Inverse semigroups are semigroups in which each element $x$ has a unique inverse $x^{-1}$, i.e. $x = xx^{-1}x$ and $x^{-1} = x^{-1}xx^{-1}$ hold. We can equip them with a natural partial order $\leq$, where $a \leq b$ holds if and only if there is an idempotent $e$ such that $a=be$. In such semigroups, two relations play an important role. These are the compatibility relation $\sim $, where $s \sim t$ if and only if $ s^{-1}t$ and $ st^{-1}$ are idempotent, and the smallest group congruence $\sigma$, where $s \mathrel{\sigma} t$ if and only if there is such a $u$ that $u \leq s,t $ holds.
A class of inverse semigroups for which the compatibility relation coincides with the smallest group congruence is called a class of $E$-unitary inverse semigroups, and a class for which every $\sigma$-class has a maximal element is called a class of $F$-inverse monoids. Thus, $F$-inverse monoids can be equipped with an additional unary operation $a \mapsto m(a)$. It turns out that $F$-inverse monoids form a variety precisely in this extended signature. Next we focus on combinatorial models of group expansions, namely the Margolis-Meakin expansion, the Birget-Rhodes expansion, and models of $F$-inverse and perfect $F$-inverse monoids. Using these models, we can describe free inverse monoids, free $F$-inverse monoids, and free perfect $F$-inverse monoids.
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