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arXiv · 2109.01371

The lattice of clones of self-dual operations collapsed

Manuel Bodirsky·Albert Vucaj·Dmitriy Zhuk
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Abstract

There are continuum many clones on a three-element set even if they are considered up to \emph{homomorphic equivalence}. The clones we use to prove this fact are clones consisting of \emph{self-dual operations}, i.e., operations that preserve the relation $\{(0,1),(1,2),(2,0)\}$. However, there are only countably many such clones when considered up to equivalence with respect to \emph{minor-preserving maps} instead of clone homomorphisms. We give a full description of the set of clones of self-dual operations, ordered by the existence of minor-preserving maps. Our result can also be phrased as a statement about structures on a three-element set, ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure $\mathfrak A$ to the polymorphism clone of a finite structure $\mathfrak B$ if and only if there is a primitive positive construction of $\mathfrak B$ in $\mathfrak A$.

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Manuel Bodirsky, Albert Vucaj, Dmitriy Zhuk. 2021-09-03. The lattice of clones of self-dual operations collapsed. https://arxiv.org/abs/2109.01371

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