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The first step when forming the polynomial hierarchies of languages is toconsider languages of the form KaL where K and L are over a finite alphabet Aand from a given variety V of languages, a being a letter from A. All suchKaL's generate the variety of languages BPol1(V).  We estimate the numerical parameters of the language KaL in terms of theirvalues for K and L. These parameters include the state complexity of theminimal complete DFA and the size of the syntactic monoids. We also estimatethe cardinality of the image of A* in the Schuetzenberger product of thesyntactic monoids of K and L. In these three cases we obtain the optimalbounds.  Finally, we also consider estimates for the cardinalities of free monoids inthe variety of monoids corresponding to BPol1(V) in terms of sizes of the freemonoids in the variety of monoids corresponding to V.

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