Varieties of restriction semigroups and varieties of categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, ^?1) by forgetting the inverse operation and retaining the two operations x⁺ = xx^?1 and x* = x^?1x. The subvariety B of strict restriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals B₂, B₂M = B] and B₀, B₀M]. Here, B₂ and B₀ are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval B₂, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson's major theorems have natural interpretations and application to the interval B₂, B] and, with modification, to the interval B₀, B₀M] that lies below it. Further exploration may lead to applications in the reverse direction.