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(L)-Semigroup Sums

Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, 241 McLean Hall, Saskatoon, SK S7N 5E6, Canada
In this note, all spaces are Hausdorff, and the term map or mapping shall always mean continuous function.
Received: 12 November 2018 / Revised: 14 December 2018 / Accepted: 17 December 2018 / Published: 22 December 2018
(This article belongs to the Collection Topological Groups)
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An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is the quotient space obtained from the union of S × { 0 } and T × { 1 } by identifying the point ( x , 0 ) in S × { 0 } with ( x , 1 ) in T × { 1 } for each x in B. It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an Abelian boundary. In particular, such sums cannot be a retract of a topological group. View Full-Text
Keywords: topological group; Lie group; compact topological semigroup; H-space; mapping cylinder; fibre bundle topological group; Lie group; compact topological semigroup; H-space; mapping cylinder; fibre bundle
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Martin, J.R. (L)-Semigroup Sums . Axioms 2019, 8, 1.

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