Clones from Creatures
Rings and Algebras
2007-05-23 v3 Logic
Abstract
A clone on a set X is a set of finitary operations on X which contains all the projections and is closed under composition. The set of all clones forms a complete lattice Cl(X) with greatest element O, the set of all finitary operations. For finite sets X the lattice is "dually atomic": every clone other than O is below a coatom of Cl(X). It was open whether Cl(X) is also dually atomic for infinite X. Assuming the continuum hypothesis, we show that there is a clone C on a countable set such that the interval of clones above C is linearly ordered, uncountable, and has no coatoms.
Keywords
Cite
@article{arxiv.math/0212379,
title = {Clones from Creatures},
author = {Martin Goldstern and Saharon Shelah},
journal= {arXiv preprint arXiv:math/0212379},
year = {2007}
}
Comments
LaTeX2e, 20 pages. Revised version: some concepts simplified, proof details added