Expletives in languages and middle units in semigroups (Q1113290)

Let \(L\subseteq A^*\) be a language over an alphabet A. A word \(x\in A^*\) is said to be an expletive for L if, for all \(y,z\in A^+\), yxz\(\in L\) \(\Leftrightarrow\) yz\(\in L\). The author studies languages L for which every letter \(a\in A\) is a prefix of some expletive ax. He shows that this condition holds if and only if the syntactic semigroup of the language is a dilation of a rectangular band of groups. The paper also includes a method for constructing all such languages L in terms of a subset B of \(A\cup \{1\}\) and mappings \(\phi\) : \(A\to G\) and \(\psi\) : \(A\times A\to G\), where G is a group. Namely, assuming that \(A\phi\) generates G as a semigroup and letting \(\pi\) : \(A^*\to G\) denote the homomorphic extension of \(\phi\), \(L=B\cup \{a_ 1...a_ n\in A^*:\) \(n\geq 2\), \((a_ 2...a_{n-1})\pi \in (a_ 1,a_ n)\psi \}\).