The aim of this paper is to show that even if the natural algebraic semantic for modal (normal) logic is modal algebra, the more general class of subordination algebras (roughly speaking, the non symmetric contact algebras) is adequate too - so leading to completeness results. This motivates for an algebraic (in the sense of universal algebra) study of those relational structures that are subordinate algebras.

Subordination algebras in modal logic

In this paper we consider the notion of subordination on distributive lattices, equivalent to that of quasi-modal operator for distributive lattices introduced by Castro and Celani in 2004 We provide topological dualities for categories of distributive lattices with a subordination and then for some categories of distributive lattices with two subordinations, structures that we name bi-subordination lattices We investigate three classes of bi-subordination lattices In particular that of positive bi-subordination lattices

Bounded Distributive Lattices with Two Subordinations

In this paper we will give suitable notions of Amalgamation and Super-amalgamation properties for the class of quasi-modal algebras intro- duced by the author in his paper Quasi-Modal algebras.

Amalgamation property in quasi-modal algebras

ABSTRACT In this work, we study the relational representation of the class of Tarski algebras endowed with a subordination, called subordination Tarski algebras. These structures were introduced in a previous paper as a generalisation of subordination Boolean algebras. We define the subordination Tarski spaces as topological spaces with a fixed basis endowed with a closed relation. We prove that there exist categorical dualities between categories whose objects are subordination Tarski algebras and categories whose objects are subordination Tarski spaces. These results extend the known dualities for modal algebras. Finally, we are going to characterise some algebraic conditions written in the quasi-modal language by means of first-order conditions.

Relational representation for subordination Tarski algebras

Subordinations on Bounded Distributive Lattices

We introduce the class of bounded distributive lattices with two operators, Δ and ∇, the first between the lattice and the set of its ideals, and the second between the lattice and the set of its filters. The results presented can be understood as a generalization of the results obtained by S. Celani.

Quasi-Modal Lattices

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Quasi-Modal Lattices