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arXiv · 1802.02420

A group-theoretical interpretation of the word problem for free idempotent generated semigroups

Yang Dandan·Igor Dolinka·Victoria Gould
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Abstract

The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup $\mathsf{IG}(\mathcal{E})$ - the `free-est' semigroup with a given biordered set $\mathcal{E}$ of idempotents. We show that when $\mathcal{E}$ is finite, the word problem for $\mathsf{IG}(\mathcal{E})$ is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of $\mathsf{IG}(\mathcal{E})$. As an application, we obtain decidability of the word problem for an important class of examples. Also, we prove that for finite $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$ is always a weakly abundant semigroup satisfying the congruence condition.

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Yang Dandan, Igor Dolinka, Victoria Gould. 2018-02-07. A group-theoretical interpretation of the word problem for free idempotent generated semigroups. https://arxiv.org/abs/1802.02420

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