Mal’tsev products of varieties, I

We investigate the Mal’tsev product V∘W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}\circ \mathcal {W}$$\end{document} of two varieties V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}$$\end{document} and W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} of the same similarity type. While such a product is usually a quasivariety, it is not necessarily a variety. We give an equational base for the variety generated by V∘W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}\circ \mathcal {W}$$\end{document} in terms of identities satisfied in V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}$$\end{document} and W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document}. Then the main result provides a new sufficient condition for V∘W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}\circ \mathcal {W}$$\end{document} to be a variety: If W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} is an idempotent variety and there are terms f(x, y) and g(x, y) such that W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} satisfies the identity f(x,y)=g(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x,y) = g(x,y)$$\end{document} and V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}$$\end{document} satisfies the identities f(x,y)=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x,y) = x$$\end{document} and g(x,y)=y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x,y) = y$$\end{document}, then V∘W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}\circ \mathcal {W}$$\end{document} is a variety. We provide a number of examples and applications of this result.


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 τ : Ω → N be a similarity type of Ω-algebras.Let B and C be subclasses of a class D of Ω-algebras.Then the Mal'tsev product B • D C of B and C relative to D consists of all algebras A in D, with a congruence θ, such that A/θ belongs to C and every congruence class of θ that is a subalgebra of A belongs to B. If D is the variety T of all Ω-algebras, then the Mal'tsev product B • T C is called simply the Mal'tsev product of B and C, and is denoted by B • C.
In the same paper, Mal'tsev proved that, if Q and R are subquasivarieties of a quasivariety K of Ω-algebras, then the Mal'tsev product Q • K R is also a quasivariety, albeit under the assumption of a finite similarity type for K.If one assumes that the second quasivariety 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 θ in the definition of Mal'tsev product of two quasivarieties may be taken to be the R-replica congruence ̺ of A, the smallest congruence of A whose induced quotient falls into the quasivariety R. (See e.g.[20,Ch. 3].)So in this case, the Mal'tsev product may be described formally as follows: (1.1) Idempotent elements of R-algebras (i.e.algebras in R) play a significant role in the theory of Mal'tsev products Q • K R of quasivarieties Q and R.
Recall that an element a of an Ω-algebra A is idempotent if {a} is a subalgebra of A. An algebra A is idempotent if every element is idempotent.A class of Ω-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 Ω-algebra with congruence θ, and let a ∈ A. Then a/θ is a subalgebra of A if and only if a/θ is an idempotent element of A/θ.
Let us note that if no non-trivial R-algebra has idempotent elements, then Q • K R contains all K-algebras, since each of them has the R-replica congruence and no corresponding congruence classes are subalgebras.So in this case Q • K R coincides with the class K.
Another extreme case concerns the situation when R is an idempotent quasivariety.In this case, each ̺-class of an algebra A in Q• K R is a subalgebra of A, and then Finally it may happen that, though R is not an idempotent quasivariety, all R-algebras have some idempotent elements.(Examples are provided by inverse semigroups.) We usually assume that Ω-algebras considered in this paper have no nullary basic operations.Note however that this assumption is not essential.
If Ω contains a symbol c of a nullary operation, one can replace it by a symbol c(x) of a unary operation as follows.If Q is a quasivariety of Ω-algebras of a type τ containing the symbol c, then instead of Q one can consider the equivalent quasivariety Q ′ of the type obtained from τ by taking the symbol c(x) in place of c, satisfying the quasi-identities true in Q and additionally the identity c(x) = c(y).Hence, in particular, the unary operation is constant on all algebras of Q ′ .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 Ω-term t such that V satisfies the identities for each ω ∈ Ω, will be called a term idempotent of V.
For varieties of algebras the following version of [5, Thm.9] holds.
Lemma 1.2.Let V be a variety of Ω-algebras of a type without nullary operation symbols.Then each algebra in V has idempotents if and only if the variety V has a term idempotent.Moreover, if V is of a plural type and has a term idempotent, then it has term idempotents of any arity n ≥ 1.
Proof.If all algebras in V have idempotents, then in particular also the free algebra XΩ 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 Ω-term t such that V satisfies (1.3), then for any algebra A in V and any a ∈ A, one has (a . . .at) . . .(a . . .at)ω = a . . .at for each ω ∈ Ω, whence a . . .at is an idempotent of A.
