Weak Bases of Boolean Co-Clones

TL;DR

This paper explores the structure of strong partial clones in Boolean co-clones, providing simple relational descriptions and weak bases that reveal a potentially simpler lattice structure than the full partial clone lattice.

Contribution

It introduces the concept of weak bases for strong partial clones and characterizes their structure within Boolean co-clones, offering insights into their lattice organization.

Findings

Weak bases have a highly regular form.

Intervals of strong partial clones are simpler than the full lattice.

Relational descriptions relate weak bases to minimal members.

Abstract

Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For classifications where primitive positive definitions are unsuitable, such as size-preserving reductions, weaker closure operations may be necessary. In this article we consider strong partial clones which can be seen as a more fine-grained framework than Post's lattice where each clone splits into an interval of strong partial clones. We investigate these intervals and give simple relational descriptions, weak bases, of the largest elements. The weak bases have a highly regular form and are in many cases easily relatable to the smallest members in the intervals, which suggests that the lattice of strong partial clones is considerably simpler than the full lattice of partial clones.