V}\circ\mathcal{W}$ decomposes as a disjoint union of subalgebras
belonging to $\mathcal{V}$ over the $\mathcal{W}$-replica of $A$. Following
the convention adopted in [4] we may call such algebras _$\mathcal{W}$ -sums
of $\mathcal{V}$-algebras_.

For the special case of $\mathcal{W}=\mathcal{S}_{\tau}$, where
$\mathcal{S}_{\tau}$ is the variety equivalent to the variety $\mathcal{S}$ of
semilattices, one obtains Corollary 3.3 of [4]. Recall that two terms are
$\mathcal{S}_{\tau}$-equivalent if and only if they contain exactly the same
variables.

The following example, provided by C. Bergman, shows that, in general, one
cannot replace the set $\mathrm{Id}(\mathcal{W})$ in the definition of the
identities true in $\mathcal{V}\circ\mathcal{W}$, by an equational base for
$\mathcal{W}$.

######  Example 2.13.

Consider the case where $\mathcal{V}=\mathcal{W}=\mathcal{LZ}$, the variety of
left-zero bands. It is defined by the unique identity $x\cdot y=x$. Every
identity of $\mathcal{LZ}$ is of the form $u=v$, where $u$ and $v$ are
groupoid words with the same first variable. By Theorem 2.10, an equational
base for $\mathsf{H}(\mathcal{LZ}\circ\mathcal{LZ})$ consists of all
identities of the form $u\cdot v=u$ again with $u$ and $v$ having the same
first variable. This is the set $\Sigma^{p}$ of Definition 2.1, if we take for
$\Sigma$ the unique identity $x\cdot y=x$ playing the role of $\sigma$.
However, if in the definition of $\sigma^{p}$, we replace
$\mathrm{Id}(\mathcal{LZ})$ by the unique identity defining $\mathcal{LZ}$,
then we will only get the identities:

$\displaystyle x\cdot x=x,$ $\displaystyle(x\cdot y)\cdot x=x\cdot y,$
$\displaystyle x\cdot(x\cdot y)=x,$ $\displaystyle(x\cdot y)\cdot(x\cdot
y)=x\cdot y.$

They are not sufficient to derive all the identities of
$\mathsf{H}(\mathcal{LZ}\circ\mathcal{LZ})$.

##  3\. A sufficient condition for $\mathcal{V}\circ\mathcal{W}$ to be a
variety

Each algebra $A$ of the same similarity type as algebras of a variety
$\mathcal{W}$ has a replica $A/\varrho$ in $\mathcal{W}$. The replica
congruence $\varrho$ is the intersection of all congruences $\theta$ of $A$
with the quotient $A/\theta$ in $\mathcal{W}$. Below we provide another
description useful in the proof of the main theorem of this section.

First, we define a binary relation $\varrho^{0}$ on $A$ as follows:

$\displaystyle(a,b)\in\varrho^{0}\,\mbox{if and only if there are}$
$\displaystyle\mbox{an identity}\,x_{1}\dots x_{n}p=x_{1}\dots x_{n}q\mbox{
true in }\mathcal{W}\,\mbox{and}$ $\displaystyle d_{1},\dots,d_{n}\in A\mbox{
such that }a=d_{1}\dots d_{n}p,\,b=d_{1}\dots d_{n}q.$

######  Lemma 3.1.

Let $\alpha$ be a reflexive and symmetric binary relation on an algebra $A$
preserving the operations of $A$. Then the transitive closure
$\operatorname{tr}(\alpha)$ of $\alpha$ is a congruence relation.

###### Proof.

The relation $\operatorname{tr}(\alpha)$ is obviously an equivalence relation.
So it remains to show that it preserves operations. Let $\omega\in\Omega$ be
an $n$-ary operation and let $a_{1},\dots,a_{n},b_{1},\dots,b_{n}\in A$ be
such that $(a_{i},b_{i})\in\operatorname{tr}(\alpha)$ for $1\leq i\leq n$.
Then for each $i$ there are elements $d_{i}^{1},\dots,d_{i}^{k_{i}}$, such
that

$a_{i}\ =\ d_{i}^{1}\ \alpha\ d_{i}^{2}\ \alpha\ \dots\ \alpha\
d_{i}^{k_{i-1}}\ \alpha\ d_{i}^{k_{i}}\ =\ b_{i}.$

Using reflexivity of $\alpha$ we can extend the sequences
$d_{i}^{1},\dots,d_{i}^{k_{i}}$ to sequences of the same length, say $k$, so
that

