Nonfinite axiomatizability of the equational theory of shuffle

Abstract

We consider language structures ${\bf L}_\Sigma = (P_\Sigma,\cdot,\otimes,+,1,0)$ , where $P_\Sigma$ consists of all subsets of the free monoid $\Sigma^*$ ; the binary operations $\cdot$ , $\otimes$ 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 ${\bf L}_\Sigma$ has no finite axiomatization. In fact we establish a stronger result: The variety Lang has no finite axiomatization over the variety of ordered algebras ${\bf Lg}_\leq$ generated by the structures $(P_\Sigma,\cdot,\otimes,1,\subseteq)$ , where $\subseteq$ denotes set inclusion.