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We show that all Sugihara monoids can be represented as algebras of binary relations, with the monoid operation given by relational composition. Moreover, the binary relations are weakening relations. The first step is to obtain an explicit relational representation of all finite odd Sugihara chains. Our construction mimics that of Maddux (2010), where a relational representation of the finite even Sugihara chains is given. We define the class of representable Sugihara monoids as those which can be represented as reducts of distributive involutive FL-algebras of binary relations. We then show that the class of representable distributive involutive FL-algebras is closed under ultraproducts. This fact is used to demonstrate that the two infinite Sugihara monoids that generate the quasivariety are also representable. From this it follows that all Sugihara monoids are representable., Comment: 29 pages, 1 figure

We give a definition of representability for distributive quasi relation algebras (DqRAs). These algebras are a generalisation of relation algebras and were first described by Galatos and Jipsen (2013). Our definition uses a construction that starts with a poset. The algebra is concretely constructed as the lattice of upsets of a partially ordered equivalence relation. The key to defining the three negation-like unary operations is to impose certain symmetry requirements on the partial order. Our definition of representable distributive quasi relation algebras is easily seen to be a generalisation of the definition of representable relations algebras by Jonsson and Tarski (1948). We give examples of representable DqRAs and give a necessary condition for an algebra to be finitely representable. We leave open the questions of whether every DqRA is representable, and also whether the class of representable DqRAs forms a variety. Moreover, our definition provides many other opportunities for investigations in the spirit of those carried out for representable relation algebras.

We discuss an ongoing line of research in the relational (non topological) semantics of non-distributive logics. The developments we consider are technically rooted in dual characterization results and insights from unified correspondence theory. However, they also have broader, conceptual ramifications for the intuitive meaning of non-distributive logics, which we explore., Comment: 36 pages

This paper continues the investigation of the logic of competing theories, be they scientific, social, political etc. We introduce a many-valued, multi-type modal language which we endow with relational semantics based on enriched reflexive graphs, inspired by Plo\v{s}\v{c}ica's representation of general lattices. We axiomatize the resulting many-valued, non-distributive modal logic of these structures and prove a completeness theorem. We illustrate the application of this logic through a case study in which we model competition among interacting political promises and social demands within an arena of political parties social groups., Comment: 24 pages

We introduce a complete many-valued semantics for basic normal lattice-based modal logic. This relational semantics is grounded on many-valued formal contexts from Formal Concept Analysis. We discuss an interpretation and possible applications of this logical framework for categorization theory to the formal analysis of multi-market competition., Comment: 24 pages

We introduce hybrid algebras as algebraic semantics for hybrid languages with nominals and, possibly, the satisfaction operator. We establish a duality between hybrid algebras and the descriptive two-sorted general frames of Ten Cate. We show that all axiomatic extensions of the basic hybrid logics, with or without the satisfaction operator, are complete with respect to their classes of hybrid algebras. Moreover, we show that by adding the usual non-orthodox rules to these logics, they become complete with respect to their classes of permeated hybrid algebras, corresponding to strongly descriptive two-sorted general frames., Comment: 25 pages