$a_{i}\ =\ d_{i}^{1}\ \alpha\ d_{i}^{2}\ \alpha\ \dots\ \alpha\ d_{i}^{k}\ =\
b_{i}.$

Since $\alpha$ preserves operations, we have

$a_{1}\dots a_{n}\omega\ \alpha\ \dots\ \alpha\ d_{j}^{1}\dots
d_{j}^{n}\omega\ \alpha\ d_{j+1}^{1}\dots d_{j+1}^{n}\omega\ \alpha\ \dots\
\alpha\ b_{1}\dots b_{n}\omega.$

Thus $(a_{1}\dots a_{n}\omega,b_{1}\dots
b_{n}\omega)\in\operatorname{tr}(\alpha)$. ∎

######  Proposition 3.2.

Let $\mathcal{W}$ be a variety and $A$ be an algebra of the same similarity
type as algebras in $\mathcal{W}$. The $\mathcal{W}$-replica congruence
$\varrho$ of $A$ coincides with the transitive closure
$\operatorname{tr}(\varrho^{0})$ of $\varrho^{0}$.

###### Proof.

We will first prove that the relation $\operatorname{tr}(\varrho^{0})$ is a
congruence relation. It is obviously reflexive and symmetric. We will show
that it preserves the operations of $\Omega$. Let $\omega\in\Omega$ be an
operation of arity $n$ and let $a_{1},\dots,a_{n},b_{1},\dots,b_{n}\in A$ be
such that $a_{i}\ \varrho^{0}\ b_{i}$ for each $1\leq i\leq n$. In particular,
this means that there are identities $x^{i}_{1}\dots
x^{i}_{k_{i}}p^{i}=x^{i}_{1}\dots x^{i}_{k_{i}}q^{i}$ satisfied in
$\mathcal{W}$, and elements $d^{i}_{1},\dots,d^{i}_{k_{i}}\in A$ such that
$a_{i}=d^{i}_{1}\dots d^{i}_{k_{i}}p^{i}$ and $b_{i}=d^{i}_{1}\dots
d^{i}_{k_{i}}q^{i}$. Then note that the identity

$(x^{1}_{1}\dots x^{1}_{k_{1}}p^{1})\dots(x^{n}_{1}\dots
x^{n}_{k_{n}}p^{n})\,\omega=(x^{1}_{1}\dots
x^{1}_{k_{1}}q^{1})\dots(x^{n}_{1}\dots x^{n}_{k_{n}}q^{n})\,\omega$

is also satisfied in $\mathcal{W}$. As $a_{1}\dots a_{n}\omega=(d^{1}_{1}\dots
d^{1}_{k_{i}}p^{1})\dots(d^{n}_{1}\dots d^{n}_{k_{n}}p^{n})\ \omega$ and
$b_{1}\dots b_{n}\omega=(d^{1}_{1}\dots
d^{1}_{k_{1}}q^{1})\dots(d^{n}_{1}\dots d^{n}_{k_{n}}q^{n})\ \omega$, it
follows that

$a_{1}\dots a_{n}\omega\ \varrho^{0}\ b_{1}\dots b_{n}\omega.$

Now by Lemma 3.1, we conclude that $\operatorname{tr}(\varrho^{0})$ is a
congruence relation.

It remains to show that $A/\operatorname{tr}(\varrho^{0})\in\mathcal{W}$, and
that $\operatorname{tr}(\varrho^{0})$ is the smallest congruence relation with
this property. However, if $x_{1}\dots x_{n}p=x_{1}\dots x_{n}q$ is an
identity true in $\mathcal{W}$, then

$\displaystyle\big{(}a_{1}/\operatorname{tr}(\varrho^{0})\big{)}\dots\big{(}a_{n}/\operatorname{tr}(\varrho^{0})\big{)}\,p=(a_{1}\dots
a_{n}p)/\operatorname{tr}(\varrho^{0})$ $\displaystyle=(a_{1}\dots
a_{n}q)/\operatorname{tr}(\varrho^{0})=\big{(}a_{1}/\operatorname{tr}(\varrho^{0})\big{)}\dots\big{(}a_{n}/\operatorname{tr}(\varrho^{0})\big{)}\,q,$

since, by definition, $(a_{1}\dots a_{n}p,a_{1}\dots
a_{n}q)\,\in\,\varrho^{0}\subseteq\operatorname{tr}(\varrho^{0})$. Thus
$A/\operatorname{tr}(\varrho^{0})$ satisfies every identity true in
$\mathcal{W}$, and hence $A/\operatorname{tr}(\varrho^{0})\in\mathcal{W}$.

