Editing Binary relation (section)

== Mathematical heaps ==
{{main|Heap (mathematics)}}
Given two sets <math>A</math> and <math>B</math>, the set of binary relations between them <math>\mathcal{B}(A,B)</math> can be equipped with a [[ternary operation]] <math>[a, b, c] = a b^\textsf{T} c</math> where <math>b^\mathsf{T}</math> denotes the [[converse relation]] of <math>b</math>. In 1953 [[Viktor Wagner]] used properties of this ternary operation to define [[Semiheap|semiheaps]], heaps, and generalized heaps.<ref>[[Viktor Wagner]] (1953) "The theory of generalised heaps and generalised groups", [[Matematicheskii Sbornik]] 32(74): 545 to 632 {{mr|id=0059267}}</ref><ref>C.D. Hollings & M.V. Lawson (2017) ''Wagner's Theory of Generalised Heaps'', [[Springer books]] {{ISBN|978-3-319-63620-7}} {{mr|id=3729305}}</ref> The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:
{{Blockquote
|text=There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other. Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) between ''different'' sets <math>A</math> and <math>B</math>, while the various types of semigroups appear in the case where <math>A = B</math>.
|author=Christopher Hollings
|title="Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"<ref>Christopher Hollings (2014) ''Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups'', page 265, History of Mathematics 41, [[American Mathematical Society]] {{ISBN|978-1-4704-1493-1}}</ref>
}}