The left restriction semigroups have arisen in a number of contexts, one being as the abstract characterization of semigroups of partial maps, another as the ‘weakly left E-ample’ semigroups of the ‘York school’, and, more recently as a variety of unary semigroups defined by a set of simple identities. We initiate a study of the lattice of varieties of such semigroups and, in parallel, of their...

We construct a modification of Churchill and Trotter’s trifree objects in e-varieties of locally E-solid semigroups, which have the property, for e-varieties of locally inverse semigroups and for e-varieties of Esolid semigroups, of being isomorphic to the bifree objects.