On $S$-Prime Element Principle

TL;DR

This paper introduces the concept of S-prime elements in V-lattices and presents an S-Prime Element Principle that provides a unified approach to establishing the existence of prime elements in multiplicative lattices.

Contribution

The paper defines S-prime elements in V-lattices and formulates a new principle that simplifies proofs of prime element existence in multiplicative lattices.

Findings

Introduced S-prime elements in V-lattices.

Established the S-Prime Element Principle.

Provided a unified approach for prime element existence.

Abstract

In this paper, we introduce $S$-prime elements in $V$-lattices, where $S$ is a multiplicatively closed subset of a $V$-lattice $L$. In addition, we introduce the $S$-Prime Element Principle to prove that certain elements in $V$-lattices are $S$-prime elements. This principle leads to a direct and uniform approach to the results on the existence of prime elements in multiplicative lattices when $S=\{1\}$.