Descriptional Complexity of the Languages KaL: Automata, Monoids and Varieties

The first step when forming the polynomial hierarchies of languages is to consider languages of the form KaL where K and L are over a finite alphabet A and from a given variety V of languages, a ∈ A being a letter. All such KaL’s generate the variety of languages BPol1(V ). We estimate the numerical parameters of the language KaL in terms of their values for K and L. These parameters include the state complexity of the minimal complete DFA and the size of the syntactic monoids. We also estimate the cardinality of the image of A in the Schützenberger product of the syntactic monoids of K and L. In these three cases we obtain the optimal bounds. Finally, we also consider estimates for the cardinalities of free monoids in the variety of monoids corresponding to BPol1(V ) in terms of sizes of the free monoids in the variety of monoids corresponding to V .