Semigroups Arising From Asynchronous Automata

We introduce a new class of semigroups arising from a restricted class of asynchronous automata. We call these semigroups "expanding automaton semigroups." We show that the class of synchronous automaton semigroups is strictly contained in the class of expanding automaton semigroups, and that the class of expanding automaton semigroups is strictly contained in the class of asynchronous automaton semigroups. We investigate the dynamics of expanding automaton semigroups acting on regular rooted trees, and show that undecidability arises in these actions. We show that this class is not closed under taking normal ideal extensions, but the class of asynchronous automaton semigroups is closed under taking these extensions. We construct every free partially commutative monoid as a synchronous automaton semigroup.Comment: 31 pages, 4 figure

The subpower membership problem for semigroups

Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) asks whether $b$ can be generated by $a_1,\ldots,a_k$. If $S$ is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in $nk$. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 letters or more is actually PSPACE-complete, while on 2 letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup $S$ embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P; otherwise it is NP-complete

Uniform decision problems in automatic semigroups

We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the problem well-defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix together with an automatic structure for its maximal subgroup. On the other hand, we show that it is undecidable in general whether a given element of a given automatic monoid has a right inverse.Comment: 19 page