Polymorphism clones of homogeneous structures: generating sets, Sierpiński rank, cofinality and the Bergman property. (English) Zbl 1402.08001
An infinite group whose connected Cayley graphs all have a finite diameter is said to have the Bergman property. The Bergman property and the notion of strong cofinality are defined for clones. In this paper, the authors show that a clone has uncountable strong cofinality if and only if it has uncountable cofinality and the Bergman property. A class of countable homogeneous structures (including the Rado graph) whose polymorphism clones have strong uncountable cofinality and in particular the Bergman property is characterized. Some open problems are presented in the last section.
Reviewer: Yuri Movsisyan (Yerevan)
MSC:
| 08A05 | Structure theory of algebraic structures |
| 03C15 | Model theory of denumerable and separable structures |
| 03C50 | Models with special properties (saturated, rigid, etc.) |
| 08A35 | Automorphisms and endomorphisms of algebraic structures |
| 08A40 | Operations and polynomials in algebraic structures, primal algebras |
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