Free products of inverse semigroups

Abstract:

A structure theorem is provided for the free product $S {\operatorname {inv}} T$ of inverse semigroups $S$ and $T$. Each element of $S {\operatorname {inv}} T$ is uniquely expressible in the form $\varepsilon (A)a$, where $A$ is a certain finite set of “left reduced” words and either $a = 1$ or $a = {a_1} \cdots {a_m}$ is a “reduced” word with $aa_m^{ - 1} \in A$. (The word ${a_1} \cdots {a_m}$ in $S {\operatorname {sgp}} T$ is called reduced if no letter is idempotent, and left reduced if exactly ${a_m}$ is idempotent; the notation $\varepsilon (A)$ stands for $\Pi \{ a{a^{ - 1}}: a \in A\}$.) Under a product remarkably similar to Scheiblich’s product for free inverse semigroups, the corresponding pairs $(A, a)$ form an inverse semigroup isomorphic with $S {\operatorname {inv}} T$. This description enables various properties of $S {\operatorname {inv}} T$ to be determined. For example $(S\:{\operatorname {inv}}\:T)\backslash (S \cup T)$ is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup of $S$ or $T$. If neither $S$ nor $T$ has a zero then $(S\:{\operatorname {inv}}\:T)$ is fundamental, but in general fundamentality itself is not preserved.