×

Functions redefined. (English) Zbl 0919.04001

Let \(R\subseteq D\times C\) be a relation with domain \(D\) and codomain \(C\). The author defines left-totality: for every \(x\in D\) there is at least one \(y\in C\) such that \(xRy\), right-uniqueness: for every \(x\in D\) there is at most one \(y\in C\) such that \(xRy\), right-totality: for every \(y\in C\) there is at least one \(x\in D\) such that \(xRy\), left-uniqueness: for every \(y\in C\) there is at most one \(x\in D\) such that \(xRy\). In a natural way for a relation \(R\) right- and left-cancellability are defined. (The reviewer would suggest to replace \(A\) by \(D\) in line 7 from below and \(A\) by \(C\) in line 6 from below on page 633.) Further, the author defines \({\mathbf R}(A):= \{y\in C\mid\) for some \(x\in A\), \(xRy\}\), and analogously \({\mathbf R}^{\leftarrow}\). The author describes relationships between the above notions, for instance: \(R\subseteq D\times C\) is a function iff \({\mathbf R}\) and \({\mathbf R}^{\leftarrow}\) establish a covariant Galois connection between \(({\mathfrak P}(D),\subseteq)\) and \(({\mathfrak P}(C),\subseteq)\), where \({\mathfrak P}\) denotes the power set.
Reviewer: E.Harzheim (Köln)

MSC:

03E20 Other classical set theory (including functions, relations, and set algebra)
PDFBibTeX XMLCite
Full Text: DOI