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COMMENTS
| An inverse semigroup S is a semigroup in which every element x in S has a unique inverse y in S in the sense that x = xyx and y = yxy. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries [Lawson 1998]. They were introduced independently by Viktor Vladimirovich Wagner [1952] in the Soviet Union, and by Gordon Preston [1954] in Great Britain. The Wagner-Preston Theorem shows that any inverse semigroup can be embedded in a symmetric inverse semigroup. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 08 2010]
The goal of this note [Araujo] is to prove the converse, that is, we prove that every unary semigroup satisfying these three identities is an inverse semigroup and the unary operation coincides with the usual inversion on such semigroups. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 23 2010]
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REFERENCES
| S. Satoh, K. Yama, M. Tokizawa, Semigroups of order 8, Semigroup Forum 49 (1994), 7-29.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, Semigroup Forum, 14 (1977), 69-79.
R. J. Plemmons, Cayley Tables for All Semigroups of Order Less Than 7. Department of Mathematics, Auburn Univ., 1965.
R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, 1998. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 08 2010]
G. B. Preston, "Inverse semi-groups". Journal of the London Mathematical Society 29: 396-403. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 08 2010]
V. V. Wagner (1952). "Generalised groups". Proceedings of the USSR Academy of Sciences 84: 1119-1122. (Russian) English translation. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 08 2010]
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LINKS
| Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Index entries for sequences related to semigroups
Joao Araujo, Michael Kinyon, An elegant 3-basis for inverse semigroups, March 21, 2010. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 23 2010]
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