3,075 research outputs found
On the Finiteness Problem for Automaton (Semi)groups
This paper addresses a decision problem highlighted by Grigorchuk,
Nekrashevich, and Sushchanskii, namely the finiteness problem for automaton
(semi)groups.
For semigroups, we give an effective sufficient but not necessary condition
for finiteness and, for groups, an effective necessary but not sufficient
condition. The efficiency of the new criteria is demonstrated by testing all
Mealy automata with small stateset and alphabet. Finally, for groups, we
provide a necessary and sufficient condition that does not directly lead to a
decision procedure
A Groupoid Approach to Discrete Inverse Semigroup Algebras
Let be a commutative ring with unit and an inverse semigroup. We show
that the semigroup algebra can be described as a convolution algebra of
functions on the universal \'etale groupoid associated to by Paterson. This
result is a simultaneous generalization of the author's earlier work on finite
inverse semigroups and Paterson's theorem for the universal -algebra. It
provides a convenient topological framework for understanding the structure of
, including the center and when it has a unit. In this theory, the role of
Gelfand duality is replaced by Stone duality.
Using this approach we are able to construct the finite dimensional
irreducible representations of an inverse semigroup over an arbitrary field as
induced representations from associated groups, generalizing the well-studied
case of an inverse semigroup with finitely many idempotents. More generally, we
describe the irreducible representations of an inverse semigroup that can
be induced from associated groups as precisely those satisfying a certain
"finiteness condition". This "finiteness condition" is satisfied, for instance,
by all representations of an inverse semigroup whose image contains a primitive
idempotent
The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable
We prove that a semigroup generated by a reversible two-state Mealy automaton
is either finite or free of rank 2. This fact leads to the decidability of
finiteness for groups generated by two-state or two-letter
invertible-reversible Mealy automata and to the decidability of freeness for
semigroups generated by two-state invertible-reversible Mealy automata
On residual finiteness of monoids, their Schützenberger groups and associated actions
RG was supported by an EPSRC Postdoctoral Fellowship EP/E043194/1 held at the University of St Andrews, Scotland.In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of Schützenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.PostprintPeer reviewe
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