Reconstructing the topology on monoids and polymorphism clones of the rationals

Polymorphism clones of homogeneous structures (Universal homogeneous polymorphisms and automatic homeomorphicity)

Every clone of functions comes naturally equipped with a topology---the topology of pointwise convergence. A clone $\mathfrak{C}$ is said to have automatic homeomorphicity with respect to a class $\mathcal{C}$ of clones, if every clone-isomorphism of $\mathfrak{C}$ to a member of $\mathcal{C}$ is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we study automatic homeomorphicity-properties for polymorphism clones of countable homogeneous relational structures. To this end we introduce and utilize universal homogeneous polymorphisms. Next to two gen…

The number of clones determined by disjunctions of unary relations

We consider finitary relations (also known as crosses) that are definable via finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite parameter set $Γ$. We prove that whenever $Γ$ contains at least one non-empty relation distinct from the full carrier set, there is a countably infinite number of polymorphism clones determined by relations that are disjunctively definable from $Γ$. Finally, we extend our result to finitely related polymorphism clones and countably infinite sets $Γ$.

On a stronger reconstruction notion for monoids and clones

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action…

关于自同态幺半群和多态克隆上拓扑重构的指南

A guide to topological reconstruction on endomorphism monoids and polymorphism clones

综述自同态幺半群和多态克隆上拓扑重构研究，介绍结构对称空间的代数与拓扑结构，回顾自同构群相关问题研究历程，阐述对上述类似问题的研究现状，还完善并扩展了一些现有结果。

The number of unary clones containing the permutations on an infinite set

We calculate the number of unary clones (submonoids of the full transformation monoid) containing the permutations, on an infinite base set. It turns out that this number is quite large, on some cardinals as large as the whole clone lattice. Moreover we find that, with one exception, even the cardinalities of the intervals between the monoid of all permutations and the maximal submonoids of the full transformation monoid are as large. Whether or not the only exception is of the same cardinality as the other intervals depends on additional axioms of set theory.

Precomplete clones on infinite sets which are closed under conjugation

We show that on an infinite set, there exist no other precomplete clones closed under conjugation except those which contain all permutations. Since on base sets of some infinite cardinalities, in particular on countably infinite ones, the precomplete clones containing the permutations have been determined, this yields a complete list of the precomplete conjugation-closed clones in those cases. In addition, we show that there exist no precomplete submonoids of the full transformation monoid which are closed under conjugation except those which contain the permutations; the monoids of the latte…