Definition 47.5.1 (Variable Words).label Let $A$ be a finite alphabet, $S = A^{+}$ be the free semigroup of all non-empty words in $A$, $T = (A \sqcup \bracsn{*})^{+}$ be the free semigroup of all non-empty words in $A \sqcup \bracsn{*}$, then the set of variable words in $A$ is the two-sided ideal $V := T \setminus S$ of all words in $A \sqcup \bracsn{*}$ containing $*$.

For each $a \in A$, the semigroup homomorphism

is the substitution map from $*$ to $a$, with $\pi_{a}|_{S} = \text{Id}_{S}$.