Some Algebraic Structures Of Languages (original) (raw)
Representation theory of finite semigroups, semigroup radicals and formal language theory
Transactions of the American Mathematical Society, 2008
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; andČerný's conjecture for an important class of automata. References 34
Algebraic and Topological Theory of Languages
Informatique Theorique Et Applications, 1995
R esum e: On dit qu'un langage est de torsion (resp. de torsion born ee, ap eriodique, ap eriodique born e) si son mono de syntaxique est de torsion (resp. de torsion born ee, ap eriodique, ap eriodique born e). Nous g en eralisons les th eor emes sur les langages rationnels de Kleene, Sch utzenberger et Straubing pour d ecrire les classes des langages de torsion, de torsion born ee, ap eriodiques et ap eriodiques born es. Ces descriptions imposent la consid eration de limites de suites de langages et d'automates pour certaines topologies d e nies par des ltrations du mono de libre. Nous donnons egalement un th eor eme concernant les langages arbitraires sur des alphabets nis.
Some algebraic structures on the generalization general products of monoids and semigroups
Arabian Journal of Mathematics
For arbitrary monoids A and B, in Cevik et al. (Hacet J Math Stat 2019:1–11, 2019), it has been recently defined an extended version of the general product under the name of a higher version of Zappa products for monoids (or generalized general product) A^{\oplus B}A⊕BA ⊕ BA⊕B_{\delta }\bowtie _{\psi }B^{\oplus A}$$ δ ⋈ ψ B ⊕ A and has been introduced an implicit presentation as well as some theories in terms of finite and infinite cases for this product. The goals of this paper are to present some algebraic structures such as regularity, inverse property, Green’s relations over this new generalization, and to investigate some other properties and the product obtained by a left restriction semigroup and a semilattice.
The Journal of Symbolic Logic, 2009
Relations between some theories of semigroups (also known as theories of strings or theories of concatenation) and arithmetic are surveyed. In particular Robinson's arithmetic Q is shown to be mutually interpretable with TC, a weak theory of concatenation introduced by Grzegorczyk. Furthermore, TC is shown to be interpretable in the theory F studied by Tarski and Szmielewa, thus confirming their claim that F is essentially undecidable. §0. Introduction. Prior to Gödel's work on the arithmetization of syntax Tarski developed a rigorous mathematical treatment of the syntax of formal languages in . He used a second-order version of the theory of concatenation on strings of symbols from a finite alphabet, or, from an algebraic viewpoint, the theory of free semigroups with a finite set of generators. Much later, after important work by Quine and Bennett, Corcoran, Frank, and Maloney investigated the relations between Tarski's theory, a generalized successor theory put forward by Hans Hermes in [10] and second-order Peano arithmetic 1 .
A note on decidability questions on presentations of word semigroups
Theoretical Computer Science, 1997
We apply automata-theoretic tools and some recently established compactness properties in the study of F-semigroups, that is, subsemigroups of free semigroups. With each F-semigroup we associate an F-presentation, which turns out to be finite for all finitely generated F-semigroups. Connections between F-presentations and ordinary presentations of semigroups are pointed out. It is also shown that it is undecidable whether two finitely generated F-semigroups satisfy a common relation in their F-presentations.
2000
New notions are introduced in algebra in order to better study the congruences in number theory. For example, the make an important such contribution.
A Note on Finitely Generated Semigroups of Regular Languages
Semigroups and Formal Languages - In honour of the 65th birthday of Donald B. McAlister - Proceedings of the International Conference, 2007
Let E = {E 1 ,. .. , E k } be a set of regular languages over a finite alphabet Σ. Consider morphism ϕ : ∆ + → (S, •) where ∆ + is the semigroup over a finite set ∆ and (S, •) = E is the finitely generated semigroup with E as the set of generators and language concatenation as a product. We prove that the membership problem of the semigroup S, the set [u] = {v ∈ ∆ + | ϕ(v) = ϕ(u)}, is a regular language over ∆, while the set Ker(ϕ) = {(u, v) | u, v ∈ ∆ + ϕ(u) = ϕ(v)} need not to be regular. It is conjectured however that every semigroup of regular languages is automatic.
On recognizable languages in divisibility monoids
Lecture Notes in Computer Science, 1999
Kleene's theorem on recognizable languages in free monoids is considered to be of eminent importance in theoretical computer science. It has been generalized into various directions, including trace and rational monoids. Here, we investigate divisibility monoids which are defined by and capture algebraic properties sufficient to obtain a characterization of the recognizable languages by certain rational expressions as known from trace theory. The proofs rely on Ramsey's theorem, distributive lattice theory and on Hashigushi's rank function generalized to our divisibility monoids. We obtain Ochmański's theorem on recognizable languages in free partially commutative monoids as a consequence.