Properties of relatively free inverse semigroups

Authors:
N. R. Reilly and P. G. Trotter

Journal:
Trans. Amer. Math. Soc. **294** (1986), 243-262

MSC:
Primary 20M07; Secondary 20M05, 20M18

DOI:
https://doi.org/10.1090/S0002-9947-1986-0819946-2

MathSciNet review:
819946

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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 is combinatorial (i.e., is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, does not contain a copy of the bicyclic semigroup) or -unitary (i.e., is the kernel of the minimum group congruence) then the relatively free inverse semigroup on the set in the variety generated by is also combinatorial, completely semisimple or -unitary, respectively.

If is a fundamental (i.e., the only congruence contained in is the identity congruence) and , then is also fundamental. may not be fundamental if . It is also shown that for any variety of groups and for , there exists a variety of inverse semigroups which is minimal with respect to the properties (i) is fundamental and (ii) , where is the variety of groups.

In the main result of the paper it is shown that there exists a variety for which is not completely semisimple, thereby refuting a long standing conjecture.

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0819946-2

Article copyright:
© Copyright 1986
American Mathematical Society