A special case concerns the situation of so-called polarized varieties.A variety V of Ω-algebras is polarized if there is a unary term idempotent of V that is constant on every member of 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 Ω-algebras is polarized if all its members are polarized algebras.(See Mal'tsev [14].)Corollary 1.3.Each algebra in V has precisely one idempotent if and only if the variety V is polarized.
In this paper we are interested in Mal'tsev products V•W of two varieties V and W of the same type τ : Ω → 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 V and W are subvarieties of a congruence permutable polarized variety K, then V • K 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 V and W are idempotent subvarieties of a congruence permutable variety K, then V • K W is a variety.
In fact, it is sufficient to assume that only the variety W is idempotent.Another special case concerns the Mal'tsev product of a strongly irregular variety V of any plural type τ and the variety of semilattices considered as the variety of the same type τ .Recall that a strongly irregular variety is a variety 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 S of semilattices may be considered as the variety S τ of any plural type τ (definitionally) equivalent to S. (See [4] for details.) Theorem 1.6.[4] If V t is a strongly irregular variety of a plural type τ and S τ is the variety of the same type τ equivalent to the variety of semilattices, then V t • S τ is a variety.
Also in [4], it was shown how to derive an equational base for the product V t • S τ from equational bases for V t and S τ , and even more generally, how to derive all identities true in the product V • S τ , for any variety V of Ω-algebras of a plural type τ .
The aim of this paper is to describe the identities true in the Mal'tsev product V • W of two varieties of a type τ 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 V • W are described in Section 2. They are obtained from the identities true in V by substituting for their variables Wequivalent term idempotents of W. In the case when the Mal'tsev product V • W is a variety, one then obtains its equational base.
Note that if U is a variety of type τ and V •W is a variety, then V • U W = (V • W) ∩ U is also a variety.However, it is possible that V • U W is a variety but V • W is not.It was shown in [4], that the Mal'tsev product S • S, where S is the variety of semilattices, is not a variety.But since S ⊆ S • S, it follows that S • S S is the variety S.
Theorem 3.3 of Section 3 provides a new sufficient condition for V • W to be a variety.We show that if W is idempotent and there are terms f (x, y) and g(x, y) such that V satisfies the identities f (x, y) = x and g(x, y) = y, and W satisfies the identity f (x, y) = g(x, y), then V • W is a variety.Some consequences of this theorem are derived.Section 4 discusses the case when the variety W is equivalent to a variety of bands.

Identities true in Mal'tsev products
In the whole paper we assume that V and W are varieties of the same type τ : Ω → N without symbols of nullary operations.Let Id(V) and Id(W) denote the sets of identities true in V and W, respectively.Members of the absolutely free Ω-algebra XΩ over a countably infinite set X, i.e.Ω-terms, are denoted x 1 . . .x k u, x 1 . . .x k v, etc. or briefly xu, xv, etc. or just u, v, etc.Note that xu may not actually contain a particular variable among x 1 , . . ., x k .We use a similar notation for elements of an Ω-algebra A. If a 1 , . . ., a k are members of A, we write au, when u is applied to elements a 1 , . . ., a k , even if a particular variable x i does not appear in u.If the identity u = v holds in a variety W, then the terms u and v will sometimes be called W-equivalent.
For a subset Σ of Id(V), define Note that the last condition of the definition of σ p implies that W |= r i . . .r i ω = r i for all 1 ≤ i ≤ k and ω ∈ Ω.If W is an idempotent variety, then this condition is always satisfied.
In other words, we may say that the identities σ p are obtained from σ by substituting for variables of σ, pairwise W-equivalent term idempotents of W. However, by Lemma 1.2, such terms can exist only if all W-algebras contain idempotent elements.Otherwise the set Σ p is empty.Lemma 2.2.Let Σ be any set of identities true in V. Then the Mal'tsev product V • W satisfies the identities Σ p .
Proof.First note that if W has no term idempotents, that means there are no terms r such that W satisfies r . . .rω = r for all ω ∈ Ω, then the set Σ p is empty, and V • W satisfies Σ p vacuously.Now we assume that there exist such terms.Let A be in V • W with the W-replica congruence ̺.Let x 1 . . .x k u = x 1 . . .x k v be an identity of Σ, and let y 1 . . .y n r 1 , . . ., y 1 . . .y n r k be pairwise W-equivalent term idempotents of W. Since A/̺ ∈ W, it follows that for any a ∈ A n , we have for every 1 ≤ i < j ≤ k.Thus all of the elements ar i , for i = 1, . . ., k, lie in the same congruence class of ̺.Since all r i are term idempotents of W, it follows that (ar 1 )/̺ . . .(ar 1 )/̺ ω = (ar 1 )/̺ for each ω ∈ Ω, that means (ar 1 )/̺ is an idempotent of A/̺.Then by Lemma 1.1, (ar 1 )/̺ is a subalgebra of A. Consequently, since A ∈ V • W, we have (ar 1 )/̺ ∈ V. Since ar 1 , . . ., ar k ∈ (ar 1 )/̺, it follows that ar 1 . . .ar k u = ar 1 . . .ar k v. Thus the identity Let us note that Lemma 2.2 holds for any equational base Σ of V.In particular, it holds for Σ = Id(V).

