Maximal clones on uncountable sets that include all permutations

The minimal clones above the permutations

We determine the atoms of the interval of the clone lattice consisting of those clones which contain all permutations, on an infinite base set. This is equivalent to the description of the atoms of the lattice of transformation monoids above the permutations.

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…

保持关系的置换克隆

Permutation clones that preserve relations

本文研究由关系定义的置换克隆，刻画有限集上其格结构，证明极大借位闭克隆的两类定义，并确定二元集上共有13个此类克隆。

All creatures great and small

Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic.

Clones on infinite sets

A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of Cl(X) for infinite X. In the first chapter we prove the combinatorial result that if X is linearly ordered, then the median functions of different arity defined by that order all generate the same clone. The second chapter deals with clones containing the almost unary functions, that is, all functions whose value is determined by one of its variables up to a…