Vol. 24, No. 2, 1968

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Topological spaces determined by left ideals of semigroups

Kenneth Derwood Magill, Jr.

Vol. 24 (1968), No. 2, 319–330

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 (T,Z) of 𝒰(T,Z) which we refer to as the realization of Z. There is a homomorphism φ from T into S((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, avbv for some v in Z.

Mathematical Subject Classification
Primary: 54.80
Secondary: 20.00
Received: 22 June 1966
Published: 1 February 1968
Kenneth Derwood Magill, Jr.