×

zbMATH — the first resource for mathematics

On relations. (English) Zbl 0682.04001
By a relation (in the general sense) R the author understands a set of mappings \(R\subseteq G^ H\), where G is the carrier set and H the index set of R. The aim of this paper is to lay foundations for the study of relations in the general sense. In § 1 (Operations with relations) and § 2 the author gives a lot of definitions which generalize the corresponding notions for binary relations, diagonal relation, composition, reflexive, symmetric, transitive etc. A typical concept is the following: If \(| H| \geq 2\) and if H is the union of three disjoint subsets \(K_ 1\), \(K_ 2\), \(K_ 3\) with \(| K_ 1| =| K_ 2| >0\), then \({\mathfrak K}=\{K_ 1,K_ 2,K_ 3\}\) is called a b-decomposition of H. Then the relation \(\{f\in G^ H|\) \(f(K_ 1)=f(K_ 2)\}\) is called diagonal with regard to \({\mathfrak K}\). § 3 deals with hulls of relations, § 4 with projections of relations. Of course, many things which are known for special cases can be transferred to the general case, and a great deal of the paper is concerned with the execution of verifications. Among others the cyclic order relations which were introduced by V. Novák [Czech. Math. J. 32(107), 460-473 (1982; Zbl 0515.06003)] are encompassed by the author’s general definition.
Reviewer: E.Harzheim

MSC:
03E20 Other classical set theory (including functions, relations, and set algebra)
06A06 Partial orders, general
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] E. Čech: Bodové množiny. (Point sets). Academia Praha, 1966.
[2] I. Chajda, V. Novák: On extensions of cyclic orders. Čas. Pěst. Mat. 110 (1985), 116-121. · Zbl 0575.06001
[3] V. Novák: Cyclically ordered sets. Czech. Math. Journ. 32 (1982), 460-473. · Zbl 0515.06003
[4] Ju. A. Schreider: Equality, resemblance and order. Mir Publishers, Moscow, 1975.
[5] J. Šlapal: On relations of type \(\alpha\). Z. Math. Logik Grundlagen Math. 34 (1988), 563 - 573. · Zbl 0668.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.