140 research outputs found

### 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
small set. We show that on X of regular cardinality, the set of such clones is
always a countably infinite descending chain. The third chapter generalizes a
result due to L. Heindorf from the countable to all uncountable X of regular
cardinality, resulting in an explicit list of all clones containing the
permutations but not all unary functions of X. Moreover, all maximal submonoids
of the full transformation monoid which contain the permutations of X are
determined, on all infinite X; this is an extension of a theorem by G. Gavrilov
for countable base sets.Comment: 70 pages; Dissertation written at the Vienna University of Technology
under the supervision of Martin Goldstern; essentially consists of the
author's papers "The clone generated by the median functions", "Clones
containing all almost unary functions, "Maximal clones on uncountable sets
that include all permutations" which are all available from arXi

### 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.Comment: 6 page

- â€¦