Positive Clifford semigroups on the plane

Author:
Reuben W. Farley

Journal:
Trans. Amer. Math. Soc. **151** (1970), 353-369

MSC:
Primary 22.05

DOI:
https://doi.org/10.1090/S0002-9947-1970-0263966-4

MathSciNet review:
0263966

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Abstract: This work is devoted to a preliminary investigation of positive Clifford semigroups on the plane. A positive semigroup is a semigroup which has a copy of the nonnegative real numbers embedded as a closed subset in such a way that 0 is a zero and 1 is an identity. A positive Clifford semigroup is a positive semigroup which is the union of groups. In this work it is shown that if *S* is a positive Clifford semigroup on the plane, then each group in *S* is commutative. Also, a necessary and sufficient condition is given in order that *S* be commutative, and an example is given of such a semigroup which is, in fact, not commutative. In addition, both the number and the structure of the components of groups in *S* is determined. Finally, it is shown that *S* is the continuous isomorphic image of a semilattice of groups.

**[1]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups*. Vol. I, Mathematical Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961. MR**24**#A2627. MR**0132791 (24:A2627)****[2]**Reuben W. Farley,*Positive commutative semigroups on the plane*, Master's Thesis, University of Tennessee, Knoxville, 1965 (unpublished).**[3]**K. H. Hofmann and P. S. Mostert,*Elements of compact semigroups*, Merrill, Columbus, Ohio, 1966. MR**35**#285. MR**0209387 (35:285)****[4]**J. G. Horne,*Real commutative semigroups on the plane*, Pacific J. Math.**11**(1961), 981-997. MR**25**#2586a. MR**0139148 (25:2586a)****[5]**-,*Real commutative semigroups on the plane*. II, Trans. Amer. Math. Soc.**104**(1962), 17-23. MR**25**#2586b. MR**0139149 (25:2586b)****[6]**P. S. Mostert,*Plane semigroups*, Trans. Amer. Math. Soc.**103**(1962), 320-328. MR**25**#1228. MR**0137779 (25:1228)****[7]**P. S. Mostert and A. L. Shields,*Semigroups with identity on a manifold*, Trans. Amer. Math. Soc.**91**(1959), 380-389. MR**21**#4204. MR**0105463 (21:4204)****[8]**L. Pontrjagin,*Topological groups*, GITTL, Moscow, 1938; English transl., Princeton Math. Series, vol. 2, Princeton Univ. Press, Princeton, N. J., 1939. MR**1**, 44.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0263966-4

Keywords:
Real semigroup,
positive semigroup,
Clifford semigroup,
semilattice of groups

Article copyright:
© Copyright 1970
American Mathematical Society