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Transactions of the American Mathematical Society

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



Free products of inverse semigroups

Author: Peter R. Jones
Journal: Trans. Amer. Math. Soc. 282 (1984), 293-317
MSC: Primary 20M05
MathSciNet review: 728714
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Abstract: A structure theorem is provided for the free product $ S\,{\operatorname{inv}}\,T$ of inverse semigroups $ S$ and $ T$. Each element of $ S\,{\operatorname{inv}}\,T$ is uniquely expressible in the form $ \varepsilon (A)a$, where $ A$ is a certain finite set of ``left reduced'' words and either $ a = 1$ or $ a = {a_1} \cdots {a_m}$ is a ``reduced'' word with $ aa_m^{ - 1} \in A$. (The word $ {a_1} \cdots {a_m}$ in $ S\,{\operatorname{sgp}}\,T$ is called reduced if no letter is idempotent, and left reduced if exactly $ {a_m}$ is idempotent; the notation $ \varepsilon (A)$ stands for $ \Pi \{ a{a^{ - 1}}:\,a \in A\} $.) Under a product remarkably similar to Scheiblich's product for free inverse semigroups, the corresponding pairs $ (A,\,a)$ form an inverse semigroup isomorphic with $ S\,{\operatorname{inv}}\,T$.

This description enables various properties of $ S\,{\operatorname{inv}}\,T$ to be determined. For example $ (S\:{\operatorname{inv}}\:T)\backslash (S \cup T)$ is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup of $ S$ or $ T$. If neither $ S$ nor $ T$ has a zero then $ (S\:{\operatorname{inv}}\:T)$ is fundamental, but in general fundamentality itself is not preserved.

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Keywords: Inverse semigroup, free product, canonical form, structural and preservational properties
Article copyright: © Copyright 1984 American Mathematical Society