New York Journal of Mathematics
Volume 25 (2019), 897-913


Neil Hindman and Dona Strauss

Some new results about the smallest ideal of βS

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Published: September 15, 2019.
Keywords: rectangular semigroup, smallest ideal, Stone-\v Cech compactification.
Subject: 54D35, 54D80, 22A30.

We provide what we believe is the first nontrivial algebraic description of the smallest ideal of the Stone-Čech compactification of a discrete semigroup. Specifically, given sets A and B, let S=R(A,B)=A x B with the discrete topology and the rectangular semigroup operation (a,b)(c,d)=(a,d). Then the smallest ideal of βS is isomorphic (but not homeomorphic) to R(βA,βB). We also determine exactly the topological center of βS. The minimal left ideals of βS all have isolated points. We derive several results about βS that must hold for any semigroup S if the minimal left ideals have isolated points.



Author information

Neil Hindman:
Department of Mathematics
Howard University
Washington, DC 20059, USA


Dona Strauss:
Department of Pure Mathematics
University of Leeds
Leeds LS2 9J2, UK