On relations.

*(English)*Zbl 0682.04001By 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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\textit{J. Šlapal}, Czech. Math. J. 39(114), No. 2, 198--214 (1989; Zbl 0682.04001)

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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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