Automatic presentations for semigroups

Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008)This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved.PostprintPeer reviewe

Unary FA-presentable semigroups

Automatic presentations, also called FA-presentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: auto-matic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally nite, but non-nitely generated unary FA-presentable semigroups may be innite. Every unary FA-presentable semigroup satises some Burnside identity.We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classication is given of the unary FA-presentable completely simple semigroups.PostprintPeer reviewe

Presentations for subsemigroups of groups

This thesis studies subsemigroups of groups from three perspectives: automatic structures, ordinary semigroup presentations, and Malcev presentaions. [A Malcev presentation is a presentation of a special type for a semigroup that can be embedded into a group. A group-embeddable semigroup is Malcev coherent if all of its finitely generated subsemigroups admit finite Malcev presentations.] The theory of synchronous and asynchronous automatic structures for semigroups is expounded, particularly for group-embeddable semigroups. In particular, automatic semigroups embeddable into groups are shown to inherit many of the pleasant geometric properties of automatic groups. It is proved that group- embeddable automatic semigroups admit finite Malcev presentations, and such presentations can be found effectively. An algorithm is exhibited to test whether an automatic semigroup is a free semigroup. Cancellativity of automatic semigroups is proved to be undecidable. Study is made of several classes of groups: virtually free groups; groups that satisfy semigroup laws (in particular [virtually] nilpotent and [virtually] abelian groups); polycyclic groups; free and direct products of certain groups; and one-relator groups. For each of these classes, the question of Malcev coherence is considered, together with the problems of whether finitely generated subsemigroups are finitely presented or automatic. This study yields closure and containment results regarding the class of Malcev coherent groups. The property of having a finite Malcev presentation is shown to be preserved under finite Rees index extensions and subsemigroups. Other concepts of index are also studied

Markov semigroups, monoids, and groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.Comment: 40 pages; 3 figure

Automatic structures for semigroup constructions

We survey results concerning automatic structures for semigroup constructions, providing references and describing the corresponding automatic structures. The constructions we consider are: free products, direct products, Rees matrix semigroups, Bruck-Reilly extensions and wreath products.Comment: 22 page

Small overlap monoids II: automatic structures and normal forms

We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C(4) monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy analogues of Kleene's theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automata-theoretic results which may be of independent interest.Comment: 17 page