International Journal of Algebra and Computation

A theory of the semidirect product of categories and the derived category of a category morphism is presented. In order to include division (≺) in this theory, the traditional setting of these constructions is expanded to include relational arrows. In this expanded setting, a relational morphism φ : M → N of categories determines an optimal decomposition

The theory of the semidirect product of varieties of categories, V * W, is developed. Associated with each variety V of categories is the collection of relational morphisms whose derived category belongs to V. The semidirect product of varieties and the composition of classes of the form are shown to stand in the relationship

Finally, it is demonstrated that all the results in the article concerning varieties of categories have pseudovariety and monoidal versions. This allows us to furnish a straightforward proof that

We introduce a family of multiplicative distributions {μ_s|s∈(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R⊆F is a function μ(R) : s → μ_s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.

It is proved that the lattice L(B_d) of quasivarieties contained in the variety B_d of idempotent semigroups contains an isomorphic copy of the ideal lattice of a free lattice on ω free generators. This result shows that a problem of Petrich [19], which calls for a description of L(B_d), is much more complex than originally expected.