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1

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

Radzikowska A. M.

Fundamenta Informaticae

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2016

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Vol. 144, nr 2

183--203

EN

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.

2

Duality via Truth for Information Algebras Based on De Morgan Lattices

Radzikowska A. M.

Fundamenta Informaticae

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2016

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Vol. 144, nr 1

45--75

EN

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.

3

Discrete Duality for Rough Relation Algebras

Düntsch I., Orłowska E.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

35--47

EN

Rough relation algebras are a generalization of relation algebras such that the underlying lattice structure is a regular double Stone algebra. Standard models are algebras of rough relations. A discrete duality is a relationship between classes of algebras and classes of relational systems (frames). In this paper we prove a discrete duality for a class of rough relation algebras and a class of frames.

4

Duality theory for state-constrained control problems governed by a first order PDE system

Pickenhain S.

Control and Cybernetics

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2002

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Vol. 31, no 3

833-845

EN

In this paper we prove weak and strong duality results for optimal control problems with multiple integrals, first-order partial differential equations and state constraints. We formulate conditions under which the sequence of canonical variables [y^epsilon] in the [epsilon]-maximum principle, proved in Pickenhain and Wagner (2000), form a maximizing sequence in the dual problem.

5

General method of relaxation. [Pt. 1], Functionals defined on BD(Omega) space

Bojarski J.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 3

289-301

EN

The aim of this paper is to find the lower semicontinuous regularization of a functional of displacement energy, with a constrains on the boundary of Omega. This functional describes the elasto-perfectly plastic energy of a solid made of a nonhomogeneous (or homogeneous) Hencky material. In this contribution we prove that the mentioned above regularization is equal to the relaxation found in [4], i.e. B** = B#*.