On the other hand, if $\theta$ is a congruence of $A$ such that
$A/\theta\in\mathcal{W}$, and $a_{1},\dots,a_{n}\in A$, then
$(a_{1}/\theta)\dots(a_{n}/\theta)\,p=(a_{1}/\theta)\dots(a_{n}/\theta)\,q$,
and hence $a_{1}\dots a_{n}p\ \theta\ a_{1}\dots a_{n}q$. Thus
$\varrho^{0}\subseteq\theta$, and we have
$\operatorname{tr}(\varrho^{0})\subseteq\operatorname{tr}(\theta)=\theta$. ∎

######  Theorem 3.3.

Let $\mathcal{V}$ and $\mathcal{W}$ be varieties of the same similarity type
without nullary operation symbols, and let $\mathcal{W}$ be idempotent. If
there exist terms $f(x,y)$ and $g(x,y)$ such that

  * (a)

$\mathcal{V}\models f(x,y)=x\ \mbox{and}\ \mathcal{V}\models g(x,y)=y$,

  * (b)

$\mathcal{W}\models f(x,y)=g(x,y)$,

then the Mal’tsev product $\mathcal{V}\circ\mathcal{W}$ is a variety.

###### Proof.

Let $A$ be a quotient of an algebra belonging to the Mal’tsev product
$\mathcal{V}\circ\mathcal{W}$, i.e.
$A\in\mathsf{H}(\mathcal{V}\circ\mathcal{W})$. Our aim is to prove that $A$
itself belongs to $\mathcal{V}\circ\mathcal{W}$.

Recall that the algebra $A$ has the $\mathcal{W}$-replica $A/\varrho$, and by
Proposition 3.2, $\varrho=\operatorname{tr}{\varrho^{0}}$. First, we will show
that the relation $\varrho^{0}$ is transitive, which implies that
$\varrho=\varrho^{0}$. Let $a,b,c\in A$, and
$a\,\varrho^{0}\,b\,\varrho^{0}\,c$. This means that there are identities
$x_{1}\dots x_{n}p_{1}=x_{1}\dots x_{n}q_{1}$ and $y_{1}\dots
y_{m}p_{2}=y_{1}\dots y_{m}q_{2}$ satisfied in $\mathcal{W}$, and elements
$d_{1},\dots,d_{n},e_{1},\dots,e_{m}\in A$ such that

$\displaystyle a=d_{1}\dots d_{n}p_{1},\ \ b=d_{1}\dots d_{n}q_{1},$
$\displaystyle b=e_{1}\dots e_{m}p_{2},\ \ c=e_{1}\dots e_{m}q_{2}.$

Now let us consider the identity

$f(x_{1}\dots x_{n}p_{1},y_{1}\dots y_{m}p_{2})=g(x_{1}\dots
x_{n}q_{1},y_{1}\dots y_{m}q_{2}).$ (3.1)

It is clear that (3.1) is satisfied in $\mathcal{W}$. On the other hand, since
$\mathcal{W}$ is idempotent, it follows by Theorem 2.10 that $A$ satisfies the
identities

$f(p_{1},q_{1})=p_{1}\,\,\mbox{and}\,\,g(p_{2},q_{2})=q_{2}.$ (3.2)

Thus

$\begin{split}f(d_{1}\dots d_{n}p_{1},e_{1}\dots e_{m}p_{2})&=f(d_{1}\dots
d_{n}p_{1},b)\\\ &=f(d_{1}\dots d_{n}p_{1},d_{1}\dots d_{n}q_{1})=d_{1}\dots
d_{n}p_{1}=a,\end{split}$

and similarly

$\begin{split}g(d_{1}\dots d_{n}q_{1},e_{1}\dots e_{m}q_{2})&=g(b,e_{1}\dots
e_{m}q_{2})\\\ &=g(e_{1}\dots e_{m}p_{2},e_{1}\dots e_{m}q_{2})=e_{1}\dots
e_{m}q_{2}=c.\end{split}$

Hence, for the elements $d_{1},\dots,d_{n},e_{1},\dots,e_{m}$, the left-hand
side of the identity (3.1) equals $a$ and its right-hand side equals $c$.
Consequently, $a\ \varrho^{0}\ c$, whence $\varrho^{0}$ is transitive, and
$\varrho=\varrho^{0}$.

