Non-generators in complete lattices and semilattices

TL;DR

This paper investigates the properties of non-generators in complete lattices and semilattices, revealing that certain structural characteristics hold in semilattices but not in lattices.

Contribution

It extends the understanding of non-generators from algebraic structures to complete semilattices and lattices, highlighting differences in their substructure properties.

Findings

Non-generators form a substructure in complete semilattices.

The same property does not hold for complete lattices.

Structural properties differ significantly between semilattices and lattices.

Abstract

As well-known, in a finitary algebraic structure the set $\Gamma$ of all the non-generators is the intersection of all the maximal proper substructures. In particular, $\Gamma$ is a substructure. We show that the corresponding statements hold for complete semilattices but fail for complete lattices, when as the notion of substructure we take complete subsemilattices and complete sublattices, respectively.