Elementary orthodox semigroups (Q798781)

A semigroup is said to be elementary if it is generated by a subset \(A\cup B\) where \(aba=a\) and \(bab=b\) for all \(a\in A\), \(b\in B\). The authors extend their earlier work [Semigroup Forum 15, 295-309 (1978; Zbl 0403.20040)], in which free elementary orthodox semigroups were defined and their existence established, by studying the free elementary orthodox semigroup FO on two mutually inverse generators. A presentation is given FO, the word problem is solved, and Green's relations are determined. The lattice of congruences on FO is investigated in terms of its sublattice of inverse congruences (which is isomorphic to the well- known congruence lattice of the free monogenic inverse semigroup).