Structure of the largest idempotent-product free sequences in semigroups. (English) Zbl 1458.11051
Summary: 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 the order induced from the sequence \(T\), 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 D. W. H. Gillam et al., Bull. Lond. Math. Soc. 4, 143–144 (1972; Zbl 0252.20064)], 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) is such that \(| \mathcal{S} \setminus E(\mathcal{S}) |\) 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.
Here, motivated by Gillam, Hall and Williams’ work D. W. H. Gillam et al., Bull. Lond. Math. Soc. 4, 143–144 (1972; Zbl 0252.20064)], 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) is such that \(| \mathcal{S} \setminus E(\mathcal{S}) |\) 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.
Citations:
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