Next, we will show that each congruence class of $\varrho$ satisfies the
identities true in $\mathcal{V}$. Since $\mathcal{W}$ is idempotent, by Lemma
1.1, all congruence classes of $\varrho$ are subalgebras of $A$. Consider the
identities

$f(x,y)=x\ \mbox{and}\ g(x,y)=y,$ (3.3)

true in the variety $\mathcal{V}$. If $x_{1}\dots x_{n}p=x_{1}\dots x_{n}q$ is
satisfied in $\mathcal{W}$, then by Theorem 2.10, $A$ satisfies the identities

$f(p,q)=p\ \mbox{and}\ g(p,q)=q.$ (3.4)

Now let $a,b\in A$ and $a\ \varrho\ b$. Since $\varrho=\varrho^{0}$, it
follows that $a=d_{1}\dots d_{n}p$ and $b=d_{1}\dots d_{n}q$ for some identity
$x_{1}\dots x_{n}p=x_{1}\dots x_{n}q$ satisfied in $\mathcal{W}$ and some
$d_{1},\dots,d_{n}\in A$. Then

$f(a,b)=f(d_{1}\dots d_{n}p,d_{1}\dots d_{n}q)=d_{1}\dots d_{n}p=a,$

and similarly,

$g(a,b)=g(d_{1}\dots d_{n}p,d_{1}\dots d_{n}q)=d_{1}\dots d_{n}q=b.$

Therefore $B$ satisfies the identities (3.3). We will use this fact to show
that $B$ satisfies any identity $x_{1}\dots x_{k}u=x_{1}\dots x_{k}v$ true in
$\mathcal{V}$. Define terms $t_{1},\dots,t_{k}$ in the following way:

$\displaystyle x_{1}\dots x_{k}t_{1}=f(f(\dots
f(f(f(x_{1},x_{2}),x_{3}),x_{4})\dots),x_{k-1}),x_{k}),$ $\displaystyle
x_{1}\dots x_{k}t_{2}=f(f(\dots
f(f(g(x_{1},x_{2}),x_{3}),x_{4})\dots),x_{k-1}),x_{k}),$ $\displaystyle
x_{1}\dots x_{k}t_{3}=f(f(\dots
f(g(f(x_{1},x_{2}),x_{3}),x_{4})\dots),x_{k-1}),x_{k}),$ $\displaystyle\cdots$
$\displaystyle x_{1}\dots x_{k}t_{k}=g(f(\dots
f(f(f(x_{1},x_{2}),x_{3}),x_{4})\dots),x_{k-1}),x_{k}).$

Since $\mathcal{W}$ satisfies the identity $f(x,y)=g(x,y)$, it also satisfies
the identities $t_{i}=t_{j}$, for every $1\leq i,j\leq k$. By Theorem 2.10, it
follows that $A$ satisfies the identity

$t_{1}\dots t_{k}u=t_{1}\dots t_{k}v.$

Now let $b_{1},\dots,b_{k}$ be elements of $B$. Since the identities (3.3)
hold in $B$, it follows that $b_{1}\dots b_{k}t_{i}=b_{i}$ for every $1\leq
i\leq k$. Thus,

$\begin{split}b_{1}\dots b_{k}u&=(b_{1}\dots b_{k}t_{1})\dots(b_{1}\dots
b_{k}t_{k})u\\\ &=(b_{1}\dots b_{k}t_{1})\dots(b_{1}\dots
b_{k}t_{k})v=b_{1}\dots b_{k}v.\end{split}$

Consequently, $B$ belongs to $\mathcal{V}$. ∎

######  Corollary 3.4.

Let $\mathcal{U}$ be any variety of $\Omega$-algebras. Let $\mathcal{V}$ and
$\mathcal{W}$ be subvarieties of $\mathcal{U}$ satisfying the assumptions of
Theorem 3.3. Then the Mal’tsev product
$\mathcal{V}\circ_{\mathcal{U}}\mathcal{W}$ is a variety.

The main result of [4] (Theorem 1.6) follows easily as a corollary of Theorem
3.3.

######  Corollary 3.5.

[4] Let $\mathcal{V}$ be a strongly irregular variety of a plural type $\tau$
and $\mathcal{S}_{\tau}$ be the variety of type $\tau$ that is equivalent to
the variety of semilattices. Then $\mathcal{V}\circ\mathcal{S}_{\tau}$ is a
variety.

