956 research outputs found
Large Aperiodic Semigroups
The syntactic complexity of a regular language is the size of its syntactic
semigroup. This semigroup is isomorphic to the transition semigroup of the
minimal deterministic finite automaton accepting the language, that is, to the
semigroup generated by transformations induced by non-empty words on the set of
states of the automaton. In this paper we search for the largest syntactic
semigroup of a star-free language having left quotients; equivalently, we
look for the largest transition semigroup of an aperiodic finite automaton with
states.
We introduce two new aperiodic transition semigroups. The first is generated
by transformations that change only one state; we call such transformations and
resulting semigroups unitary. In particular, we study complete unitary
semigroups which have a special structure, and we show that each maximal
unitary semigroup is complete. For there exists a complete unitary
semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by
transformations that map a non-empty subset of states to a single state; we
call such transformations and semigroups semiconstant. In particular, we
examine semiconstant tree semigroups which have a structure based on full
binary trees. The semiconstant tree semigroups are at present the best
candidates for largest aperiodic semigroups.
We also prove that is an upper bound on the state complexity of
reversal of star-free languages, and resolve an open problem about a special
case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table
Some structural properties of the free profinite aperiodic semigroup
Profinite semigroups provide powerful tools to understand properties of classes of regular languages. Until very recently however, little was known on the structure of "large" relatively free profinite semi- groups. In this paper, we present new results obtained for the class of all finite aperiodic (that is, group-free) semigroups. Given a finite alphabet X, we focus on the following problems: (1) the word problem for ω-terms on X evaluated on the free pro-aperiodic semigroup, and (2) the computation of closures of regular languages in the ω-subsemigroup of the free pro-aperiodic semigroup generated by X.FCT through the Centro de Matemática da Universidade do Minho and the Centro de Matemática da Universidade do PortoEuropean Community Fund FEDERESF programme “Automata: from Mathematics to Applications (AutoMathA)”Pessoa Portuguese-French project Egide-Grices 11113Y
Nica-Toeplitz algebras associated with product systems over right LCM semigroups
We prove uniqueness of representations of Nica-Toeplitz algebras associated
to product systems of -correspondences over right LCM semigroups by
applying our previous abstract uniqueness results developed for
-precategories. Our results provide an interpretation of conditions
identified in work of Fowler and Fowler-Raeburn, and apply also to their
crossed product twisted by a product system, in the new context of right LCM
semigroups, as well as to a new, Doplicher-Roberts type -algebra
associated to the Nica-Toeplitz algebra. As a derived construction we develop
Nica-Toeplitz crossed products by actions with completely positive maps. This
provides a unified framework for Nica-Toeplitz semigroup crossed products by
endomorphisms and by transfer operators. We illustrate these two classes of
examples with semigroup -algebras of right and left semidirect products.Comment: Title changed from "Nica-Toeplitz algebras associated with right
tensor C*-precategories over right LCM semigroups: part II examples". The
manuscript accepted in J. Math. Anal. App
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