Clones above the unary clone

TL;DR

This paper constructs large families of pairwise incomparable clones on a countable set, all sharing the same unary operations, demonstrating the vast diversity of clone structures with fixed unary parts.

Contribution

It introduces the first known large families of incomparable clones with identical unary fragments, expanding understanding of clone lattice complexity.

Findings

Constructed 2^c incomparable clones with the same unary fragment.

For each n, constructed 2^c clones sharing the same n-ary fragment.

Demonstrated the diversity of clone structures with fixed unary operations.

Abstract

Let c be the cardinality of the continuum. We give a family of pairwise incomparable clones (on a countable base set) 2^c members, all with the same unary fragment, namely the set of all unary operations. We also give, for each n, a family of 2^c clones all with the same n-ary fragment, and all containing the set of all unary operations.