###### Proof.

Let $t(x,y)=x$ be a strongly irregular identity satisfied in $\mathcal{V}$.
Define terms $f(x,y)=t(x,y)$ and $g(x,y)=t(y,x)$. Since $\mathcal{S}_{\tau}$
satisfies all regular identities of type $\tau$, it also satisfies
$f(x,y)=g(x,y)$. As $\mathcal{V}$ satisfies $f(x,y)=x$ and $g(x,y)=y$, and the
variety $\mathcal{S}_{\tau}$ is idempotent, it follows by Theorem 3.3 that
$\mathcal{V}\circ\mathcal{S}_{\tau}$ is a variety. ∎

Let us note that the satisfaction of the three identities of Theorem 3.3
implies that the variety $\mathcal{V}\cap\mathcal{W}$ is trivial. That
suggests the following conjecture: If $\mathcal{V}\cap\mathcal{W}$ is trivial,
then the Mal’tsev product $\mathcal{V}\circ\mathcal{W}$ is a variety. The
following example shows that this conjecture is false. The example was
originally provided by C. Bergman, however with a different proof.

######  Example 3.6.

Let $\mathcal{V}$ be any variety of commutative groupoids, and let
$\mathcal{LZ}$ be the variety of left-zero bands. Then the intersection
$\mathcal{V}\cap\mathcal{LZ}$ is the trivial variety $\mathcal{E}$.

Let $A$ be the groupoid whose multiplication table is given below.

$\cdot$ | 0 | 1 | 2 | 3
---|---|---|---|---
0 | 0 | 0 | 0 | 0
1 | 0 | 1 | 0 | 0
2 | 2 | 2 | 2 | 2
3 | 2 | 3 | 2 | 3

The $\mathcal{LZ}$-replica congruence of $A$ is the congruence $\varrho$ with
the congruence classes $\\{0,1\\}$ and $\\{2,3\\}$ which are semilattices.
Hence $A$ is a member of $\mathcal{V}\circ\mathcal{LZ}$.

The groupoid $A$ has also a congruence $\theta$ with three congruence classes
$\\{0,2\\},\\{1\\},\\{3\\}$. The quotient $B=A/\theta$ has only one proper
non-trivial congruence $\alpha$ with the classes $\\{\\{0,2\\},\\{3\\}\\}$ and
$\\{\\{1\\}\\}$. The groupoid $B$ is neither commutative nor a left-zero band,
and the quotient $B/\alpha$ is not a left-zero-band. Hence $B$ is not a member
of $\mathcal{V}\circ\mathcal{LZ}$.

The following corollaries show that triviality of $\mathcal{V}\cap\mathcal{W}$
combined with some types of congruence permutability is sufficient for the
existence of the two terms required by Theorem 3.3. First recall that two
congruences $\alpha$ and $\beta$ are $3$-_permutable_ (or $3$-_permute_) if

$\alpha\circ\beta\circ\alpha=\beta\circ\alpha\circ\beta.$

An algebra is $3$-permutable if any two of its congruences $3$-permute, and a
variety is $3$-permutable if all of its algebras are $3$-permutable. Note as
well the following.

######  Lemma 3.7.

[16, Lemma 4.66] Let $\alpha$ and $\beta$ be congruences of an algebra $A$.
Then $\alpha\circ\beta\circ\alpha\supseteq\beta\circ\alpha\circ\beta$ if and
only if $\alpha\vee\beta=\alpha\circ\beta\circ\alpha$.

We will also need one more lemma.

######  Lemma 3.8.

Let $\theta,\theta_{1},\theta_{2}$ be congruences of an $\Omega$-algebra $A$
such that $\theta\subseteq\theta_{1},\theta_{2}$. If the quotient $A/\theta$
is congruence $3$-permutable, then the congruences $\theta_{1}$ and
$\theta_{2}$ are $3$-permutable.

