Congruences on equivalence algebras (Q2475557)

A groupoid \(\langle G,\cdot\rangle\) is said to be quasi-trivial if \(x\cdot y\in \{x,y\}\) for all \(x,y\in G\). A quasi-trivial groupoid \(\langle G,\cdot\rangle\) is called an equivalence algebra whenever the binary relation \(\{\langle x,y\rangle\in G\times G\); \(x\cdot y= x\}\) is an equivalence relation on \(G\). A characterization of congruence lattices of equivalence algebras is given in the paper. Namely it is shown that such congruence lattices are semisimple, semimodular and atomic.