Expletive languages (Q1281274)

Let \(A\) be an alphabet and \(L\subseteq A^+\). A word \(u\in A^+\) is called expletive of \(L\) if for any \(x,y\in A^+\), \(xuy\in L\) is equivalent to \(xy\in L\) [\textit{E. Tully}, Semigroup Forum 38, No. 1, 77-84 (1989; Zbl 0662.20043)]. The set of all expletives of \(L\) is denoted by \(\text{Exp}(L)\). \(L\) is called an expletive language if \(L=\text{Exp}(L')\) for some \(L'\subseteq A^+\) and \(L\) is called full expletive if \(\text{Exp}(L)=A^+\). It is shown that \(L\subseteq\text{Exp}(L)\) for every expletive language \(L\). The connection with the pseudovariety MID of all finite midunit semigroups (defined by the identity \(abc=ac\)) is worked out. In particular, it is shown that a language \(L\) over \(A\) is full expletive iff its syntactic semigroup is a (finite) midunit semigroup. Also, it is proved that the class FEL of all full expletive languages is a \(+\)-variety, which implies that there is a one-to-one correspondence between FEL and MID. Finally, all full expletive languages over \(A\) are described and it is shown that their number is finite if \(A\) is finite.