###### Proof.

Let $a,b\in A$ and $a\ \theta_{1}\circ\theta_{2}\circ\theta_{1}\ b$. Then

$a/\theta\quad(\theta_{1}/\theta)\circ(\theta_{2}/\theta)\circ(\theta_{1}/\theta)\quad
b/\theta.$

By $3$-permutability of $A/\theta$, we also have

$a/\theta\quad(\theta_{2}/\theta)\circ(\theta_{1}/\theta)\circ(\theta_{2}/\theta)\quad
b/\theta.$

So there are elements $c,d\in A$ such that

$a/\theta\quad(\theta_{2}/\theta)\quad c/\theta\quad(\theta_{1}/\theta)\quad
d/\theta\quad(\theta_{2}/\theta)\quad b/\theta.$

Thus $a\ \theta_{2}\ c\ \theta_{1}\ d\ \theta_{2}\ b$, whence finally

$\theta_{1}\circ\theta_{2}\circ\theta_{1}\subseteq\theta_{2}\circ\theta_{1}\circ\theta_{2}.$

The opposite inclusion is derived analogously. ∎

######  Corollary 3.9.

Let $\mathcal{V}$ and $\mathcal{W}$ be subvarieties of a variety $\mathcal{U}$
of $\Omega$-algebras, and let $\mathcal{W}$ be idempotent. If
$\mathcal{V}\cap\mathcal{W}$ is the trivial variety and $\mathcal{U}$ is
congruence $3$-permutable, then $\mathcal{V}\circ\mathcal{W}$ is a variety.

###### Proof.

Consider the $\mathcal{V}$-replica congruence $\varrho_{\mathcal{V}}$, the
$\mathcal{W}$-replica congruence $\varrho_{\mathcal{W}}$ and the
$\mathcal{U}$-replica congruence $\varrho_{\mathcal{U}}$ of the free algebra
$X\Omega$ for $X=\\{x,y\\}$. Then
$\varrho_{\mathcal{U}}\subseteq\varrho_{\mathcal{V}},\varrho_{\mathcal{W}}$.
By assumption $X\Omega/\varrho_{\mathcal{U}}$ is congruence $3$-permutable.
Hence, by Lemma 3.8, the congruences $\varrho_{\mathcal{V}}$ and
$\varrho_{\mathcal{W}}$ are $3$-permutable. Thus

$\varrho_{\mathcal{V}}\vee\varrho_{\mathcal{W}}=\varrho_{\mathcal{V}}\circ\varrho_{\mathcal{W}}\circ\varrho_{\mathcal{V}}.$

The quotient $X\Omega/(\varrho_{\mathcal{V}}\vee\varrho_{\mathcal{W}})$ lies
in $\mathcal{V}\cap\mathcal{W}$, so it is a trivial algebra. It follows that
all the elements of $X\Omega$ form one congruence class. In particular,

$x\,\varrho_{\mathcal{V}}\circ\varrho_{\mathcal{W}}\circ\varrho_{\mathcal{V}}\,y.$

This means that there are terms $f(x,y)$ and $g(x,y)$ such that

$x\,\,\varrho_{\mathcal{V}}\,\,f(x,y)\,\,\varrho_{\mathcal{W}}\,\,g(x,y)\,\,\varrho_{\mathcal{V}}\,\,y.$

By definition of the congruences $\varrho_{\mathcal{V}}$ and
$\varrho_{\mathcal{W}}$, the variety $\mathcal{W}$ satisfies the identity
$f(x,y)=g(x,y)$ and the variety $\mathcal{V}$ satisfies the identities
$f(x,y)=x$ and $g(x,y)=y$. By Theorem 3.3, $\mathcal{V}\circ\mathcal{W}$ is a
variety. ∎

In the proof of the next corollary, we need Theorem $7.3(2)$ of [6]. We state
it below as a lemma.

######  Lemma 3.10.

Let $A$ be an algebra with a congruence $\theta$. If $P(x,y,z)$ is a term
which is Mal’tsev on each congruence class of $\theta$, then for any other
congruence $\psi$ of $A$,

$\theta\circ\psi\circ\theta\,\subseteq\,\psi\circ\theta\circ\psi.$

######  Corollary 3.11.

Let $\mathcal{V}$ be a congruence permutable variety and $\mathcal{W}$ be an
idempotent variety. If $\mathcal{V}\cap\mathcal{W}$ is the trivial variety,
then $\mathcal{V}\circ\mathcal{W}$ is a variety.

