-Prime and -Primary Elements in Multiplicative Lattices

We investigate -prime and -primary elements in a compactly generated multiplicative lattice . By a counterexample, it is shown that a -primary element in need not be primary. Some characterizations of -primary and -prime elements in are obtained. Finally, some results for almost prime and almost primary elements in with characterizations are obtained. 1. Introduction A multiplicative lattice is a complete lattice provided with commutative, associative, and join distributive multiplication in which the largest element 1 acts as a multiplicative identity. An element is called compact if, for , implies the existence of a finite number of elements in such that . The set of compact elements of will be denoted by . A multiplicative lattice is said to be compactly generated if every element of it is a join of compact elements. Throughout this paper denotes a compactly generated multiplicative lattice with 1 compact in which every finite product of compact elements is compact. An element is said to be proper if . A proper element is called a prime element if implies or , where , and is called a primary element if implies or for some , where . A proper element is said to be weakly prime if implies either or , where , and is called weakly primary if implies or for some , where . For , . The radical of is denoted by and is defined as . An element is called semiprimary if is a prime element and is called semiprime if . An element is called join irreducible if implies or . A proper element is said to be a maximal element if for any other proper element . An element is said to be nilpotent if for some . An element is called a zero divisor if for some and is called an idempotent if . A multiplicative lattice is said to be a domain if it is without zero divisors and is said to be quasi-local if it contains a unique maximal element. A quasi-local multiplicative lattice with maximal element is denoted by . An element is called meet principal if for all . An element is called join principal if for all . An element is called weak meet principal if for all . An element is called weak join principal if for all . An element is called principal if is both meet principal and join principal. An element is called weak principal if is both weak meet principal and weak join principal. A multiplicative lattice is a Noether lattice if it is modular, principally generated (every element is a join of principal elements) and satisfies the ascending chain condition. A Noether lattice is local if it contains precisely one maximal prime. In a Noether lattice , an element is said to satisfy