Structure of the largest idempotent-product free sequences in semigroups

Abstract

Let S be a finite semigroup, and let E(S) be the set of all idempotents of S. Gillam, Hall and Williams proved in 1972 that every S-valued sequence T of length at least |S| − |E(S)|+1 is not (strongly) idempotent-product free, in the sense that it contains a nonempty subsequence the product of whose terms, in their natural order in T , is an idempotent, which affirmed a question of Erdős. They also showed that the value |S| − |E(S)| + 1 is best possible.Here, motivated by Gillam, Hall and Williams' work, we determine the structure of the idempotent-product free sequences of length |S \ E(S)| when the semigroup S (not necessarily finite) satisfies |S \ E(S)| is finite, and we introduce a couple of structural constants for semigroups that reduce to the classical Davenport constant in the case of finite abelian groups.

A stronger connection between the Erdős-Burgess and Davenport constants

“…The connection between the Erdős-Burgess and Davenport constants first appeared in a recent paper of Wang [18] on maximal sequences over semigroups that avoid idempotent products. When S is a finite abelian group, for instance, the identity is the only idempotent element, so I(S) = D(S) trivially.…”

Lower bound for the Erdős-Burgess constant of finite commutative rings

“…Hall and N.H. Williams [10] proved that a sequence T over any finite semigroup S of length at least |S \ E(S)| + 1 must contain one or more terms whose product, in the order induced from the sequence T , is an idempotent, and therefore, completely answered the Erdős' question. The Gillam-Hall-Williams Theorem was extended to infinite semigroups by the author [17] in 2019. It was also remarked that although the bound |S \ E(S)| + 1 is optimal for general semigroups S, that better bound can be obtained, at least in principle, for specific classes of semigroups.…”

“…Definition A. ( [17], Definition 4.1) For a commutative semigroup S, define the Erdős-Burgess constant of S, denoted I(S), to be the least ℓ ∈ N ∪ {∞} such that every sequence T of terms from S and of length ℓ must contain one or more terms such that their product is an idempotent.…”

Existence of Erdős-Burgess constant in commutative rings

“…Hall and N.H. Williams [10] proved that a sequence T over any finite semigroup S of length at least |S \ E(S)| + 1 must contain one or more terms whose product, in the order induced from the sequence T , is an idempotent, and therefore, completely answered Erdős' question. The Gillam-Hall-Williams Theorem was extended to infinite semigroups by the author [19]. It was also remarked that the bound |S \ E(S)| + 1, although is optimal for general semigroups S, can be improved, at least in principle, for specific classed of semigroups.…”