###### Proof.

Let $\mathcal{U}=\mathsf{H}(\mathcal{V}\circ\mathcal{W})$, and let congruences
$\varrho_{\mathcal{V}}$, $\varrho_{\mathcal{W}}$ and $\varrho_{\mathcal{U}}$
of the free algebra $X\Omega$ for $X=\\{x,y\\}$ be defined the same way as in
the proof of Corollary 3.9. By Proposition 2.9 and Theorem 2.10, it follows
that the free $\mathcal{U}$-algebra $X\mathcal{U}$ on $X$ belongs to
$\mathcal{V}\circ\mathcal{W}$. Recall that $X\mathcal{U}$ is isomorphic to
$X\Omega/\varrho_{\mathcal{U}}$. The $\mathcal{W}$-replica of $X\mathcal{U}$
is idempotent, so the congruence classes of the $\mathcal{W}$-replica
congruence $\varrho_{\mathcal{W}}/\varrho_{\mathcal{U}}$ of $X\mathcal{U}$ are
subalgebras of $X\mathcal{U}$ belonging to the congruence permutable variety
$\mathcal{V}$. Thus there exists a term that is Mal’tsev on each congruence
class of $\varrho_{\mathcal{W}}/\varrho_{\mathcal{U}}$. Then by Lemma 3.10,

$\varrho_{\mathcal{W}}/\varrho_{\mathcal{U}}\circ\varrho_{\mathcal{V}}/\varrho_{\mathcal{U}}\circ\varrho_{\mathcal{W}}/\varrho_{\mathcal{U}}\subseteq\varrho_{\mathcal{V}}/\varrho_{\mathcal{U}}\circ\varrho_{\mathcal{W}}/\varrho_{\mathcal{U}}\circ\varrho_{\mathcal{V}}/\varrho_{\mathcal{U}}.$

This implies that

$\varrho_{\mathcal{W}}\circ\varrho_{\mathcal{V}}\circ\varrho_{\mathcal{W}}\subseteq\varrho_{\mathcal{V}}\circ\varrho_{\mathcal{W}}\circ\varrho_{\mathcal{V}}.$

Therefore, by Lemma 3.7,

$\varrho_{\mathcal{V}}\vee\varrho_{\mathcal{W}}=\varrho_{\mathcal{V}}\circ\varrho_{\mathcal{W}}\circ\varrho_{\mathcal{V}}.$

The remaining part of the proof proceeds exactly as in the last part of the
proof of Corollary 3.9. ∎

##  4\. Band sums of algebras

In this section we provide some applications of Theorem 3.3. We first consider
$\Omega$-algebras of a plural type $\tau$ with binary and possibly unary
operations. Recall that a _band_ $(S,\cdot)$ is an idempotent semigroup. (For
properties of bands see [8].) It can be considered as an algebra of the same
type $\tau$ by defining, for each $\omega\in\Omega$, $xy\omega:=x\cdot y$ in
the case $\omega$ is binary, and $x\omega=x$ if $\omega$ is unary.

In all examples below, $f(x,y)$ and $g(x,y)$ denote the terms one needs in
order to apply Theorem 3.3.

######  Example 4.1.

(Lattices and bands). Let $\mathcal{L}$ be any variety of lattices
$(L,+,\cdot)$ and $\mathcal{B}$ any variety of bands considered as algebras
$(B,+,\cdot)$ satisfying $x+y=x\cdot y$. Let

$f(x,y)=x+xy\,\,\mbox{and}\,\,g(x,y)=xy+y.$

Then the variety of all lattices satisfies $f(x,y)=x$ and $g(x,y)=y$, and the
variety of all bands satisfies $f(x,y)=x\cdot xy=xy=xy\cdot y=g(x,y)$. By
Theorem 3.3, it follows that $\mathcal{L}\circ\mathcal{B}$ is a variety.

A small alteration shows that the same holds for the Mal’tsev product of the
variety of Boolean algebras and a variety of bands. This time we consider
Boolean algebras as algebras $(A,+,\cdot,^{\prime})$ with $0$ defined by
$x\cdot x^{\prime}=y\cdot y^{\prime}$ and $1$ defined by
$x+x^{\prime}=y+y^{\prime}$, and bands as algebras of the same type satisfying
$x+y=x\cdot y$ and $x^{\prime}=x$. The same terms $f(x,y)$ and $g(x,y)$ as for
lattices show that the assumptions of Theorem 3.3 are satisfied.

######  Example 4.2.

(Quasigroups and bands). We consider quasigroups as algebras
$(Q,\cdot,/,\setminus)$ with three binary operations, and bands as algebras of
the same type with three equal binary band operations. Let

$f(x,y)=(x\cdot y)/y\,\,\mbox{and}\,\,g(x,y)=x\setminus(x\cdot y).$

Then the varieties of quasigroups satisfy $f(x,y)=x$ and $g(x,y)=y$ and the
varieties of bands satisfy $f(x,y)=g(x,y)$. Theorem 3.3 shows that for any
variety $\mathcal{Q}$ of quasigroups and any variety $\mathcal{B}$ of bands,
the Mal’tsev product $\mathcal{Q}\circ\mathcal{B}$ is a variety.

