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Content available remote Duality for Quasilattices and Galois Connections
100%
Romanowska A. B. , Smith J. D. H.
Fundamenta Informaticae
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2017
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tom Vol. 156, nr 3/4
331--359
EN
The primary goal of the paper is to establish a duality for quasilattices. The main ingredients are duality for semilattices and their representations, the structural analysis of quasilattices as Płonka sums of lattices, and the duality for lattices developed by Hartonas and Dunn. Lattice duality treats the identity function on a lattice as a Galois connection between its meet and join semilattice reducts, and then invokes a duality between Galois connections and polarities. A second goal of the paper is a further examination of this latter duality, using the concept of a pairing to provide an algebraic equivalent to the relational structure of a polarity.
2
Content available remote Embedding modes into semimodules, part I
100%
Pilitowska A. , Romanowska A. B.
Demonstratio Mathematica
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2011
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tom Vol. 44, nr 3
523-534
EN
By recent results of M. Stronkowski, it is known that not all modes embed as subreducts into semimodules over commutative unital semirings. Related to this problem is the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provide a general construction of such semirings, along with basic examples and some general properties. The second part of the paper will deal with applications of the general construction to some selected varieties of modes, and will provide a description of semirings determining varieties of semimodules having algebras from these varieties as idempotent subreducts.
3
Content available remote Embedding modes into semimodules, part II
100%
Pilitowska A. , Romanowska A. B.
Demonstratio Mathematica
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2011
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tom Vol. 44, nr 4
781-790
EN
The first part of this paper specified the semi-affinization semiring of a mode variety as the universal scalar semiring for semimodules whose idempotent subreducts lie in the given variety of modes. The current part of the paper focusses on some selected varieties of modes (affine spaces, barycentric algebras, semilattice modes), and computes the semi-affinization semirings of these varieties.
4
Content available remote Embedding modes into semimodules, part III
100%
Pilitowska A. , Romanowska A. B.
Demonstratio Mathematica
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2011
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tom Vol. 44, nr 4
791-800
EN
In the first part of this paper, we considered the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provided a general construction of such semirings, along with basic examples and some general properties. In the second part of the paper we discussed some selected varieties of modes, in particular, varieties of affine spaces, varieties of barycentric algebras and varieties of semilattice modes, and described the semirings determining their semi-linearizations, the varieties of semimodules having these algebras as idempotent subreducts. The third part is devoted to varieties of differential groupoids and more general differential modes, and provides the semirings of the semi-linearizations of these varieties.
5
Content available remote Abstract Barycentric Algebras
80%
Romanowska A. B. , Smith J. D. H. , Orłowska E.
Fundamenta Informaticae
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2007
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tom Vol. 81, nr 1-3
257-273
EN
This paper presents a new approach to the study of (real) barycentric algebras, in particular convex subsets of real affine spaces. Barycentric algebras are cast in the setting of two-sorted algebras. The real unit interval indexing the set of basic operations of a barycentric algebra is replaced by an LP-algebra, the algebra of ukasiewicz Product Logic. This allows one to define barycentric algebras abstractly, independently of the choice of the unit real interval. It reveals an unexpected connection between barycentric algebras and (fuzzy) logic. The new class of abstract barycentric algebras incorporates barycentric algebras over any linearly ordered field, the B-sets of G. M. Bergman, and E. G. Manes' if-then-else algebras over Boolean algebras.
6
Content available remote Embedding sums of cancellative modes into functorial sums
80%
Romanowska A. B. , Stronkowski M. , Zamojska-Dzienio A.
Demonstratio Mathematica
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2011
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tom Vol. 44, nr 3
557-569
EN
The paper discusses a representation of modes (idempotent and entropic algebras)as subalgebras of so called functorial sums of cancellative algebras.We show that each mode that has a homomorphism onto an algebra satisfying a certain additional condition,with corresponding cancellative congruence classes,embeds into a functorial sum of cancellative algebras.We also discuss typical applications of this result.
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