POLYNOMIAL CLOSURE AND TOPOLOGY
Benjamin Steinberg · International Journal of Algebra and Computation · 2000
Suppose H is a pseudovariety of groups. This paper studies the pseudovariety of monoids corresponding to the variety of languages generated by the polynomial closure of the variety of H-languages (which is shown to be J*H via a short syntactic argument) and the pseudovariety of ordered monoids corresponding to the polynomial closure of the variety of H-languages. We also explore alternative descriptions of these pseudovarieties under additional hypotheses. Decidability results are given. In addition, we study the positive varieties of pro-V open and closed recognizable sets for a pseudovariety V of monoids. In particular, we obtain a basis of ordered pseudoidentities for such positive varieties. For pseudovarieties of groups H, we relate these positive varieties to the polynomial closure of the H-languages. Again, decidability results are obtained.