Properties of relatively free inverse semigroups
Norman R. Reilly, Peter G. Trotter · Transactions of the American Mathematical Society · 1986
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