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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
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##### 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
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