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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $ 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 $ \vert X\vert \geqslant {\aleph _0}$, then $ F{\mathcal{V}_X}$ is also fundamental. $ F{\mathcal{V}_X}$ may not be fundamental if $ \vert X\vert < {\aleph _0}$. It is also shown that for any variety of groups $ \mathcal{U}$ and for $ \vert X\vert \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.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0819946-2
Article copyright: © Copyright 1986 American Mathematical Society