Automatic Homeomorphicity of Locally Moving Clones

2015/12/01 by Robert Barham, Barham, Robert · 2 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #math.LO #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1512.00251

openalex publication_date 2015/12/01 · arxiv created 2016/07/26 · arxiv updated 2016/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend the work of M. Rubin on locally moving groups to clones, showing that a locally moving polymorphism clone has automatic homeomorphicity with respect to the class of all polymorphism clones. We show that if Pol(M,L) is the polymorphism clone of a reduct of (ℚ,<) or (\mathbbL,C) such that Aut(M,L) \not= Aut(M,=) and End(M,L) = Emb(M,L) then Pol(M,L) is locally moving (and hence has automatic homeomorphicity with respect to the class of all polymorphism clones), where ℚ is the rationals, (\mathbbL,C) is the infinite binary-branching homogeneous C-relation. We also show that if M=(\mathbbB, ∪ , ∩, ^c, 1, 0), the Fraïssè Generic Boolean algebra, then Pol(M) is locally moving.

Cited by

• Reconstruction theorems for semigroups of functions which contain all transpositions of a set and for clones with the same property
• Reconstructing the topology of the elementary self-embedding monoids of countable saturated structures

Related

• Polymorphism clones of homogeneous structures (Universal homogeneous polymorphisms and automatic homeomorphicity)
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• The lattice of clones of self-dual operations collapsed
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• Permutation clones that preserve relations