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Inverse Semigroup


A semigroup S is said to be an inverse semigroup if, for every a in S, there is a unique b (called the inverse of a) such that a=aba and b=bab. This is equivalent to the condition that every element has at least one inverse and that the idempotents of S commute (Lawson 1999). Note that if b is an inverse of a, then ba is an idempotent.


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Semigroup

This entry contributed by Nicolas Bray

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References

Clifford, A. H. and Preston, G. B. The Algebraic Theory of Semigroups, Vol. 1. Providence, RI: Amer. Math. Soc., 1961.Clifford, A. H. and Preston, G. B. The Algebraic Theory of Semigroups, Vol. 2. Providence, RI: Amer. Math. Soc., 1967.Lawson, M. V. Inverse Semigroups: The Theory of Partial Symmetries. Singapore: World Scientific, 1999.Lyapin, E. S. Semigroups. Providence, RI: Amer. Math. Soc., 1974.Shevrin, L. N. "Inversion Semi-Group." In Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia," Vol. 5 (Managing Ed.M. Hazewinkel). Dordrecht, Netherlands: Reidel, pp. 184-185, 1988.Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.

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Inverse Semigroup

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Bray, Nicolas. "Inverse Semigroup." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InverseSemigroup.html

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