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 S is combinatorial (i.e., H \mathcal {H} is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, S S does not contain a copy of the bicyclic semigroup) or E E -unitary (i.e., E ( S ) E(S) is the kernel of the minimum group congruence) then the relatively free inverse semigroup F V X F{\mathcal {V}_X} on the set X X in the variety V \mathcal {V} generated by S S is also combinatorial, completely semisimple or E E -unitary, respectively. If S S is a fundamental (i.e., the only congruence contained in H \mathcal {H} is the identity congruence) and | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , then F V X F{\mathcal {V}_X} is also fundamental. F V X F{\mathcal {V}_X} may not be fundamental if | X | > ℵ 0 |X| > {\aleph _0} . It is also shown that for any variety of groups U \mathcal {U} and for | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , there exists a variety of inverse semigroups V \mathcal {V} which is minimal with respect to the properties (i) F V X F{\mathcal {V}_X} is fundamental and (ii) V ∩ G = U \mathcal {V} \cap \mathcal {G} = \mathcal {U} , where G \mathcal {G} is the variety of groups. In the main result of the paper it is shown that there exists a variety V \mathcal {V} for which F V X F{\mathcal {V}_X} is not completely semisimple, thereby refuting a long standing conjecture.