Decision problems for word-hyperbolic semigroups

2 Word Hyperbolic Semigroups

The study of automatic and word hyperbolic groups is a prominent topic in geometric group theory. The definition of automatic groups extends naturally to semigroups [3, 8], but word hyperbolic groups are defined by a geometric condition, the thin triangle condition, whose extension is not immediate. The problem is that for semigroups the metric properties of the Cayley diagram are not as closel...

Hyperbolic Groups and Completely Simple Semigroups

We begin with a brief introduction to the theory of word hyperbolic groups. We then consider four possible conditions which might reasonably be used as definitions or partial definitions of hyperbolicity in semigroups: having a hyperbolic Cayley graph; having hyperbolic Schützenberger graphs; having a context-free multiplication table; or having word hyperbolic maximal subgroups. Our main resul...

Word Hyperbolic Semigroups 32

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...

Decision and Separability Problems for Baumslag-Solitar Semigroups

We show that the semigroups Sk,` having semigroup presentations 〈a, b : abk = b`a〉 are residually finite and finitely separable. Generally, these semigroups have finite separating images which are finite groups and other finite separating images which are semigroups of order-increasing transformations on a finite partially ordered set. These semigroups thus have vastly different residual and se...