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TL;DR

This paper investigates the structure of clone lattices on large sets, showing that under certain set-theoretic assumptions, these lattices are not dually atomic, revealing complex properties of algebraic clones.

Contribution

It demonstrates that the clone lattice on a set of uncountable regular cardinal size is not dually atomic under specific set-theoretic conditions.

Findings

Clone lattice on a set of size kappa is not dually atomic.

Results depend on the assumption 2^kappa=kappa^+.

Provides insights into the structure of algebraic clones on large sets.

Abstract

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.