Corollary 2.3. For any equational base
The following simple examples illustrate how the set Σ p is built from the set Σ.
Example 2.4.Let V be the variety LZ of left-zero semigroups defined by the identity x • y = x, and let W be the variety S of semilattices.It is known that r = s is an identity true in S precisely if the terms r and s have the same sets of variables.As the set Σ we take the one-element set {σ}, where σ is the single identity defining LZ.Since semilattices are idempotent, the set Σ p consists of all identities of the form where both r and s contain precisely the same variables x 1 , . . ., x m and m = 1, 2, . . . .Note that Σ p contains the idempotent law x • x = x, but neither the commutative law nor the associative law are contained in Σ p .By Lemma 2.2, the Mal'tsev product LZ • S satisfies the identities Σ p .It follows by results of [4] that in fact these identities define LZ • S.
Example 2.5.Let V be the variety LZ and let W be the variety CS of constant semigroups defined by the identity x • y = z • t.It is easy to see that r = s is a (non-trivial) identity true in 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 • r 2 and s = s 1 • s 2 .
As the set Σ we take the one-element set {σ}, where σ is the identity defining LZ.To describe Σ p , note that for r Consequently the last condition of the definition of σ p is satisfied.It follows that the set Σ p consists of the identities of the form where r 1 , r 2 , s 1 , s 2 are any groupoid terms.If we define c := x • x, then Σ p may be written as follows.
Note that Σ p contains neither the idempotent nor the associative, nor the commutative law.By Lemma 2.2, the Mal'tsev product LZ • CS satisfies the identities Σ p .
Example 2.6.In this example V will be again the variety LZ with Σ and σ defined as in the previous examples.And W will be the variety GP of groups, considered as algebras (G, •, −1 ) with one binary and one unary operation, where the group identity e is defined as x • 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 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 σ p consists of the identities of the form r • s = r, where r = s are the identities satisfied in GP such that r • r = r and r −1 = r.However, the last two conditions are satisfied precisely if r = e.It follows that Σ p = {r • s = r | GP |= r = s = e}.By Lemma 2.2, the Mal'tsev product LZ • GP satisfies the identities Σ p .
In the next example, algebras in W have more than one idempotent.
Example 2.7.In this example V, σ and Σ are defined as in Example 2.6.The variety W is the regularisation GP of the variety GP of groups, that means it is defined by the regular identities true in GP. (See e.g.[20, ch. 4] and [4].)This class consists precisely of the P lonka 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 P lonka sums and to [13] for properties of inverse semigroups.)It is clear that GP is contained in GP • S. Let A = i∈I A i be the P lonka 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 ∈ 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 The elements of the form aa −1 form all idempotents of the semigroup A. Note also the following identities true in all Clifford semigroups It follows that σ p consists of the identities of the form r • s = r, where r = s are identities satisfied in GP such that r • r = r and r −1 = r.Identities satisfied in GP are described by P lonka's Theorem (see [20,Ch. 4] and [4]), whereas the last two conditions hold precisely if r = rr By Lemma 2.2, the Mal'tsev product LZ • GP satisfies the identities Σ p .(See [12] again.)Now let v = w be the unique identity defining a proper subvariety V of MU.As the set Σ we take the one element set consisting of this identity.Let W be any of the varieties U k or U 1,k .Then the identities of Σ p are obtained from v = w by substituting for variables in v = w, W-equivalent term idempotents of W. Finally note that the varieties U n,k for n > 1 have no term idempotents, so algebras of these varieties have no idempotent elements, hence the set Σ p is empty.In this case the variety H(V • U n,k ) generated by the Mal'tsev product V • U n,k coincides with MU.
If U is the variety defined by Σ p for any equational base Σ of V, then By basic properties of free algebras in quasivarieties (see e.g.[20, §3.3]) it is known that, if ̺ is the W-replica congruence of the free U-algebra XU over a set X, then where XW is the free W-algebra over X.
