A semigroup 
is said to be an inverse semigroup if, for every 
in 
, there is a unique 
(called the inverse of 
) such that 
and 
. This is equivalent to the condition that every element
has at least one inverse and that the idempotents
of 
commute (Lawson 1999). Note that if 
is an inverse of 
, then 
is an idempotent.
Inverse Semigroup
See also
SemigroupThis 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.Referenced on Wolfram|Alpha
Inverse SemigroupCite this as:
Bray, Nicolas. "Inverse Semigroup." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InverseSemigroup.html