A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set R of finitary algebraic relations yields a duality on a class of algebras A = ISP(M), those subsets R′ of R which yield optimal dualities are characterised. Further, the manner in which the relations in R are constructed from those in R′ is revealed in...

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective canonical fo...

The present paper surveys the history of algebraic represetations of complete directed graphs, known in graph theory as tournaments, or equivalently, relational structures with a trichotomous binary relation. Essentially, two kinds of algebraizations of tournaments were studied in the literature: algebras with one binary operation (called groupoids of tournaments) and algebras with two binary o...

An algebra with two binary operations · and + that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice satisfying Birkhoff’s equation x · (x + y) = x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation + and one for the operation ·. A bisemilattice for which each of these semilattices is...

Let Var(Mplan) denote the variety generated by the class Mplan of planar modular lattices. In 1977, based on his structural investigations, R. Freese proved that Var(Mplan) has continuumly many subvarieties. The present paper provides a new approach to this result utilizing lattice identities. We also show that each subvariety of Var(Mplan) is generated by its planar (subdirectly irreducible) m...

A hoop is a naturally ordered pocrim (i.e., a partially ordered com-mutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasivariety, by its nite members.

Recently, the first two authors characterized in Di Nola and Dvurečenskij (2009) [1] subdirectly irreducible state-morphism MV-algebras. Unfortunately, the main theorem (Theorem 5.4(ii)) has a gap in the proof of Claim 10, as the example below shows. We now present a correct characterization and its correct proof. © 2010 Elsevier B.V. All rights reserved.