Proposition 2.9.Let U be the variety defined by Σ p , where Σ is any equational base of V. Let XU be the free U-algebra over a set X. Then Proof.We already know that the W-replica ̺ of F := XU is isomorphic to XW.Let a ∈ F be such that the congruence class B := a/̺ is a subalgebra of F .We will show that B belongs to V.
Let σ be an identity x 1 . . .x k u = x 1 . . .x k v from Σ and let r 1 , . . ., r k ∈ B. Since B is a single ̺-class, it follows that r i /̺ = r j /̺.As F/̺ is free in W, we get that W satisfies the identities r i = r j .Similarly, since B is a subalgebra of F , we conclude that (r 1 . . .r 1 ω)/̺ = r 1 /̺ for each ω ∈ Ω, whence W satisfies the identities r 1 . . .r 1 ω = r 1 .Thus the identity r 1 . . .r k u = r 1 . . .r k v belongs to σ p ⊆ Σ p .So it is satisfied in U.In particular in the free U-algebra F we have the equality This shows that B satisfies the identities of Σ, so it belongs to V. Theorem 2.10.Let V and W be varieties of the same similarity type and let Σ be an equational base for V.The variety H(V • W) generated by the Mal'tsev product V • W is defined by the identities Σ p .
Proof.Let U be the variety defined by Σ p .By Corollary 2.3, we know that H(V •W) ⊆ U, and by Proposition 2.9, that XU ∈ V •W ⊆ H(V •W).Thus the varieties H(V • W) and U share the same free algebras.Since free algebras are uniquely defined and determine the variety, it follows that H(V • W) = U.
Corollary 2.11.If not all W-algebras contain idempotent elements, then H(V • W) does not depend on V, and coincides with the variety T of all Ω-algebras.
Corollary 2.12.If Σ is an equational base for V and the Mal'tsev product V • W is a variety, then Σ p is an equational base for V • W.
Let us recall that if W is an idempotent variety, then all blocks of congruences of algebras in V • W are subalgebras, and the last condition in the definition of σ p is redundant.Note, as well, that if W is idempotent, then each algebra A in V • W decomposes as a disjoint union of subalgebras belonging to V over the W-replica of A. Following the convention adopted in [4] we may call such algebras W-sums of V-algebras.
For the special case of W = S τ , where S τ is the variety equivalent to the variety S of semilattices, one obtains Corollary 3.3 of [4].Recall that two terms are S τ -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 Id(W) in the definition of the identities true in V • W, by an equational base for W.
Example 2.13.Consider the case where V = W = LZ, the variety of left-zero bands.It is defined by the unique identity x • y = x.Every identity of 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 H(LZ • LZ) consists of all identities of the form u • v = u again with u and v having the same first variable.This is the set Σ p of Definition 2.1, if we take for Σ the unique identity x • y = x playing the role of σ.However, if in the definition of σ p , we replace Id(LZ) by the unique identity defining LZ, then we will only get the identities: They are not sufficient to derive all the identities of H(LZ • LZ).

A sufficient condition for V • W to be a variety
Each algebra A of the same similarity type as algebras of a variety W has a replica A/̺ in W. The replica congruence ̺ is the intersection of all congruences θ of A with the quotient A/θ in W. Below we provide another description useful in the proof of the main theorem of this section.
First, we define a binary relation ̺ 0 on A as follows: (a, b) ∈ ̺ 0 if and only if there are an identity x 1 . . .x n p = x 1 . . .x n q true in W and Using reflexivity of α we can extend the sequences d 1 i , . . ., d ki i to sequences of the same length, say k, so that  Proof.We will first prove that the relation tr(̺ 0 ) is a congruence relation.It is obviously reflexive and symmetric.We will show that it preserves the operations of Ω.Let ω ∈ Ω be an operation of arity n and let a 1 , . . ., a n , b 1 , . . ., b n ∈ A be such that a i ̺ 0 b i for each 1 ≤ i ≤ n.In particular, this means that there are identities x i 1 . . .x i ki p i = x i 1 . . .x i ki q i satisfied in W, and elements Now by Lemma 3.1, we conclude that tr(̺ 0 ) is a congruence relation.
