Subshift semigroups

Given a one-sided subshift $X$ on a finite alphabet, we consider the semigroup $S_X =L_X \cup \{0\}$, where $L_X $ is the language of $X $, equipped with the multiplication operation given by concatenation, when allowed, and set to vanish otherwise. We then study the inverse hull $H(S_X )$, relating it with C*-algebras that have been discussed in the literature in association with subshifts.