882 research outputs found
Self-Automaton Semigroups
After reviewing automaton semigroups, we introduce Cayley Automata and the
corresponding Cayley Automaton semigroups. We investigate which semigroups are
isomorphic to their Cayley Automaton semigroup and give some results for
special classes of semigroups. We answer a question posed by Cain relating to
the dual construction.Comment: 18 page
Enumeration of idempotents in planar diagram monoids
We classify and enumerate the idempotents in several planar diagram monoids:
namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The
classification is in terms of certain vertex- and edge-coloured graphs
associated to Motzkin diagrams. The enumeration is necessarily algorithmic in
nature, and is based on parameters associated to cycle components of these
graphs. We compare our algorithms to existing algorithms for enumerating
idempotents in arbitrary (regular *-) semigroups, and give several tables of
calculated values.Comment: Majorly revised (new title, new abstract, one additional author), 24
pages, 6 figures, 8 tables, 5 algorithm
Profinite Groups Associated to Sofic Shifts are Free
We show that the maximal subgroup of the free profinite semigroup associated
by Almeida to an irreducible sofic shift is a free profinite group,
generalizing an earlier result of the second author for the case of the full
shift (whose corresponding maximal subgroup is the maximal subgroup of the
minimal ideal). A corresponding result is proved for certain relatively free
profinite semigroups. We also establish some other analogies between the kernel
of the free profinite semigroup and the \J-class associated to an irreducible
sofic shift
Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory
In this paper we characterize the congruence associated to the direct sum of
all irreducible representations of a finite semigroup over an arbitrary field,
generalizing results of Rhodes for the field of complex numbers. Applications
are given to obtain many new results, as well as easier proofs of several
results in the literature, involving: triangularizability of finite semigroups;
which semigroups have (split) basic semigroup algebras, two-sided semidirect
product decompositions of finite monoids; unambiguous products of rational
languages; products of rational languages with counter; and \v{C}ern\'y's
conjecture for an important class of automata
On union ultrafilters
We present some new results on union ultrafilters. We characterize stability
for union ultrafilters and, as the main result, we construct a new kind of
unordered union ultrafilter
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