Topological spaces determined by left ideals of semigroups

Kenneth Derwood Magill, Jr.

Vol. 24 (1968), No. 2, 319–330

DOI: 10.2140/pjm.1968.24.319

Abstract

In this paper, we show how each (proper or improper) left ideal Z of a semigroup T determines, in a natural way, a topological space. The space will be denoted by 𝒰(T.Z) and will be referred to as the structure space of the pair (T,Z). Any such structure space is compact and T₁, although it need not be Hausdorff. If T contains a left identity, then the ideal Z corresponds, in a natural way, to a subspace ℛ(T,Z) of 𝒰(T,Z) which we refer to as the realization of Z. There is a homomorphism φ from T into S(ℛ(T,Z)) (for any space X, S(X) denotes the semigroup, under composition, of all continuous functions mapping X into X). Moreover, φ is injective if and only if for every pair of distinct elements a and b of T, av≠bv for some v in Z.

Mathematical Subject Classification

Primary: 54.80

Secondary: 20.00

Milestones

Received: 22 June 1966

Published: 1 February 1968