Английская Википедия:Demonic composition

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In mathematics, demonic composition is an operation on binary relations that is similar to the ordinary composition of relations but is robust to refinement of the relations into (partial) functions or injective relations.

Unlike ordinary composition of relations, demonic composition is not associative.

Definition

Suppose <math>R</math> is a binary relation between <math>X</math> and <math>Y</math> and <math>S</math> is a relation between <math>Y</math> and <math>Z.</math> Their Шаблон:Visible anchor <math>R \textbf{;}^{\to} S</math> is a relation between <math>X</math> and <math>Z.</math> Its graph is defined as <math display=block>\{(x, z) \ : \ x \mathrel{(S \circ R)} z \text{ and for all } y \in Y \ (x \mathrel{R} y \text{ implies } y \mathrel{S} z)\}.</math>

Conversely, their Шаблон:Visible anchor <math>R \textbf{;}^{\leftarrow} S</math> is defined by <math display=block>\{(x, z) \  : \ x \mathrel{(S \circ R)} z \text{ and for all } y \in Y \ (y \mathrel{S} z \text{ implies } x \mathrel{R} y)\}.</math>

References

• Шаблон:Citation.