On the other hand, if θ is a congruence of A such that A/θ ∈ W, and a 1 , . . ., a n ∈ A, then (a 1 /θ) . . .(a n /θ) p = (a 1 /θ) . . .(a n /θ) q, and hence a 1 . . .a n p θ a 1 . . .a n q.Thus ̺ 0 ⊆ θ, and we have tr(̺ 0 ) ⊆ tr(θ) = θ.Theorem 3.3.Let V and W be varieties of the same similarity type without nullary operation symbols, and let W be idempotent.If there exist terms f (x, y) and g(x, y) such that Proof.Let A be a quotient of an algebra belonging to the Mal'tsev product V • W, i.e.A ∈ H(V • W).Our aim is to prove that A itself belongs to V • W.
Hence, for the elements d 1 , . . ., d n , e 1 , . . ., e m , the left-hand side of the identity (3.1) equals a and its right-hand side equals c.Consequently, a ̺ 0 c, whence ̺ 0 is transitive, and ̺ = ̺ 0 .Next, we will show that each congruence class of ̺ satisfies the identities true in V. Since W is idempotent, by Lemma 1.1, all congruence classes of ̺ are subalgebras of A. Consider the identities f (x, y) = x and g(x, y) = y, true in the variety V.If x 1 . . .x n p = x 1 . . .x n q is satisfied in W, then by Theorem 2.10, A satisfies the identities f (p, q) = p and g(p, q) = q. (3.4) Now let a, b ∈ A and a ̺ b.Since ̺ = ̺ 0 , it follows that a = d 1 . . .d n p and b = d 1 . . .d n q for some identity x 1 . . .x n p = x 1 . . .x n q satisfied in W and some d 1 , . . ., d n ∈ A. Then and similarly, Therefore B satisfies the identities (3.3).We will use this fact to show that B satisfies any identity x 1 . . .x k u = x 1 . . .x k v true in V. Define terms t 1 , . . ., t k in the following way: Since W satisfies the identity f (x, y) = g(x, y), it also satisfies the identities t i = t j , for every 1 ≤ i, j ≤ k.By Theorem 2.10, it follows that A satisfies the identity Consequently, B belongs to V. Corollary 3.4.Let U be any variety of Ω-algebras.Let V and W be subvarieties of U satisfying the assumptions of Theorem 3.3.Then the Mal'tsev product V • U 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 V be a strongly irregular variety of a plural type τ and S τ be the variety of type τ that is equivalent to the variety of semilattices.Then V • S τ is a variety.
Proof.Let t(x, y) = x be a strongly irregular identity satisfied in V. Define terms f (x, y) = t(x, y) and g(x, y) = t(y, x).Since S τ satisfies all regular identities of type τ , it also satisfies f (x, y) = g(x, y).As V satisfies f (x, y) = x and g(x, y) = y, and the variety S τ is idempotent, it follows by Theorem 3.3 that V • S τ is a variety.
Let us note that the satisfaction of the three identities of Theorem 3.3 implies that the variety V ∩ W is trivial.That suggests the following conjecture: If V ∩ W is trivial, then the Mal'tsev product V • 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 V be any variety of commutative groupoids, and let LZ be the variety of left-zero bands.Then the intersection V ∩ LZ is the trivial variety E.
Let A be the groupoid whose multiplication table is given below.
The following corollaries show that triviality of V ∩ 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 α and β are 3-permutable (or 3-permute) if 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 α and β be congruences of an algebra A.
The opposite inclusion is derived analogously.Corollary 3.9.Let V and W be subvarieties of a variety U of Ω-algebras, and let W be idempotent.If V ∩ W is the trivial variety and U is congruence 3-permutable, then V • W is a variety.
Proof.Consider the V-replica congruence ̺ V , the W-replica congruence ̺ W and the U-replica congruence ̺ U of the free algebra XΩ for X = {x, y}.Then ̺ U ⊆ ̺ V , ̺ W .By assumption XΩ/̺ U is congruence 3-permutable.Hence, by Lemma 3.8, the congruences ̺ V and ̺ W are 3-permutable.Thus The quotient XΩ/(̺ V ∨ ̺ W ) lies in V ∩ W, so it is a trivial algebra.It follows that all the elements of XΩ form one congruence class.In particular, This means that there are terms f (x, y) and g(x, y) such that By definition of the congruences ̺ V and ̺ W , the variety W satisfies the identity f (x, y) = g(x, y) and the variety V satisfies the identities f (x, y) = x and g(x, y) = y.By Theorem 3.3, V • 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 θ.If P (x, y, z) is a term which is Mal'tsev on each congruence class of θ, then for any other congruence Corollary 3.11.Let V be a congruence permutable variety and W be an idempotent variety.If V ∩ W is the trivial variety, then V • W is a variety.
