Algebraic functions in Łukasiewicz implication algebras

In this article we study algebraic functions in [Formula: see text]-subreducts of MV-algebras, also known as Łukasiewicz implication algebras. A function is algebraic on an algebra [Formula: see text] if it is definable by a conjunction of equations on [Formula: see text]. We fully characterize algebraic functions on every Łukasiewicz implication algebra belonging to a finitely generated variety. The main tool to accomplish this is a factorization result describing algebraic functions in a subproduct in terms of the algebraic functions of the factors. We prove a global representation theorem for finite Łukasiewicz implication algebras which extends a similar one already known for Tarski algebras. This result together with the knowledge of algebraic functions allowed us to give a partial description of the lattice of classes axiomatized by sentences of the form [Formula: see text] within the variety generated by the 3-element chain.

A finitely generated variety of combinatorial inverse semigroups having uncountably many subvarieties

A finite combinatorial inverse semigroup Θ of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Θ possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Θ, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.