Word Hyperbolic Semigroups
A.J. Duncan and R.H. Gilman
Status Math. Proc. Camb. Phil. Soc., 136 (2004) no. 3, 513--524.
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 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 closely tied to the
algebraic structure as they are in the case of groups. We propose a
language--theoretic definition of word hyperbolic semigroup which for
groups is equivalent to the original definition.
We also consider briefly word problems of semigroups. The standard
word problem of a group is the language of all words representing the
identity. As it stands this definition makes no sense for semigroups,
but there is a variation which does.
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28th April 2005.