86 research outputs found
A large class of sofic monoids
We prove that a monoid is sofic, in the sense recently introduced by
Ceccherini-Silberstein and Coornaert, whenever the J-class of the identity is a
sofic group, and the quotients of this group by orbit stabilisers in the rest
of the monoid are amenable. In particular, this shows that the following are
all sofic: cancellative monoids with amenable group of units; monoids with
sofic group of units and finitely many non-units; and monoids with amenable
Schutzenberger groups and finitely many L-classes in each D-class. This
provides a very wide range of sofic monoids, subsuming most known examples
(with the notable exception of locally residually finite monoids). We conclude
by discussing some aspects of the definition, and posing some questions for
future research
Word problems recognisable by deterministic blind monoid automata
We consider blind, deterministic, finite automata equipped with a register
which stores an element of a given monoid, and which is modified by right
multiplication by monoid elements. We show that, for monoids M drawn from a
large class including groups, such an automaton accepts the word problem of a
group H if and only if H has a finite index subgroup which embeds in the group
of units of M. In the case that M is a group, this answers a question of Elston
and Ostheimer.Comment: 8 pages, fixed some typos and clarified ambiguity in the abstract,
results unchange
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
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