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The potential $ \mathcal{J}$-relation and amalgamation bases for finite semigroups

Authors: T. E. Hall and Mohan S. Putcha
Journal: Proc. Amer. Math. Soc. 95 (1985), 361-364
MSC: Primary 20M10
MathSciNet review: 806071
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Abstract: Let $ S$ be a finite semigroup, $ a,b \in S$. When does there exist a finite semigroup $ T$ containing $ S$ such that $ a\mathcal{J}b$ in $ T$? This problem was posed to the second named author by John Rhodes in 1974. We show here that if $ a$, $ b$ are regular, then such a semigroup $ T$ exists if and only if either $ a\mathcal{J}b$ in $ S$, or $ a \notin SbS$ and $ b \notin SaS$. We use this result to show that analgamation bases for the class of finite semigroups have linearly ordered $ \mathcal{J}$-classes.

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Article copyright: © Copyright 1985 American Mathematical Society