This paper studies finitely generated quasivarieties of Sugihara algebras. These quasivarieties provide complete algebraic semantics for certain propositional logics associated with the relevant logic R-mingle. The motivation for the paper comes from the study of admissible rules. Recent earlier work by the present authors, jointly with Freisberg and Metcalfe, laid the theoretical foundations for a feasible approach to this problem for a range of logics-the Test Spaces Method. The method, based on natural duality theory, provides an algorithm to obtain the algebra of minimum size on which admissibility of sets of rules can be tested. (In the most general case a set of such algebras may be needed rather than just one.) The method enables us to identify this 'admissibility algebra' for each quasivariety of Sugihara algebras which is generated by an algebra whose underlying lattice is a finite chain. To achieve our goals, it was first necessary to develop a (strong) duality for each of these quasivarieties. The dualities promise also to also provide a valuable new tool for studying the structure of Sugihara algebras more widely.

This paper presents a systematic study of coproducts. This is carried out principally, but not exclusively, for finitely generated quasivarieties A that admit a (term) reduct in the variety D of bounded distributive lattices. In this setting we present necessary and sufficient conditions on A for the forgetful functor U A from A to D to preserve coproducts. We also investigate the possible behaviours of U A as regards coproducts in A under weaker assumptions. Depending on the properties exhibited by the functor, different procedures are then available for describing these coproducts. We classify a selection of well-known varieties within our scheme, thereby unifying earlier results and obtaining some new ones.The paper's methodology draws heavily on duality theory. We use Priestley duality as a tool and our descriptions of coproducts are given in terms of this duality. We also exploit natural duality theory, specifically multisorted piggyback dualities, in our analysis of the behaviour of the forgetful functor into D. In the opposite direction, we reveal that the type of natural duality that the class A can possess is governed by properties of coproducts in A and the way in which the classes A and U A (A) interact. Take, as above, A to be a D-based and finitely generated quasivariety. In this section we outline the results on which our analysis of coproducts in A will rest. Underlying our strategy throughout will be duality theory, in two forms. Our main results are obtained by operating with these two forms in tandem, and toggling between them. First we briefly discuss the role played by Priestley duality as a platform on top of which dualities for classes of Dbased algebras can be built. For many such classes, this technique gives a valuable, set-based, representation theory. However, although isolated results exist in the literature for particular classes, such representations do not lend themselves well in general to the description of coproducts. Secondly, we venture a little way into the theory of (multisorted) natural dualities as it applies to finitely generated D-based quasivarieties, since such dualities, in common with Priestley duality itself, have good categorical properties. The key theorem on which we shall rely is the Multisorted Piggyback Duality Theorem. We present the bare minimum of the theory necessary to state it (Theorem 2.1). It allows us immediate access to coproducts, via dual structures which are (multisorted) cartesian products. This would be of little assistance were it not possible directly to retrieve the coproduct, or at least the Priestley dual space of its D-reduct, from the dual structure. Theorem 2.3 provides exactly the translation tool we need. (We remark that the usefulness of Theorem 2.3 potentially extends well beyond the applications to coproducts given in this paper.)Priestley duality for (bounded) distributive lattices establishes a dual equivalence between the category D and the category P of Priestley spaces, that is, compact totally order-disconnected spaces with con...

Abstract. This paper provides a fresh perspective on the representation of distributive bilattices and of related varieties. The techniques of natural duality are employed to give, economically and in a uniform way, categories of structures dually equivalent to these varieties. We relate our dualities to the product representations for bilattices and to pre-existing dual representations by a simple translation process which is an instance of a more general mechanism for connecting dualities based on Priestley duality to natural dualities. Our approach gives us access to descriptions of algebraic/categorical properties of bilattices and also reveals how 'truth' and 'knowledge' may be seen as dual notions.

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.

Abstract. An ℓ-group G is an abelian group equipped with a translation invariant lattice order. Baker and Beynon proved that G is finitely generated projective iff it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e., an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (á la Whitehead) will be combined in this paper with the W lodarczyk-Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.