On the Bifree Locally Inverse Semigroup

AbstractA class of regular semigroups closed under taking direct products, regular subsemigroups, and homomorphic images is an existence variety, (or e-variety) of regular semigroups. For an e-variety V of locally inverse or E-solid regular semigroups, the bifree objectBFV(X) on a set X is the natural concept of a "free object" in V. Its existence has been proved by Y. T. Yeh. Using canonical forms of the elements of the bifree completely simple semigroup BFCS(X) we shall present two models of the bifree locally inverse semigroup BFLI(X). One is by means of a semidirect product of a semilattice by the bifree completely simple semigroup and is an analogue to Scheiblich′s model of the free inverse semigroup. The other description is in terms of canonical forms which are strongly related to Schein′s canonical forms for the elements of the free inverse semigroup. The proofs use the concept of bi-identities and are syntactical. As an application we get (new proofs of) the following results: (i) each locally inverse semigroup divides a semidirect product of a semilattice and a completely simple semigroup, (ii) the e-variety of all locally inverse semigroups is generated by either of the Mal′cev products S∘CS and I∘RB where S = semilattices, CS = completely simple semigroups, I = inverse semigroups, RB = rectangular bands