On the Theory of Partial Transformations

Abstract

In this short communication to the Academy of Sciences, Wagner confined his attention to binary relations between the elements of a single set A, denoting the collection of all such relations by $\mathfrak{P}(A \times A)$. He noted that the latter forms a semigroup under composition of binary relations; this semigroup is ordered by set inclusion and, moreover, has a natural involution, via which any binary relation is sent to its inverse. Wagner called a subset of $\mathfrak{P}(A \times A)$ symmetric if it is closed under this involution; he identified the most important of the symmetric subsets of $\mathfrak{P}(A \times A)$ as being $\mathfrak{M}(A \times A)$, the collection of all one-to-one partial transformations of A. He proved that within $\mathfrak{M}(A \times A)$ both the order relation and the involution may be expressed in terms of composition of transformations. Wagner went on to relate $\mathfrak{M}(B \times B)$ to the group $\mathfrak{G}(A \times A)$ of all bijections of A, for some A ⊃ B.