Proof.Let U = H(V • W), and let congruences ̺ V , ̺ W and ̺ U of the free algebra XΩ 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 U-algebra XU on X belongs to V • W. Recall that XU is isomorphic to XΩ/̺ U .The W-replica of XU is idempotent, so the congruence classes of the W-replica congruence ̺ W /̺ U of XU are subalgebras of XU belonging to the congruence permutable variety V. Thus there exists a term that is Mal'tsev on each congruence class of ̺ W /̺ U .Then by Lemma 3.10, Therefore, by Lemma 3.7, The remaining part of the proof proceeds exactly as in the last part of the proof of Corollary 3.9.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, +, •, ′ ) with 0 defined by x • x ′ = y • y ′ and 1 defined by x + x ′ = y + y ′ , and bands as algebras of the same type satisfying x + y = x • y and x ′ = 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, •, /, \) with three binary operations, and bands as algebras of the same type with three equal binary band operations.Let f (x, y) = (x • y)/y and g(x, y) = x \ (x • 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 Q of quasigroups and any variety B of bands, the Mal'tsev product Q • 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 \ y.
As quasigroups and loops are congruence permutable, and the intersecion Q ∩ 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, •, −1 ) with one binary and one unary operation, where the identity 1 is defined as x • 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 • y −1 • y and g(x, y) = x • x −1 • y.
It is easy to see that these terms satisfy the assumptions of Theorem 3.3.
A similar argument shows that if V is a variety of rings or modules, and bands are considered as algebras of the same type as V, then V • B is a variety.Examples 4.1-4.3above may be extended to varieties of any plural type.First we show that bands may be considered as algebras of any plural type τ .Let (A, •) be any band.For a given (plural) type τ , define operations of this type on A as follows.For each n-ary ω ∈ Ω, define the operation ω on A as x 1 . . .x n ω := x 1 • . . .• x n .In this way, one obtains an algebra (A, Ω) of type τ .The semigroup operation • is recovered via x • y = xy . . .yω for any at least binary ω ∈ Ω.It satisfies the associative law, and each at least binary operation ω may be obtained by composition from this binary operation.On the other hand, let (A, Ω) be an idempotent algebra of a (plural) type τ such that for any two at least binary ω and ω ′ in Ω, one has xy . . .yω = xy . . .yω ′ and this binary operation is associative.Then by defining x • y := xy . . .yω, one obtains a band (A, •).It is easy to see that the varieties of bands and Ωbands are (definitionally) equivalent.If (A, •) belongs to some special variety B of bands, identities defining B may be "translated" into the language of Ω-algebras using the binary operation • derived from Ω-operations.In this way one obtains the variety of Ω-bands equivalent to the variety B. Theorem 4.4.Let V be a variety of some plural type τ , and let B be a variety of Ω-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 V |= f (x, y) = x and V |= g(x, y) = y, then V • 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 V • B of Theorem 4.4 are band sums of V-algebras.In particular, these varieties of band sums contain as a subvariety the variety V • S τ of semilattice sums of V-algebras (provided that B is not trivial).
−1 a = b −1 b for elements a and b in one summand of A, recall that a, b ∈ A i for some i ∈ 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.

Example 2 . 8 .
Consider the variety MU of monounary algebras (A, u).Recall that all terms of monounary type have the form u m (x) for m ∈ N.There are two types of proper subvarieties of MU: the varieties U k , for k ∈ N, each defined by the identity u k (x) = u k (y), and the varieties U n,k , each defined by the identity u n+k (x) = u k (x), for k ∈ N and n ∈ 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 MU is the variety U 1,0 defined by the identity u(x) = x.A term idempotent u m (x) of a subvariety of MU must satisfy u 1+m (x) = u m (x).The only subvarieties of MU having term idempotents are the varieties U k and U 1,k for k ∈ N. Term idempotents have the form u k+l (x) for k, l ∈ N.

Proposition 3 . 2 .
Let W be a variety and A be an algebra of the same similarity type as algebras in W. The W-replica congruence ̺ of A coincides with the transitive closure tr(̺ 0 ) of ̺ 0 .

Example 4 . 1 .
(Lattices and bands).Let L be any variety of lattices (L, +, •) and B any variety of bands considered as algebras (B, +, •) satisfying x + y = x • y.Let f (x, y) = x + xy 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 • xy = xy = xy • y = g(x, y).By Theorem 3.3, it follows that L • B is a variety.