Automatic semigroup

Abstract: The notion of automaticity has been widely studied in groups and some progress has been made in understanding the notion in the wider context of semigroups The purpose of this paper is to study automatic completely-simple semigroups We show that, if \(S\) is a completely-simple semigroup \(M[H;I;J;P]\) (with \(I\) and \(J\) finite), then \(S\) is automatic if and only if the group \(H\) is automatic As a consequence, we deduce that automatic completely-simple semigroups are finitely presented We also show that automatic completely-simple semigroups are characterized by the fellow traveller property and also that the existence of an automatic structure is independent of the choice of generating set

Abstract: We consider a Rees matrix semigroup S = M[U; I, J; P] over a semigroup U, with I and J finite index sets, and relate the automaticity of S with the automaticity of U. We prove that if U is an automatic semigroup and S is finitely generated then S is an automatic semigroup. If S is an automatic semigroup and there is an entry p in the matrix P such that pU 1 = U then U is automatic. We also prove that if S is a prefix-automatic semigroup, then U is a prefix-automatic semigroup.

Abstract: The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S∖T under the natural actions of T on S via right and left multiplication. This partitions the complement S∖T into T-relative -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schutzenberger group. If the Rees index |S∖T| is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schutzenberger groups, and vice versa. We also give a method for constructing a presentation for S from presentations of T and the Schutzenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups).

Abstract: We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup,namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication.The framework embraces many of the well-known classes of (automatic) semigroups: finite monoids, free semigroups, free commutative monoids, trace or divisibility monoids, braid or Artin-Tits or Krammer or Garside monoids, Baumslag-Solitar semigroups, etc.Like plactic monoids or Chinese monoids, some neither left- nor right-cancellative automatic semigroups are also investigated, as well as some residually finite variations of the bicyclic monoid. It provides what appears to be the first known connection from a class of automatic semigroupsto a class of automaton semigroups. It is worthwhile noting that, in all these cases, "being an automatic semigroup" and "being an automaton semigroup" become dual properties in a very automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as a cornerstone of our construction.