The varieties of semilattice-ordered semigroups satisfying $$x^3\approx x$$
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The varieties of semilattice-ordered semigroups satisfying x 3 ≈ x and x y ≈ yx Miaomiao Ren1 · Xianzhong Zhao1
Published online: 8 March 2016 © Akadémiai Kiadó, Budapest, Hungary 2016
Abstract The aim of this paper is to study the varieties of semilattice-ordered Burnside semigroups satisfying x 3 ≈ x and x y ≈ yx. It is shown that the collection of all such varieties forms a distributive lattice of order 9. Also, all of them are finitely based and finitely generated. This gives a generalization and expansion of the results obtained by McKenzie and Romanowska (Contrib Gen Algebra Proc Klagenf Conf 1978 1:213–218, 1979). Keywords Semilattice-ordered Burnside semigroup · Lattice · Subdirectly irreducible member · Variety · 0-Group Mathematics Subject Classification
16Y60 · 08B05 · 08B15 · 20M07
1 Introduction and preliminaries By a semilattice-ordered semigroup (see [4,9]) we mean an algebra (S, +, ·) such that: • the additive reduct (S, +) is a semilattice; • the multiplicative reduct (S, ·) is a semigroup; • (S, +, ·) satisfies the identities x(y + z) ≈ x y + x z and (x + y)z ≈ x z + yz. Such an algebra is an additively idempotent semiring (see [18–20,24]). The class of all semilattice-ordered semigroups forms a variety and is denoted by SLOS. Semilattice-ordered semigroups P(S) and P f (S) play key roles in the study of the variety SLOS, where S is a semigroup, P(S) (P f (S), respectively) is the set of all non-empty subsets
In loving memory of Miaomiao’s grandmother Xiulan Li.
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Xianzhong Zhao [email protected] Miaomiao Ren [email protected]
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School of Mathematics, Northwest University, Xi’an 710127, Shaanxi, People’s Republic of China
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The varieties of semilattice-ordered semigroups…
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(finite subsets, respectively) of S, equipped with an addition and a multiplication as follows: A + B = A ∪ B,
A ◦ B = {ab|a ∈ A, b ∈ B}.
In fact, if X + denotes the free semigroup on a non-empty set X, then P f (X + ) is free in SLOS on X (see [9]). Recall that a semigroup is called a Burnside semigroup if it satisfies the identity x n ≈ x m with m < n (see [2,3,17]). A semilattice-ordered semigroup is called a semilattice-ordered Burnside semigroup if its multiplicative reduct is a Burnside semigroup. The variety of semilattice-ordered Burnside semigroups (Burnside semigroups, respectively) defined by the identity x n ≈ x m will be denoted by Sr(n, m) (Sg(n, m), respectively). There is a rich literature on Burnside semigroups and semilattice-ordered Burnside semigroups (see [2–6,8–12,17–27]). In 1979 McKenzie and Romanowska [10] studied the varieties of so-called ·-distributive bisemilattices, i.e., the subvarieties of Sr(2, 1) satisfying the additional identity x y ≈ yx. They showed that the collection of all such subvarieties forms a five-element lattice: the trivial variety T, the variety D of distributive lattices, the variety M of semilattice-ordered semigroups whose reducts are equal semilattices, the variety D ∨ M and the subvariety Bi of Sr(2, 1) determined by the identity x y ≈ yx. Note that
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