82 research outputs found
Properties and special filters of pseudocomplemented posets
Investigating the structure of pseudocomplemented lattices started ninety
years ago with papers by V. Glivenko, G. Birkhoff and O. Frink and this
structure was essentially developed by G. Gr\"atzer. In recent years, some
special filters in pseudocomplemented and Stone lattices have been studied by
M. Sambasiva Rao. However, in some applications, in particular in non-classical
logics with unsharp logical connectives, pseudocomplemented posets instead of
lattices are used. This motivated us to develop an algebraic theory of
pseudocomplemented posets, i.e. we derive identities and inequalities holding
in such posets and we use them in order to characterize the so-called Stone
posets. Then we adopt several concepts of special filters and we investigate
their properties in pseudocomplemented posets. Moreover, we show how properties
of these filters influence algebraic properties of the underlying
pseudocomplemented posets
Information completeness in Nelson algebras of rough sets induced by quasiorders
In this paper, we give an algebraic completeness theorem for constructive
logic with strong negation in terms of finite rough set-based Nelson algebras
determined by quasiorders. We show how for a quasiorder , its rough
set-based Nelson algebra can be obtained by applying the well-known
construction by Sendlewski. We prove that if the set of all -closed
elements, which may be viewed as the set of completely defined objects, is
cofinal, then the rough set-based Nelson algebra determined by a quasiorder
forms an effective lattice, that is, an algebraic model of the logic ,
which is characterised by a modal operator grasping the notion of "to be
classically valid". We present a necessary and sufficient condition under which
a Nelson algebra is isomorphic to a rough set-based effective lattice
determined by a quasiorder.Comment: 15 page
Rough Sets Determined by Quasiorders
In this paper, the ordered set of rough sets determined by a quasiorder
relation is investigated. We prove that this ordered set is a complete,
completely distributive lattice. We show that on this lattice can be defined
three different kinds of complementation operations, and we describe its
completely join-irreducible elements. We also characterize the case in which
this lattice is a Stone lattice. Our results generalize some results of J.
Pomykala and J. A. Pomykala (1988) and M. Gehrke and E. Walker (1992) in case
is an equivalence.Comment: 18 pages, major revisio
Modal Pseudocomplemented De Morgan Algebras
summary:Modal pseudocomplemented De Morgan algebras (or -algebras for short) are investigated in this paper. This new equational class of algebras was introduced by A. V. Figallo and P. Landini ([Figallo, A. V., Landini, P.: Notes on -valued modal algebras Preprints del Instituto de Ciencias Básicas, Univ. Nac. de San Juan 1 (1990), 28–37.]) and they constitute a proper subvariety of the variety of all pseudocomplemented De Morgan algebras satisfying . Firstly, a topological duality for these algebras is described and a characterization of -congruences in terms of special subsets of the associated space is shown. As a consequence, the subdirectly irreducible algebras are determined. Furthermore, from the above results on the -congruences, the principal ones are described. In addition, it is proved that the variety of -algebras is a discriminator variety and finally, the ternary discriminator polynomial is described
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