Again a small alteration shows that the Mal’tsev product of any variety of
loops and any variety of bands is a variety. Loops are considered here as
algebras of the same type as quasigroups with the identity $1$ defined by
$x/x=y\setminus y$.

As quasigroups and loops are congruence permutable, and the intersecion
$\mathcal{Q}\cap\mathcal{B}$ is trivial, the statements of this example may
also be proved using Corollary 3.11.

######  Example 4.3.

(Groups and bands). Groups are loops with associative multiplication. So by
Example 4.2 it follows that the Mal’tsev product of the variety of groups and
the variety of bands is a variety. However, instead of algebras with three
binary operations, groups may be considered as algebras $(G,\cdot,^{-1})$ with
one binary and one unary operation, where the identity $1$ is defined as
$x\cdot x^{-1}$. (See e.g. [4].) Bands are considered as algebras of the same
type. As terms of Theorem 3.3 we take

$f(x,y)=x\cdot y^{-1}\cdot y\,\,\mbox{and}\,\,g(x,y)=x\cdot x^{-1}\cdot y.$

It is easy to see that these terms satisfy the assumptions of Theorem 3.3.

A similar argument shows that if $\mathcal{V}$ is a variety of rings or
modules, and bands are considered as algebras of the same type as
$\mathcal{V}$, then $\mathcal{V}\circ\mathcal{B}$ is a variety.

Examples 4.1–4.3 above may be extended to varieties of any plural type. First
we show that bands may be considered as algebras of any plural type $\tau$.
Let $(A,\cdot)$ be any band. For a given (plural) type $\tau$, define
operations of this type on $A$ as follows. For each $n$-ary $\omega\in\Omega$,
define the operation $\omega$ on $A$ as $x_{1}\dots
x_{n}\omega:=x_{1}\cdot\ldots\cdot x_{n}$. In this way, one obtains an algebra
$(A,\Omega)$ of type $\tau$. The semigroup operation $\cdot$ is recovered via
$x\cdot y=xy\dots y\omega$ for any at least binary $\omega\in\Omega$. It
satisfies the associative law, and each at least binary operation $\omega$ may
be obtained by composition from this binary operation. On the other hand, let
$(A,\Omega)$ be an idempotent algebra of a (plural) type $\tau$ such that for
any two at least binary $\omega$ and $\omega^{\prime}$ in $\Omega$, one has
$xy\dots y\omega=xy\dots y\omega^{\prime}$ and this binary operation is
associative. Then by defining $x\cdot y:=xy\dots y\omega$, one obtains a band
$(A,\cdot)$. It is easy to see that the varieties of bands and $\Omega$-bands
are (definitionally) equivalent. If $(A,\cdot)$ belongs to some special
variety $\mathcal{B}$ of bands, the identities defining $\mathcal{B}$ may be
“translated” into the language of $\Omega$-algebras using the binary operation
$\cdot$ derived from $\Omega$-operations. In this way one obtains the variety
of $\Omega$-bands equivalent to the variety $\mathcal{B}$.

######  Theorem 4.4.

Let $\mathcal{V}$ be a variety of some plural type $\tau$, and let
$\mathcal{B}$ be a variety of $\Omega$-bands of the same type. Let $f(x,y)$
and $g(x,y)$ be terms containing both variables $x$ and $y$, and such that the
first variable of both terms is the same and the last variable of both terms
is the same. If

$\mathcal{V}\models f(x,y)=x\ \mbox{and}\ \mathcal{V}\models g(x,y)=y,$

then $\mathcal{V}\circ\mathcal{B}$ is a variety.

###### Proof.

First recall that the free band on two generators $x$ and $y$ consists of the
following elements: $x,y,xy,yx,xyx,yxy$. Hence if $f$ and $g$ contain both
variables $x$ and $y$ and have the same first and the same last variable, then
the idempotence of bands implies that $f(x,y)=g(x,y)$. Thus the assumptions of
Theorem 3.3 are satisfied. ∎

Let us note that algebras in varieties $\mathcal{V}\circ\mathcal{B}$ of
Theorem 4.4 are band sums of $\mathcal{V}$-algebras. In particular, these
varieties of band sums contain as a subvariety the variety
$\mathcal{V}\circ\mathcal{S}_{\tau}$ of semilattice sums of
$\mathcal{V}$-algebras (provided that $\mathcal{B}$ is not trivial).

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