Clones from Creatures

Details Clones from Creatures Clones from Creatures

2002-12-30

arxiv.org/abs/math/0212379v3

Mathematics

Rings and Algebras

LaTeX2e, 20 pages. Revised version: some concepts simplified, proof details added

Scientific paper

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.

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