Positive Clifford semigroups on the plane

Farley, Reuben W.

doi:10.1090/S0002-9947-1970-0263966-4

Positive Clifford semigroups on the plane
HTML articles powered by AMS MathViewer

by Reuben W. Farley PDF

Trans. Amer. Math. Soc. 151 (1970), 353-369 Request permission

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.

References

A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791

Positive commutative semigroups on the plane

Karl Heinrich Hofmann and Paul S. Mostert, Elements of compact semigroups, Charles E. Merrill Books, Inc., Columbus, Ohio, 1966. MR 0209387
J. G. Horne Jr., Real commutative semigroups on the plane, Pacific J. Math. 11 (1961), 981–997. MR 139148
J. G. Horne Jr., Real commutative semigroups on the plane, Pacific J. Math. 11 (1961), 981–997. MR 139148
Paul S. Mostert, Plane semigroups, Trans. Amer. Math. Soc. 103 (1962), 320–328. MR 137779, DOI 10.1090/S0002-9947-1962-0137779-3
Paul S. Mostert and Allen L. Shields, Semigroups with identity on a manifold, Trans. Amer. Math. Soc. 91 (1959), 380–389. MR 105463, DOI 10.1090/S0002-9947-1959-0105463-8

Topological groups

1