On the Theory of Partial Transformations

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 𝔓 ( A × A ) $$\mathfrak{P}(A \times A)$$ . 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 𝔓 ( A × A ) $$\mathfrak{P}(A \times A)$$ symmetric if it is closed under this involution; he identified the most important of the symmetric subsets of 𝔓 ( A × A ) $$\mathfrak{P}(A \times A)$$ as being 𝔐 ( A × A ) $$\mathfrak{M}(A \times A)$$ , the collection of all one-to-one partial transformations of A. He proved that within 𝔐 ( A × A ) $$\mathfrak{M}(A \times A)$$ both the order relation and the involution may be expressed in terms of composition of transformations. Wagner went on to relate 𝔐 ( B × B ) $$\mathfrak{M}(B \times B)$$ to the group 𝔊 ( A × A ) $$\mathfrak{G}(A \times A)$$ of all bijections of A, for some A ⊃ B.