Semigroups of tolerance relations (Q1820876)

A binary relation \(\rho\) between elements of a set A called partly reflexive if \((a_ 1,a_ 2)\in \rho \Rightarrow (a_ 1,a_ 1)\in \rho\) and \((a_ 2,a_ 2)\in \rho\) for all \(a_ 1,a_ 2\in A\). It is called a (partial) tolerance relation if it is both symmetric and (partly) reflexive; it is called a multipermutation if, for every \(a\in A\), there exist \(a_ 1,a_ 2\in A\) such that \((a_ 1,a)\in \rho\) and \((a,a_ 2)\in \rho\). A binary relation \(\rho\) between the elements of a universal algebra A is called stable if \(\rho\) is a (possibly, empty) subalgebra of \(A\times A.\) Theorem 1. A semigroup S is isomorphic to a semigroup of symmetric binary relations if and only if S is commutative and satisfies the condition \(x=xy^ 2z^ 2\Rightarrow x=xy^ 2\) for all x,y,z\(\in S.\) Theorem 2. The following conditions are equivalent for any semigroup S: S is isomorphic to a semigroup of stable symmetric binary relations on (A) a Boolean algebra; (B) a ring; (C) a group; (D) a vector space; (E) a module. The stable symmetric binary relations on these structures can be chosen to be multipermutations and, if S is finite, the structures (A)- (E) can be chosen to be finite. A semigroup S satisfies any (or all) of the equivalent conditions (A)-(E) if and only if S is commutative and \(x=x^ 3\) for every \(x\in S\). It is shown that a semigroup S is isomorphic to a semigroup of (partial) tolerance relations if and only if S is commutative and for all x,y,z\(\in S\), \(x=xyz\Rightarrow x=xy\).