Any mapping S : C → K, where K is a field and C = (C,Σ) is a Σ-algebra, is called a KC-series. These series are natural generalizations of both formal series on strings over a field and the tree series introduced by Berstel and Reutenauer [1]. We consider various operations on KC-series and their effects on the linear extensions of the series. We also study some algebraic aspects of theKΣ-algeb...

This is an expository account of R. McKenzie’s recent refutation of the RS conjecture. We adopt the usual conventions: if S is an algebra then S denotes the universe of S, |S| denotes the cardinality of S, and |S| denotes the successor cardinal to |S|. For an algebra A, let Vsi(A) denote the class of all nontrivial subdirectly irreducible members of V(A), and (following McKenzie) define κ(A) = ...

The present paper is based on [11], where a number of conjectures are made concerning the structure of the lattice Ext(Kt) of normal extensions of the tense logic Kt. That paper was mainly dealing with splittings of Ext(Kt) and some sublattices, and this is what I will concentrate on here as well. The main tool in analysing the splittings of Ext(Kt) will be the splitting theorem of [7]. In [11]...

By J onsson's Lemma, the variety V(K) generated by a nite lattice has only nitely many subvarieties. This led to the conjecture that, conversely, if a lattice variety has only nitely many subvarieties, then it is generated by a nite lattice. A stronger form of the conjecture states that a nitely generated variety V(K) has only nitely many covers in the lattice of lattice varieties, and that eac...

The main purpose of this paper is to axiomatize the join of the variety DPCSHC of dually pseudocomplemented semi-Heyting algebras generated by chains and the variety generated by D2, the De Morgan expansion of the four element Boolean Heyting algebra. Toward this end, we first introduce the variety DQDLNSH of dually quasi-De Morgan linear semi-Heyting algebras defined by the linearity axiom and...

In this work we develop a categorical duality for certain classes of manysorted algebras, called many-sorted lattices because each sort admits a structure of distributive lattice. This duality is strongly based on the Priestley duality for distributive lattices developed in [3] and [4] and on the representation of many sorted lattices with operators given by Sofronie-Stokkermans in [6]. In this...

We consider the families L of propositional superintuitionistic logics (s.i.l.) andNE(K) of normal modal logics (n.m.l.). It is well known that there is a duality between L and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also NE(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig’s int...

In this work we solve the following problem: Characterize the subclasses of implication algebras that can be axiomatized by sentences of the form 89! ^ p = q. In the process we obtain a representation result for …nite implication algebras, and as a by-product of our solution a number of interesting classes of implication algebras arise. We also obtain a characterization of the congruence permut...

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the→-free reducts of Heyting algebras while the variety of implicative semilattices by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that ge...

By a tournament we mean a directed graph (T,→) such that whenever x, y are two distinct elements of T , then precisely one of the two cases, either x→ y or y → x, takes place. There is a one-to-one correspondence between tournaments and commutative groupoids satisfying ab ∈ {a, b} for all a and b: set ab = a if and only if a→ b. This makes it possible to identify tournaments with their correspo...