THE LATTICE OF VARIETIES OF STRICT LEFT RESTRICTION SEMIGROUPS

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety B of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety B-R of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices L(B) and L(B-R) of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval [D, D boolean OR M], every subvariety of B is induced from a member of B-R and vice versa. Here D is generated by the three-element left restriction semigroup D and M is the variety of monoids. The analogues hold for pseudovarieties.