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

DOI:
https://doi.org/10.1090/S0002-9947-97-01868-0

MathSciNet review:
1407479

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Abstract: We show that for a discrete semigroup there exists a uniquely determined complete Boolean algebra - the algebra of clopen subsets of . is the phase space of the universal minimal dynamical system for 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 is either atomic or atomless; that is weakly homogenous provided has a minimal left ideal; and that for countable semigroups 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 of a group-like semigroup can be constructed via universal minimal dynamical system for and, moreover, and are the same.

**[A]**J. Auslander,*Minimal Flows and their Extensions*, North-Holland, 1988. MR**89m:54050****[BB]**B. Balcar and A. Blaszczyk,*On minimal dynamical systems on Boolean algebras*, Comment. Math. Univ. Carolinae**31**(1) (1990), 7-11. MR**91h:54059****[BD]**B. Balcar and A. Dow,*Dynamical systems on compact extremally disconnected spaces*, Topology and its Application**41**(1991), 41-56. MR**92k:54048****[BF]**B. Balcar and F. Franek,*Independent Families in Complete Boolean Algebras*, Trans. Amer. Math. Soc.**274**(2) (1982), 607-617. MR**83m:06020****[BJZ]**B. Balcar, T. Jech, and J. Zapletal,*Semi-Cohen Boolean algebras*, Ann. Pure Appl. Logic - submitted.**[BS]**B. Balcar and P. Simon,*Appendix on General Topology*, Handbook of Boolean algebras (J.D. Monk and R. Bonnet, eds.), North-Holland, 1989. MR**90k:06004****[CN]**W. W. Comfort and S. Negrepontis,*The theory of ultrafilters*, Springer-Verlag, Berlin, Heidelberg, 1974. MR**53:135****[CvM]**W.W. Comfort and J. van Mill,*Some topological groups with, and some without, proper dense subgroups*, Topology and its Application**41**(1991), 3-15. MR**92h:54050****[CP]**A.H. Clifford and G.B. Preston,*The Algebraic Theory of Semigroups*, vol. 1, American Mathematical Society, 1961. MR**24:A2627****[vD]**E. K. van Douwen,*The maximal totally bounded group topology on and the biggest minimal -space, for Abelian groups*, Topology and its Application**34**(1990), 69-91. MR**91d:54044****[E]**R. Ellis,*Lectures on Topological Dynamics*, W.A. Benjamin, Inc., 1969. MR**42:2463****[En]**R. Engelking,*General Topology*, PWN, Warsaw, 1977. MR**58:18316b****[F]**H. Furstenberg,*Recurrence in Ergodic Theory and Combinatorial Number Theory*,

Princeton University Press, Princeton, New Jersey, 1981. MR**82j:28010****[Fu]**S. Fuchino,*-Cohen algebras*, preprint (1992).**[G]**S. Glasner,*Proximal Flows*, Springer-Verlag (Lecture Notes in Mathematics No. 517), 1976. MR**57:13890****[Ga]**J. Gait,*Transformation groups with no equicontinuous minimal set*, Compositio Mathematica**25**(1972), 87-92. MR**47:4233****[H]***Handbook of Boolean Algebras*(J.D. Monk and R. Bonnet, eds.), North-Holland, 1989. MR**90k:06004****[Hi]**N. Hindman,*Ultrafilters and Ramsey theory - an update*, Set Theory and its Applications Proceedings, Ontario 1987 (J. Steprans and S. Watson, eds.), Springer-Verlag (Lecture Notes in Mathematics No. 1401), 1989. MR**91b:54003****[Hi1]**N. Hindman,*The existence of certain ultrafilters on and a conjecture of Graham and Rothschild*, Proc. Amer. Math. Soc.**36**(1972), 341-346. MR**46:7041****[HiS]**N. Hindman and D. Strauss,*Algebraic and Topological Equivalences in the Stone-\v{C}ech compactification of a Discrete Senigroup - preprint*.**[HS]**H. Herrlich and G.E. Strecker,*Category Theory*, Allyn and Bacon, Inc., Boston, 1973. MR**50:2284****[Ko]**S. Koppelberg,*A lattice structure on the isomorphism types of complete Boolean algebras*, Set Theory and Model Theory, Springer-Verlag (Lecture Notes in Mathematics No. 872), 1981. MR**83b:03005****[Ko1]**S. Koppelberg,*Characterizations of Cohen algebras*, to appear in the proceedings volume of the conference in honour of M. E. Rudin s retirement (Topology conference 1991). MR**95e:06035****[KS]**S. Koppelberg and S. Shelah,*Subalgebras of Cohen algebras need not be Cohen*, preprint.**[L]**S. Lang,*Algebra*, Addison-Wesley, 1984. MR**86j:00003****[P]**A.T. Paterson,*Amenability*, Mathematical Surveys and Monographs (29), American Mathematical Society, 1988. MR**90e:43001****[T]**S. Turek,*A note on universal minimal dynamical systems*, Comment. Math. Univ. Carolinae**32**(4) (1991), 781-783. MR**93f:54053****[deV]**J. de Vries,*Elements of Topological Dynamics*, Kluwer Academic Publishers, 1993. MR**94m:54098****[W]**S. Wagon,*The Banach-Tarski Paradox*, Cambridge University Press, 1985. MR**87e:04007**

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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

Email:
balcar@beba.cesnet.cz

**Frantisek Franek**

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

Email:
franek@mcmail.cis.mcmaster.ca

DOI:
https://doi.org/10.1090/S0002-9947-97-01868-0

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