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1

Duality for Quasilattices and Galois Connections

Romanowska A. B., Smith J. D. H.

Fundamenta Informaticae

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2017

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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

One-sided quantum quasigroups and loops

Smith J. D. H.

Demonstratio Mathematica

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2015

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Vol. 48, nr 4

620--636

EN

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. This paper examines their one-sided analogues, which are not self-dual. Just as quantum quasigroups are the “quantum” version of quasigroups, so one-sided quantum quasigroups are the “quantum” version of left or right quasigroups.

3

Modes, modals, and barycentric algebras: a brief survey and an additivity theorem

Smith J. D. H.

Demonstratio Mathematica

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2011

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Vol. 44, nr 3

571-587

EN

Modes are idempotent and entropic algebras.Modals are both join semi lattices and modes,where the mode structure distributes over the join.Barycentric algebras are equipped with binary operations from the open unit interval,satisfying idempo tence,skew commutativity,and skew associativity.The article aims to give a brief survey of these structures and some of their applications.Special attention is devoted to hierar chical statistical mechanics and the modeling of complex systems.An additivity theorem for the entropy of independent combinations of systems is proved.

4

Abstract Barycentric Algebras

Romanowska A.B., Smith J.D.H., Orłowska E.

Fundamenta Informaticae

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2007

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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.