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1

Decomposition of Relations and Concept Lattices

Berghammer R., Winter M.

Fundamenta Informaticae

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2013

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Vol. 126, nr 1

37--82

EN

We introduce the decomposition of an arbitrary relation into a sequential composition of three relations, viz. of a mapping with a partial order and then the transpose of a mapping. After presenting some basic properties, we investigate the specific classes of junkfree, irreducible and minimal decompositions and show that for all relations a minimal decomposition exists. We also study decompositions with regard to DedekindMacNeille completions and concept lattices. These constructions are closely related to decompositions of relations. In our setting the fundamental theorem of concept lattices states that concept lattices are minimal-complete decompositions and all such decompositions are isomorphic. As a further main result we prove that the cutDedekindMacNeille completion of the order that belongs to the minimal decomposition of a relation is isomorphic to the concept lattice of that relation. Instead of considering binary relations on sets, we will work point-free within the general framework of allegories. This complement-free approach implies that the results of the paper can be applied to all models of these algebraic structures, including, for instance, lattice-valued fuzzy relations.

2

Centralizer clones are preserved by category equivalences

Supaporn W.

Demonstratio Mathematica

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2011

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Vol. 44, nr 4

819-822

EN

In this paper it will be proved that if a clone C is category equivalent to a centralizer clone then C is a centralizer clone.

3

Rough Relation Algebras Revisited

Düntsch I., Winter M.

Fundamenta Informaticae

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2006

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Vol. 74, nr 2,3

283-300

EN

Rough relation algebras arise from Pawlak's information systems by considering as object ordered pairs on a fixed set X. Thus, the subsets to be approximated are binary relations over X, and hence, we have at our disposal not only the set theoretic operations, but also the relational operators ;, ˇ , and the identity relation 1˘. In the present paper, which is a continuation of [6], we further investigate the structure of abstract rough relation algebras.

4

Representability of pairing relation algebras depends on your ontology

Kurucz A., Németi I.

Fundamenta Informaticae

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2000

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Vol. 44, Nr 4

397-420

EN

We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are pairing (or projection) algebras of algebraic logic and fork algebras of computer science. It is proved by Sain and Németi that there is no `strong' representation theorem for all abstract pairing algebras in most set theories including ZFC as well as most non-well-founded set theories. Such a `strong' representation theorem would state that every abstract pairing algebra is isomorphic to a set relation algebra having projection elements which are defined with the help of the real (set theoretic) pairing function. Here we show that, by choosing an appropriate (non-well-founded) set theory as our metatheory, pairing algebras and fork algebras admit such `strong' representation theorems.