# Products of threads in topological semigroups

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## Abstract

A thread is a semigroup S on a metric arc in which one endpoint is an identity for S and the other endpoint is a zero for S. This paper is concerned generally with a study of the product of two threads in a topological semigroup. Particular emphasis is placed on determining sufficient conditions for a product of two threads to contain a two-cell. The definitions which are used throughout the work as well as some preliminary concepts and theorems from the literature are given first. Next, several theorems are presented which give sufficient conditions for the multiplication function in a topological semigroup to be monotone when restricted to certain subsets of products of two threads. One of the main theorems in this paper states that in any topological semigroup the product of two threads is either a point, an arc or contains a two-cell. Furthermore the product of an M-thread, a thread under min multiplication, and an arbitrary thread which have only the zero and identity in common is a two-cell. Conditions are established which imply that the product of two usual threads, a thread iseomorphic to the real unit interval, is a two-cell. Finally, products of threads in a compact, uniquely divisible semigroup which has codimension two are investigated. If S is a semigroup in which e is the identity, z is the zero, E(S) = H = {e,z}, and multiplication is monotone when restricted to the product of any two threads, then S is a two-cell. In a semigroup in which the set of idempotents consists of the identity e and zero z, the multiplication function is locally one-to-one at (e,e). This results in a local cancellation about the identity.