Properties of relatively free inverse semigroups

Abstract:

The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if $S$ is combinatorial (i.e., $\mathcal {H}$ is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, $S$ does not contain a copy of the bicyclic semigroup) or $E$-unitary (i.e., $E(S)$ is the kernel of the minimum group congruence) then the relatively free inverse semigroup $F{\mathcal {V}_X}$ on the set $X$ in the variety $\mathcal {V}$ generated by $S$ is also combinatorial, completely semisimple or $E$-unitary, respectively. If $S$ is a fundamental (i.e., the only congruence contained in $\mathcal {H}$ is the identity congruence) and $|X| \geqslant {\aleph _0}$, then $F{\mathcal {V}_X}$ is also fundamental. $F{\mathcal {V}_X}$ may not be fundamental if $|X| < {\aleph _0}$. It is also shown that for any variety of groups $\mathcal {U}$ and for $|X| \geqslant {\aleph _0}$, there exists a variety of inverse semigroups $\mathcal {V}$ which is minimal with respect to the properties (i) $F{\mathcal {V}_X}$ is fundamental and (ii) $\mathcal {V} \cap \mathcal {G} = \mathcal {U}$, where $\mathcal {G}$ is the variety of groups. In the main result of the paper it is shown that there exists a variety $\mathcal {V}$ for which $F{\mathcal {V}_X}$ is not completely semisimple, thereby refuting a long standing conjecture.