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Transactions of the American Mathematical Society

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Structural properties of universal minimal dynamical systems for discrete semigroups

Authors: Bohuslav Balcar and Frantisek Franek
Journal: Trans. Amer. Math. Soc. 349 (1997), 1697-1724
MSC (1991): Primary 54H20, 03G05; Secondary 06E05
MathSciNet review: 1407479
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Abstract: We show that for a discrete semigroup $S$ there exists a uniquely determined complete Boolean algebra $B(S)$ - the algebra of clopen subsets of $M(S)$. $M(S)$ is the phase space of the universal minimal dynamical system for $S$ and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that $B(S)$ is either atomic or atomless; that $B(S)$ is weakly homogenous provided $S$ has a minimal left ideal; and that for countable semigroups $B(S)$ is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection $G(S)$ of a group-like semigroup $S$ can be constructed via universal minimal dynamical system for $S$ and, moreover, $B(S)$ and $B(G(S))$ are the same.

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Additional Information

Bohuslav Balcar
Affiliation: Mathematical Institute of AV ČR, Žitná 25, 115 67 Praha 1, Czech Republic, and Center for Theoretical Study, Jilská 1, 110 00 Praha 1, Czech Republic

Frantisek Franek
Affiliation: Department of Computer Science and Systems, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Keywords: Semigroup, group, dynamical system, minimal dynamical system, universal minimal dynamical system, ultrafilter, ultrafilter dynamical system, Boolean algebra, Cohen alegbra, extremally disconnected compact space.
Received by editor(s): July 5, 1994
Additional Notes: The research of the first author was supported by AV ČR research grant 119401, while the research of the second author was supported by NSERC research grant OGP0025112.
Article copyright: © Copyright 1997 American Mathematical Society

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