6,025 research outputs found
Binary patterns in the Prouhet-Thue-Morse sequence
We show that, with the exception of the words and , all
(finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can
actually be found in that sequence as segments (up to exchange of letters in
the infinite case). This result was previously attributed to unpublished work
by D. Guaiana and may also be derived from publications of A. Shur only
available in Russian. We also identify the (finitely many) finite binary
patterns that appear non trivially, in the sense that they are obtained by
applying an endomorphism that does not map the set of all segments of the
sequence into itself
Commutative positive varieties of languages
We study the commutative positive varieties of languages closed under various
operations: shuffle, renaming and product over one-letter alphabets
On the group of a rational maximal bifix code
We give necessary and sufficient conditions for the group of a rational
maximal bifix code to be isomorphic with the -group of , when
is recurrent and is rational. The case where is uniformly
recurrent, which is known to imply the finiteness of , receives
special attention.
The proofs are done by exploring the connections with the structure of the
free profinite monoid over the alphabet of
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 iterated semidirect products of finite semilattices
AbstractLet Sl denote the class of all finite semilattices and let Sln be the pseudovariety of semigroups generated by all semidirect products of n finite Semilattices. The main result of this paper shows that, for every n ⩾ 3, Sln is not finitely based. Nevertheless, a simple basis of identities is described for each Sln. We also show that Sln is generated by a semigroup with 2n generators but, except for n ⩽ 2, we do not know whether the bound 2n is optimal
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