Nonfinite axiomatizability of the equational theory of shuffle

Abstract

We consider language structures LΣ = (PΣ,·, ⊗, +, 1,0), where PΣ consists of all subsets of the free monoid Σ*; the binary operations ⊗, and + are concatenation, shuffle product and union, respectively, and where the constant 0 is the empty set and the constant 1 is the singleton set containing the empty word. We show that the variety Lang generated by the structures LΣ has no finite axiomatization.