Binary and ternary relations. (English) Zbl 0772.04005

Novák, Vítězslav; Novotný, Miroslav

Math. Bohem. 117, No. 3, 283-292 (1992).

Let \(G\) be a set, \(\rho\) a binary relation on \(G\). Further, let \(r\) be a binary relation on the set \(\rho\) with the property \(\alpha=(x,y)\in\rho\), \(\beta=(z,u)\in\rho\), \((\alpha,\beta)\in r\Rightarrow y=z\). Then \(r\) is called a binding relation on \(\rho\), and \((G,\rho,r)\) is called a double binary structure. A ternary structure is a pair \((G,t)\) where \(t\subseteq G^ 3\) holds. Let \(\mathcal B\) be the class of all double binary structures and \(\mathcal T\) the class of all ternary structures. The authors define two operators \({\mathfrak T}: {\mathcal B}\to{\mathcal T}\) and \({\mathfrak B}: {\mathcal T}\to{\mathcal B}\) as follows: Let \({\mathfrak G}=(G,\rho,r)\in{\mathcal B}\). Then define a ternary relation \(t\) on \(G\) by \((x,y,z)\in t\Leftrightarrow(x,y)=\alpha\in\rho\), \((y,z)=\beta\in\rho\) and \((\alpha,\beta)\in r\). Then \({\mathfrak T}({\mathfrak G}):=(G,t)\in{\mathcal T}\). The operator \({\mathfrak B}\) is so defined that \(({\mathfrak T}\circ{\mathfrak B})({\mathfrak G})={\mathfrak G}\) for \({\mathfrak G}\in {\mathcal T}\). Then the authors define several properties for ternary structures (symmetric, asymmetric, cyclic, transitive, weakly transitive, cyclically ordered) and for double binary structures (inversely symmetric, inversely asymmetric, reversely transitive, transferable) and establish several theorems of the following form: A ternary (resp. double binary) structure \(\mathfrak G\) has a property of one of the above lists iff \({\mathfrak B}({\mathfrak G})\) (resp. \({\mathfrak T}({\mathfrak G}))\) has a property of the other list.

Reviewer: E.Harzheim (Köln)

MSC:

03E20	Other classical set theory (including functions, relations, and set algebra)
06A06	Partial orders, general

Keywords:

cyclically ordered set; binary relation; binding relation; double binary structure; ternary structure; ternary relation