Locality of DS and associated varieties
Journal of Pure and Applied Algebra
We prove that the pseudovariety DS, of all finite monoids, each of whose regular D" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 14.4px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">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 first author and Pustejovsky together with the description by 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 second author's example of a variety (and a pseudovariety) of completely regular monoids that is not local.