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Published in:
2017 | OriginalPaper | Chapter
6. On the Theory of Partial Transformations
Authors : Christopher D. Hollings, Mark V. Lawson
Published in: Wagner’s Theory of Generalised Heaps
Publisher: Springer International Publishing
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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.