TAMENESS OF THE PSEUDOVARIETY LS1

Abstract

The notion of κ-tameness of a pseudovariety was introduced by Almeida and Steinberg and is a strong property which implies decidability of pseudovarieties. In this paper we prove that the pseudovariety LSl, of local semilattices, is κ-tame.A finite non-empty set A is called an alphabet and its elements are called letters. A (finite) word on A is a finite sequence w = (a 1 , . . . , a n ) of elements of A, usually written w = a 1 · · · a n . The integer n is called the length of w. The empty sequence, called the empty word, is denoted by 1 and its length is 0. The length of a word w is denoted by |w|. The product of two words w = a 1 a 2 · · · a n and z = b 1 b 2 · · · b m is the word wz = a 1 a 2 · · · a n b 1 b 2 · · · b m . We denote by A * the set of words on A and by A + the set of non-empty words. The set A * (resp. A + ) endowed with the product is a monoid (resp. semigroup) whose identity is the empty word and is called the free monoid (resp. the free semigroup) generated by A.A word w ∈ A + is said to be primitive if it is not a power of another word; that is, if w = u n for some u ∈ A * and n ∈ N implies w = u (and n = 1). Two words w and z are said to be conjugate if there exist words u, v ∈ A * such that w = uv and z = vu. We notice that, if w is a primitive word and z is a conjugate of w, then z is also primitive. Let an order be fixed for the letters of the alphabet A. A Lyndon word is a primitive word which is minimal, with respect to the lexicographic ordering, in its conjugation class.A bi-infinite (resp. right-infinite, left-infinite) word on A is a sequence w = (a n ) n of letters of A indexed by Z (resp. N, −N), also written

Complete reducibility of systems of equations with respect to R

“…The pseudovariety Ab of all finite Abelian groups is completely tame [14]. The pseudovariety LSl of all finite local semilattices is tame [28]. The pseudovariety R of all finite semigroups in which the Green relation R is trivial is tame, which entails the reducibility of several associated pseudovarieties such as R ∨ G [11].…”

Towards a pseudoequational proof theory

“…Proof. The ω-reducibility of LSl for the equation x = y was first proved in [24], where graph systems of equations are also considered. 3 The following basis of identities for the variety LSl may be found in [21]:…”

Complete Reducibility of Pseudovarieties

“…Here are some examples: the pseudovariety G of all finite groups is κ-reducible [17] 5 but not completely κ-reducible [24]; for a prime p, the pseudovariety G p of all finite p-groups is not κ-reducible but it is σ-reducible for a certain infinite signature σ [2]; the pseudovariety Ab of all finite Abelian groups is completely κ-reducible [9]; the pseudovariety OCR of all finite orthodox completely regular semigroups is κ-reducible [13]; the pseudovariety CR of all finite completely regular semigroups is κ-reducible [14] 6 ; the pseudovariety LSl of all finite semigroups S whose local subsemigroups eSe are semilattices is κ-reducible [23]; the pseudovariety R is κ-reducible [5]; the pseudovariety J of all finite J -trivial semigroups is completely κ-reducible [3]. The κ-reducibility of the pseudovariety A of all finite aperiodic semigroups was announced by J. Rhodes in 1997 but no proof has yet been published.…”