More on cancellative semigroups on manifolds. (English) Zbl 0635.22003

Hofmann, Karl Heinrich; Weiss, W.

Semigroup Forum 37, No. 1, 93-111 (1988).

The paper deals with cancellative topological semigroups which, as topological spaces, are connected, paracompact, finite dimensional and locally Euclidean. D. R. Brown and R. S. Houston [Semigroup Forum 35, 279-302 (1987; Zbl 0626.22001)] proved that such a semigroup S can be locally embedded into a simply connected Lie group G(S). This embedding is achieved by constructing local groups of one sided quotients. In the same paper Brown and Houston use the explicit construction of this embedding to provide the semigroup with an analytic structure and give some functorial properties of the semigroups in question. The present paper shows how to obtain the analytic structure of the semigroup and various functorial properties using only the topological and algebraic properties of the Brown-Houston local embedding. The main step is to equip the direct product \(S\times G(S)\) with a topology for which both projections, \(S\times G(S)\to S\) and \(S\times G(S)\to G(S)\) are Hausdorff sheaves of topological spaces. From these sheaves various invariants are constructed which, in the future, may be helpful in deciding whether S can be embedded globally into G(S).

Reviewer: J.Hilgert

MSC:

22A15	Structure of topological semigroups
22E05	Local Lie groups

Keywords:

cancellative topological semigroups; simply connected Lie group; local groups; analytic structure; local embedding; Hausdorff sheaves of topological spaces

Citations:

Zbl 0626.22001

References:

[1]	[BH] Brown, D.R. and R.S. Houston,Cancellativesemigroupsonmanifolds, Semigroup Forum, to appear.
[2]	Carruth, J. H.; Hildebrandt, J. A.; Koch, R. J., Thetheoryoftopologicalsemigroups I II (1983), New York and Basel: M. Dekker, New York and Basel · Zbl 0515.22003
[3]	[CP] Clifford, A.H. and G.B. Preston,Thealgebraictheoryofsemigroups I, II, Providence 1961 and 1967. · Zbl 0111.03403
[4]	Comfort, W. W., Asurveyofcardinalinvariants, (Gen.) Top. and its Appl., 1, 163-199 (1971) · Zbl 0216.19002
[5]	Davey, B. A., Sheafspacesandsheavesofuniversalalgebras, Math. Z., 134, 275-290 (1973) · Zbl 0259.08002 · doi:10.1007/BF01214692
[6]	Dugundji, J., Topology (1966), Boston: Allyn and Bacon, Boston · Zbl 0144.21501
[7]	Hochschild, G., ThestructureofLiegroups (1965), San Francisko et al.: Holden Day, San Francisko et al. · Zbl 0131.02702
[8]	Jacoby, R., Sometheoremsonthestructureoflocallycompactlocalgroups, Ann. Math., 66, 36-69 (1957) · Zbl 0084.03202 · doi:10.2307/1970116
[9]	Tennison, B. R., Sheaftheory (1975), Cambridge et al.: Cambridge Univ. Press, Cambridge et al. · Zbl 0313.18010 · doi:10.1017/CBO9780511661761
[10]	Spanier, E. H., AlgebraicTopology (1966), New York et al.: McGraw Hill, New York et al. · Zbl 0145.43303
[11]	Weinert, H. J., Semigroupsofrightquotientsoftopologicalsemigroups, Trans. Amer. Math. Soc., 147, 333-348 (1970) · Zbl 0201.02702 · doi:10.1090/S0002-9947-1970-0252551-6