The Kernel Category and Variants of the Concatenation Product

For each variety V of finite monoids, we consider three varieties [Formula: see text] of languages defined in the following way: for each alphabet [Formula: see text] is the Boolean algebra generated by the languages of the form L or L0aL1, where [Formula: see text], a ∈ A and the product L0aL1 is deterministic (resp. codeterministic, bideterministic). We present a description of the corresponding varieties of finite monoids. Such descriptions are done in terms of categories, using the kernel category of a relational morphism. We also give some connections with the positive varieties of languages and with the varieties of finite ordered monoids.

THE BIDETERMINISTIC CONCATENATION PRODUCT

This paper is devoted to the study of the bideterministic concatenation product, a variant of the concatenation product. We give an algebraic characterization of the varieties of languages closed under this product. More precisely, let V be a variety of monoids, [Formula: see text] the corresponding variety of languages and [Formula: see text] the smallest variety containing [Formula: see text] and the bideterministic products of two languages of [Formula: see text]. We give an algebraic description of the variety of monoids [Formula: see text] corresponding to [Formula: see text]. For instance, we compute [Formula: see text] when V is one of the following varieties: the variety of idempotent and commutative monoids, the variety of monoids which are semilattices of groups of a given variety of groups, the variety of ℛ-trivial and idemptotent monoids. In particular, we show that the smallest variety of languages closed under bideterministic product and containing the language {1}, corresponds to the variety of [Formula: see text]-trivial monoids with commuting idempotents. Similar results were known for the other variants of the concatenation product, but the corresponding algebraic operations on varieties of monoids were based on variants of the semidirect product and of the Malcev product. Here the operation [Formula: see text] makes use of a construction which associates to any finite monoid M an expansion [Formula: see text] with the following properties: (1) M is a quotient of [Formula: see text] (2) the morphism [Formula: see text] induces an isomorphism between the submonoids of [Formula: see text] and of M generated by the regular elements and (3) the inverse image under π of an idempotent of M is a 2-nilpotent semigroup.