On the Complete Join of Permutative Combinatorial Rees-sushkevich Varieties

A semigroup variety is a Rees-Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. The collection of all permutative combinatorial Rees-Sushkevich varieties constitutes an incomplete lattice that does not contain the complete join J of all its varieties. The objective of this article is to investigate the subvarieties of J. It is shown that J is locally finite, non-finitely generated, and contains only finitely based subvarieties. The subvarieties of J are precisely the combinatorial Rees-Sushkevich varieties that do not contain a certain semigroup of order four.


Introduction
Recall that a semigroup is 0-simple if it does not contain any nontrivial proper ideals.The class of 0-simple semigroups is one of the most important classes of semigroups.Indeed, as each finite semigroup can be obtained from finite 0-simple semigroups by a sequence of ideal extensions, the role that finite 0simple semigroups play in semigroup theory is comparable to the role that finite simple groups play in group theory.Naturally, the varieties generated by 0-simple semigroups and their subvarieties deserve special attention.
Following Kublanovsky [4], any subvariety of a periodic variety generated by 0-simple semigroups is said to be a Rees-Sushkevich variety.Investigation of the lattice of Rees-Sushkevich varieties has recently been initiated by Reilly, Volkov, and the author (see [5]- [10], [12]- [14], and [19]).In particular, several aspects of the lattice C of combinatorial Rees-Sushkevich varieties have been considered in [5]- [7], [10], and [19].Recall that a semigroup variety is combinatorial if all groups in it are trivial.
A semigroup variety is permutative if it satisfies some permutation identity.Denote by P the set of all permutative varieties in C. It is easy to show that P is a lattice (Proposition 5).However, the complete join J of all varieties in P is not permutative (Proposition 11) so that P is an incomplete sublattice of the subvariety lattice L(J) of J.The structure of L(J) is quite complex, for it follows from a result of Vernikov and Volkov [17] that every finite lattice is embeddable in L(J).
The objective of the present article is to investigate the variety J and its subvarieties.Specifically, it is shown that the variety J is locally finite, nonfinitely generated, and contains only finitely based subvarieties.Consequently, L(J) is a countably infinite lattice.It is also shown that the subvarieties of J are precisely the varieties in C that do not contain a certain semigroup of order four.The aforementioned properties of J are presented in Section 4.

Background
Let X + and X * respectively be the free semigroup and free monoid over a countably infinite alphabet X. Elements of X are referred to as letters, and elements of X + and X * are referred to as words.
The head and tail of a word u are respectively the first and last letters occurring in u and are denoted by h(u) and t(u).The length of u is the number |u| of letters occurring in u counting multiplicity.The content of u is the set of letters occurring in u and is denoted by C(u).The set of length-two We write u = v when u and v are identical words and write u ≈ v to stand for a semigroup identity.Let Σ be a set of identities.We write Σ u ≈ v or u Σ ≈ v if the identity u ≈ v is derivable from the identities in Σ.The variety defined by Σ is the class of all semigroups that satisfy all identities in Σ and is denoted by [Σ].If V is a variety with V = [Σ], then Σ is said to be a basis for V.A variety is finitely based if it possesses a finite basis.
A permutation identity is an identity of the form x mα where x 1 , . . ., x m are distinct letters and α is a nontrivial permutation on {1, . . ., m}.A permutative variety is a variety that satisfies some permutation identity.
We refer the reader to [3] and [2] respectively for undefined terminology in semigroup theory and universal algebra.

The subvariety lattice L(A 2 ) of A 2
Recall that C denotes the lattice of all combinatorial Rees-Sushkevich varieties and P denotes the set of all permutative varieties in C.
Denote by A 2 the idempotent-generated 0-simple semigroup of order five and by B 2 the Brandt semigroup of order five: These two semigroups play very important roles in the theory of semigroup and especially in the theory of semigroup varieties.They appeared or were investigated in, for example, [1], [3]- [16], and [19].Denote by A 2 the variety generated by A 2 .

Proposition 2 ([10, Proposition 1.2]) The variety A 2 is the largest combinatorial Rees-Sushkevich variety. Consequently, the lattice C coincides with the subvariety lattice
In view of Proposition 2, any variety V in C is defined within A 2 by some set Σ of identities, that is, Lemma 3 (Trahtman [15,16]) The identities A word of length at least two is said to be connected if it cannot be written as a product of two nonempty words with disjoint contents.
and t(u) = t(v).Suppose σ is an identity that does not hold in the semigroup B 2 .Then the identity u ≈ v holds in the variety A 2 ∩ [σ].In particular, the following identities hold in any subvariety of A 2 that does not contain B 2 : Proof.Without loss of generality, suppose  Proof.Suppose U, V ∈ P. Clearly U ∩ V is a variety in C that satisfies all (permutation) identities of U and V so that U ∩ V ∈ P. By Lemma 1, there exist i, j ≥ 1 such that the permutation identities π i and π j hold in U and V respectively.Then U ∨ V is a variety in C that satisfies the identity π m where m = max{i, j}.Therefore U ∨ V ∈ P. 2
A word is simple if all letters occurring in it have multiplicity one.Suppose X is (alphabetically) ordered by <.
Clearly an ordered word is necessarily simple.A word u is said to be in canonical form if any of the following conditions hold: (A) u = xvx for some ordered word v ∈ X * with x / ∈ C(v); (B) u = xyvxy for some ordered word v ∈ X * with x, y / ∈ C(v) and x = y.
Note that a word in canonical form is necessarily connected.

