Structure of the largest idempotent-product free sequences in semigroups

Abstract

Let $\mathcal{S}$ be a finite semigroup, and let $E(\mathcal{S})$ be the set of all idempotents of $\mathcal{S}$. Gillam, Hall and Williams proved in 1972 that every $\mathcal{S}$-valued sequence $T$ of length at least $|\mathcal{S}|-|E(\mathcal{S})|+1$ 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 $|\mathcal{S}|-|E(\mathcal{S})|+1$ is best possible. Here, motivated by Gillam, Hall and Williams' work, we determine the structure of the idempotent-product free sequences of length $|\mathcal{S}\setminus E(\mathcal{S})|$ when the semigroup $\mathcal{S}$ (not necessarily finite) satisfies $|\mathcal{S}\setminus E(\mathcal{S})|$ 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.

12 pages

References in corpus (2)

• A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of $\F_2[x]$
• Additively irreducible sequences in commutative semigroups

Cited by in corpus (5)

• A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of $\F_2[x]$
• Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$
• Davenport constant of the multiplicative semigroup of the ring ${\mathbb{Z}}_{n_1}\oplus \cdots \oplus {\mathbb{Z}}_{n_r}$
• Additively irreducible sequences in commutative semigroups
• Davenport constant for semigroups II