Partial clones containing all Boolean monotone self-dual partial functions

TL;DR

This paper investigates the structure of partial clones on the Boolean domain, demonstrating that the set of all partial clones containing all monotone self-dual partial functions is uncountably infinite.

Contribution

It proves that the collection of partial clones containing all monotone self-dual functions has continuum cardinality, extending understanding of clone lattice complexity.

Findings

The set of partial clones containing all monotone self-dual functions is of continuum size.

Maximal partial clones on  ext{ are well-studied, but their intersections are complex.

The structure of partial clones containing these functions is highly rich and uncountably infinite.

Abstract

The study of partial clones on $\mathbf{2}:=\{0,1\}$ was initiated by R. V. Freivald. In his fundamental paper published in 1966, Freivald showed, among other things, that the set of all monotone partial functions and the set of all self-dual partial functions are both maximal partial clones on $\mathbf{2}$. Several papers dealing with intersections of maximal partial clones on $\mathbf{2}$ have appeared after Freivald work. It is known that there are infinitely many partial clones that contain the set of all monotone self-dual partial functions on $\mathbf{2}$, and the problem of describing them all was posed by some authors. In this paper we show that the set of partial clones that contain all monotone self-dual partial functions is of continuum cardinality on $\mathbf{2}$.