Lemma 7
Let u be a connected word.Then there exists a unique word u in canonical form such that C(u) = C(u ), h(u) = h(u ), and t(u) = t(u ).

Further, the identity u ≈ u holds in the variety
Proof.The existence and uniqueness of u is easy to verify.Since any word in canonical form is connected, the identity u ≈ u holds in A 2 ∩ [(5a)] by Lemma 4. 2 Lemma 8 A non-simple word u is equivalent within A 2 ∩ [(5a)] to a word pw q where (i) p, q ∈ X * are simple words; (ii) w ∈ X + is a connected word; (iii) C(p), C(w), C(q) are pairwise disjoint sets.

Proof.
By assumption, we may write u = pvq where p, q ∈ X * are simple, v ∈ X + is non-simple with h = h(v) and t = t(v) each occurring at least twice in v, and C(p), C(v), C(q) are pairwise disjoint sets.(Note that h = t is possible).The words v and v 2 are equivalent within Hence, by Lemma 7, the words u and pw q are equivalent within where the word w = v 2 is connected.2 Proposition 9 Every subvariety of A 2 ∩ [(5a)] is finitely based.

Proof. The variety
)] by some set Σ of identities.By Lemma 8, we may assume that all identities in Σ are formed by words that are either simple or of the form pw q.It follows from [18] that V is finitely based. 2

Main results
Recall that J denotes the complete join of all varieties in P.This section presents several properties of J and its subvarieties.
Proof.It is easy to see that within A 2 , the identity (5a) is a consequence of the identity π m for any m ≥ 1. Therefore . By Proposition 9, V is defined within A 2 ∩ [(5a)] by some finite set Σ of identities.We may assume that the identities in Σ do not hold in A 2 ∩ [(5a)].By Lemma 8, we may assume that all identities in Σ are formed by words that are either simple or of the form pw q.Further, since J contains semilattices, each identity in Σ is formed by a pair of words with identical content.Let σ : u ≈ v be an identity in Σ.
Case 1: Suppose both u and v are simple.Then σ is a permutation identity.
Case 2: Suppose u is simple and v is non-simple (of the form pw q).
Hence V satisfies the permutation identity uϕ ≈ uχ, and we arrive at the same contradiction in Case 1.
Therefore Cases 1 and 2 are both impossible, whence all identities in Σ are formed by non-simple words of the form pw q.Suppose τ : u 1 ≈ u 2 is such an identity in Σ, say u i = p i w i q i for i ∈ {1, 2}.It is easy to show that if then u 1 = u 2 so that the identity τ is contradictorily satisfied by Thus at least one of the four equalities in (6) do not hold, whence {(1), π m } τ for any m ≥ max{|u 1 |, |u 2 |}.But this contradicts the fact that Consequently, the identity τ , and hence V, do not exist, whence Proposition 11 The variety J is not permutative.Consequently, the lattice P is incomplete.
Proof.By referring to the identity basis for J in Proposition 10, it is easy to show that J does not satisfy any of the identities π m and hence cannot be permutative by Lemma 1. Therefore P does not contain the complete join J of its varieties, whence it is a lattice (Proposition 5) that is incomplete. 2

Proposition 12
The variety J is locally finite and non-finitely generated.
Proof.The variety J is locally finite since A 2 is finitely generated.Let S be a semigroup in J with |S| < m.For 1 ≤ i ≤ m, let a, b, g i , h i ∈ S. Since the list g 1 , . . ., g m contains an element (say g i ) that appears at least twice and that S satisfies the identities (4), we have By an identical argument, The identities ( 2) and (5b) also hold in S so that Hence S satisfies the identity π m .But J does not satisfy π m (Proposition 11) and so cannot be generated by S. 2 Proposition 13 Every subvariety of J is finitely based.Consequently, L(J) is a countably infinite lattice.
Proof.The first part follows from Propositions 9 and 10, while the second part holds since only countably many finite sets of identities exist up to relabelling of letters. 2 The last result of this article involves the semigroup Y = e, f, s : e 2 = e, f 2 = f, ef = f e = 0, es = sf = s of order four.It is easy to show that Y is isomorphic to a subsemigroup of the 0-simple semigroup B 2 and so belongs to A 2 by Proposition 2.

Theorem 14
The following statements on a variety V in C are equivalent.
(i) V is contained in J; (ii) V does not contain Y.
Consequently, J is the largest variety in C that does not contain Y.
Proof.Since e 2 sef 2 = 0 = s = e 2 esf 2 , the identity (5a) of J does not hold in the semigroup Y so that statement (i) implies statement (ii).
We have thus shown that the identity (5b) holds in every finite semigroup of the locally finite variety V and hence must also hold in V. Consequently, V ⊆ A 2 ∩ [(5b)] = J by Lemma 6 and Proposition 10. 2

2 Proposition 5
7] that they belong to the same D-class D, which must be a rectangular band.Since C 2 (uv) = C 2 (vu), h(uv) = h(vu), and t(uv) = t(vu), the identity uv ≈ vu holds in A 2 by Lemma 3. Therefore u and v are commuting elements in the rectangular band D and so must coincide in F n .The set P constitutes a sublattice of L(A 2 ).
x n }.Denote by F n the free object of A 2 ∩[σ] over the generators {x 1 , . . ., x n }.Since A 2 is locally finite, F n is a finite semigroup.It follows from [1, Exercise 8.1.6]that the regular D-classes of F n are subsemigroups.Now u and v are regular elements of F n by [10, Proposition 2.2], and it follows from [1, Theorem 8.1.