An Element $\phi$-$\delta$-Primary to another Element in Multiplicative Lattices

TL;DR

This paper introduces and characterizes a new type of element called $oldsymbol{\phi ext{-}oldsymbol{\delta} ext{-primary}}$ in multiplicative lattices, exploring its properties and relationships with existing concepts.

Contribution

It defines the $oldsymbol{\phi ext{-}oldsymbol{\delta} ext{-primary}}$ element in multiplicative lattices and investigates its properties and conditions relating to $oldsymbol{\delta}$-primary elements.

Findings

Counterexample shows $oldsymbol{\phi ext{-}oldsymbol{\delta} ext{-primary}}$ does not imply $oldsymbol{\delta}$-primary.

Conditions identified when $oldsymbol{{b}}$ is $oldsymbol{\delta}$-primary given $oldsymbol{{b}}$ is $oldsymbol{\phi ext{-}oldsymbol{\delta} ext{-primary}}$.

Many properties and characterizations of $oldsymbol{\phi ext{-}oldsymbol{\delta} ext{-primary}}$ elements are established.

Abstract

In this paper, we introduce an element $\phi$-$\delta$-primary to another element in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between these structures. By a counter example, it is shown that if an element $b\in L$ is $\phi$-$\delta$-primary to a proper element $p\in L$ then $b$ need not be $\delta$-primary to $p$ and found conditions under which an element $b\in L$ is $\delta$-primary to a proper element $p\in L$ if $b$ is $\phi$-$\delta$-primary to $p$.