31 research outputs found
Varieties of Languages in a Category
Eilenberg's variety theorem, a centerpiece of algebraic automata theory,
establishes a bijective correspondence between varieties of languages and
pseudovarieties of monoids. In the present paper this result is generalized to
an abstract pair of algebraic categories: we introduce varieties of languages
in a category C, and prove that they correspond to pseudovarieties of monoids
in a closed monoidal category D, provided that C and D are dual on the level of
finite objects. By suitable choices of these categories our result uniformly
covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer,
respectively, and yields new Eilenberg-type correspondences
Recognisable languages over monads
The principle behind algebraic language theory for various kinds of
structures, such as words or trees, is to use a compositional function from the
structures into a finite set. To talk about compositionality, one needs some
way of composing structures into bigger structures. It so happens that category
theory has an abstract concept for this, namely a monad. The goal of this paper
is to propose monads as a unifying framework for discussing existing algebras
and designing new algebras
Locally countable pseudovarieties
The purpose of this paper is to contribute to the theory of profinite
semigroups by considering the special class consisting of those all of whose
finitely generated closed subsemigroups are countable, which are said to be
locally countable. We also call locally countable a pseudovariety V (of finite
semigroups) for which all pro-V semigroups are locally countable. We
investigate operations preserving local countability of pseudovarieties and
show that, in contrast with local finiteness, several natural operations do not
preserve it. We also investigate the relationship of a finitely generated
profinite semigroup being countable with every element being expressable in
terms of the generators using multiplication and the idempotent (omega) power.
The two properties turn out to be equivalent if there are only countably many
group elements, gathered in finitely many regular J-classes. We also show that
the pseudovariety generated by all finite ordered monoids satisfying the
inequality is locally countable if and only if
Locally countable pseudovarieties
The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressible in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J -classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality 1 6 x n is locally countable if and only if n = 1