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Representation Theorems for Lattice-ordered Modal Algebras and their Axiomatic Extensions

Radzikowska A. M.

Fundamenta Informaticae

2016

Vol. 144, nr 2

183--203

In this paper we present relational representation theorems for lattice-based modal algebras and their axiomatic extensions taking into account well-known schemas of modal logics. The underlying algebraic structures are bounded, not necessarily distributive lattices. Our approach is based on the Urquhart’s result for non-distributive lattices and Allwein and Dunn developments for algebras of liner logics.

Duality via Truth for Information Algebras Based on De Morgan Lattices

Radzikowska A. M.

Fundamenta Informaticae

2016

Vol. 144, nr 1

45--75

Duality via truth is a kind of correspondence between a class of algebras and a class of relational systems (frames). These classes are viewed as two kinds of semantics for some logic: algebraic semantics and Kripke-style semantics, respectively. Having defined the notion of truth, the duality principle states that a sequent/formula is true in one semantics if and only if it is true in the other one. In consequence, the algebras and their corresponding frames express equivalent notion of truth. In this paper we develop duality via truth between modal algebras based on De Morgan lattices and their corresponding frames. Some axiomatic extensions of these algebras are considered. Basing on these results we present duality via truth between some classes of latticebased information algebras and their corresponding frames.

Algebraic Properties of Generalized Rough Sets

Kondo M.

Fundamenta Informaticae

2009

Vol. 94, nr 1

41-48

In this paper we consider the problem whether for a map φ : A(B) →B there is a relation R on A(B) such that properties of a map φ* : B → B are reflected to R. This is a converse problem that J.Järvinen proved in [5]. We give an affirmative answer to the problem. Moreover we give equational bases which characterize properties of the map &phi:. This means that the classes of some types of algebras form varieties. Hence, there are full correspondence between properties of relations and varieties of modal algebras.