Locality of Ds and Associated Varieties

We prove that the pseudovariety DS, of all nite monoids each of whose regular D-classes is a subsemigroup, is local. (A pseudovariety (or variety) V is local if any category whose local monoids belong to V divides a member of V). The proof uses the \kernel theorem" of the rst two authors together with the description by P. Weil of DS as an iterated \block product". The one-sided analogues of these methods provide wide new classes of local pseudovarieties of completely regular monoids. We conclude, however, with the third author's example of a variety (and a pseudovariety) of completely regular monoids that is not local. A pseudovariety V of monoids is said to be local if every nite category C whose local (or \loop") monoids belong to V divides a monoid in V. Locality has important applications to both the theory of pseudovarieties and to its connection with varieties of formal languages 20, 10]. The rst author showed in 4] and with M. B. Szendrei in 7] that many pseudovarieties that consist of completely regular monoids or, equivalently, satisfy an identity x n+1 = x, are local. For instance the pseudovariety CR of all nite completely regular monoids is local 4]. Any nontrivial subpseudovariety consisting of orthogroups (those in which the idempotents form a submonoid) is local. These results are here delimited in one way, but extended in two ways. An example is given of a nontrivial pseudovariety (and a nontrivial variety) of completely regular monoids that is not local, answering a question posed in 7]. However, it is shown that many well-studied operators on completely regular pseudovarieties preserve locality. Most importantly, it is shown that if V is a local completely regular pseudovariety then GV,