The endotopism semigroups of an equivalence relation. (Q2921147)

Let \(X\) be a non-empty set and \(\rho\) a binary relation on \(X\). An \textit{endotopism} of \(\rho\) is a pair \((\varphi,\psi)\) of transformations of \(X\) such that \((x,y)\in\rho\) implies \((\varphi(x),\psi(y))\in\rho\) for all \(x,y\in X\).NEWLINENEWLINE The paper under review investigates six different types of semigroups consisting of some endotopisms of a fixed binary relation. The main results include: necessary and sufficient conditions for existence of all these different types of endotopisms and conditions which guarantee regularity and coregularity for all these semigroups. The authors introduce the notion of endotype for a binary relation, which, roughly speaking, encodes cardinalities for all these six semigroups of endotopisms for this particular binary relation, and calculate this endotype for all binary relations.