Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in  that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in . Other authors have extended this result to the cases of (...) k-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature. (shrink)
We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.
We present a systematic study of join-extensions and join-completions of partially ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from properties of the Dedekind–MacNeille completion to the proof of the finite embeddability property for a number of varieties of lattice-ordered algebras.
The central result of this paper provides a simple equational basis for the join, IRLLG, of the variety LG of lattice-ordered groups (-groups) and the variety IRL of integral residuated lattices. It follows from known facts in universal algebra that IRLLG=IRL×LG. In the process of deriving our result, we will obtain simple axiomatic bases for other products of classes of residuated structures, including the class IRL×s LG, consisting of all semi-direct products of members of IRL by members of LG. We (...) conclude the paper by presenting a general method for constructing such semi-direct products, including wreath products. (shrink)
In this paper, we explore the effects of certain forbidden substructure conditions on preordered sets. In particular, we characterize in terms of these conditions those preordered sets which can be represented as the supremum of a well-ordered ascending chain of lowersets whose members are constructed by means of alternating applications of disjoint union and ordinal sums with chains. These decompositions are examples of ordinal decompositions in relatively normal lattices as introduced by Snodgrass, Tsinakis, and Hart. We conclude the paper with (...) an application to information systems. (shrink)
The starting point of the present study is the interpretation of intuitionistic linear logic in Petri nets proposed by U. Engberg and G. Winskel. We show that several categories of order algebras provide equivalent interpretations of this logic, and identify the category of the so called strongly coherent quantales arising in these interpretations. The equivalence of the interpretations is intimately related to the categorical facts that the aforementioned categories are connected with each other via adjunctions, and the compositions of the (...) connecting functors with co-domain the category of strongly coherent quantales are dense. In particular, each quantale canonically induces a Petri net, and this association gives rise to an adjunction between the category of quantales and a category whose objects are all Petri nets. (shrink)