Geodesic
Transformation semigroups preserving a~binary relation
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 129-135.

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We investigate the semigroups of full and partial transformations of a set $X$ which preserve a binary relation $\sigma$ defined on $X$. We consider in detail the case where $\sigma$ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of $\sigma$ for the semigroup of binary relations. It is proved that subsets of $\mathrm B(X)$ preserving $\sigma$ are semigroups in each case. We give the condition of regularity of $\mathrm B_\sigma(X)$ in the case where $\sigma(X)$ is a quasi-order. 
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I. B. Kozhukhov; V. A. Yaroshevich. Transformation semigroups preserving a~binary relation. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 129-135. http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a9/

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