An Element Weakly Primary to Another Element

We introduce the concept “An element weakly primary to another element” and using this concept we have generalized some result proved by Manjarekar and Chavan (2004). It is shown that if is a family of elements weakly primary to a in L, then is weakly primary to a.


Introduction
Multiplicative lattice is a complete lattice provided with commutative, associative, and join distributive multiplication for which the largest element 1 acts as a multiplicative identity. A proper element of is called prime element if ⩽ ⇒ ⩽ or ⩽ for , ∈ and is called primary element if ⩽ implies ⩽ or ⩽ for some ∈ + . An element of is called compact if ⩽ ∨ , and ⊆ implies the existence of finite number of elements 1 , 2 , 3 , . . . , of such that ⩽ 1 ∨ 2 ∨ 3 ∨ ⋅ ⋅ ⋅ ∨ . Throughout this paper, denotes compactly generated multiplicative lattice with 1 compact and every finite product of compact elements is compact. Let * be the set of all compact elements in . Also, ( : ) is the greatest element in such that ⩽ . An element ∈ is join principal if ∨ ( : ) = ( ∨ ) : and meet principle if ∧ = (( : ) ∧ ) , for all , ∈ . An element is principle if it is both join and meet principle. For ∈ , √ = ∨{ ∈ * | ⩽ for some ∈ + }. An element ∈ is called semiprimary if √ is primary element. is said to satisfy the condition ( * ) if every semiprimary element is primary element.
An element ∈ is said to be strong join principle element if is compact and join principle. An element ∈ is p-primary if is primary and √ = and ∈ is semiprime if √ = . An element of is called zero divisor if ∃0 ̸ = ∈ such that = 0, and if has no zero divisor then will be called lattice domain or simply a domain. * denotes the set of compact elements of . The concept of weakly prime element is studied by Ç allialp et al. [1]. The concept of weakly primary element is introduced by Sachin and Vilas [2]. For other definitions and simple properties of multiplicative lattice, one can refer to Dilworth [3].

Definition 1.
Weakly primary element is defined as follows.
An element ∈ is said to be a weakly primary element if for , ∈ * , 0 ̸ = ⩽ implies ⩽ or ⩽ for some ∈ + .
In the lattice of Example 2, an element is weakly primary element. From Definition 1, it is clear that every weakly prime element is weakly primary element Converse need not be true. Since in Example 2, is weakly primary element but it is not weakly prime element. Further, if is a weakly primary element, then √ is a weakly prime element. Because if for compact element and such that 0 ̸ = ⩽ √ then = ( ) ⩽ for some ∈ + . As is a weakly primary element, either ⩽ or ( ) = ⩽ for some ∈ + . Consequently, ⩽ √ or ⩽ √ . Thus, √ is a weakly prime 2 Chinese Journal of Mathematics  element. This implies that every weakly primary element is a weakly semiprimary element. It need not be true that is always weakly prime or is always weakly semiprimary. In Example 2, the least element 0 is not semiprimary as √ 0 = is not a weakly prime element. The concept of "An element primeto another element" is introduced in [4]. An element ∈ is primeto an element if for ∈ , ⩽ implies ⩽ . Now, we define the following.
Definition 3. An element weakly prime to another element is defined as follows. An element, ∈ , is called weakly prime to an element ∈ if for any ∈ , 0 ̸ = ⩽ implies ⩽ . In Example 2, the element is weakly prime to an element , but is not weakly prime to any other element of . This follows directly from the fact that an element is weakly prime to an element if and only if : = .

An Element Weakly Primary to Another Element
Now we introduce the following main concept which is a generalization of the concept introduced by Manjarekar and Chavan [5].
Definition 4. An element is said to be weakly primary to another element in if for ∈ * , 0 ̸ = ⩽ implies ⩽ for some ∈ + .
In Example 2, the element is weakly primary to , but note that is not weakly prime to . This follows directly from Corollary 9, and note that ( : ) = = √ and < . Evidently, if is weakly prime to , then is weakly primary to in . Now if is a weakly prime element and , then is weakly prime to , and if is a weakly primary element and a compact element , then is a weakly primary element to .
Thus, from this, it is clear that elements weakly primary to another element exist in the lattice . Since is compactly generated multiplicative lattice with 1 compact, weakly prime element and hence weakly primary element exists in . Hereafter, will be a domain. We prove some interesting results including characterizations.

Theorem 5.
No proper nonzero element is weakly prime or weakly primary to itself in .
Proof. If is a proper nonzero element in and is weakly primary to itself, then 0 ̸ = .1 = implies that 1 = , a contradiction. Therefore, no proper nonzero element is weakly prime or weakly primary to itself in . Now we prove some characterizations of an element weakly primary to .
Theorem 6. Let ∈ be a semiprime element. Then 0 ̸ = is weakly primary to if and only if is weakly prime to .
Theorem 11. Let be a lattice domain. Let , ∈ . Then, 0 ̸ = is weakly prime to if and only if b is weakly prime to ( : ) for every 0 ̸ = ∈ .
Theorem 12. Let , ∈ and let be a lattice domain. If is weakly primary to and is a semiprime element in , then is weakly primary to ( : ) for every ∈ .
Proof. It follows from Theorems 6 and 11.
Theorem 13. Let , ∈ . Then 0 ̸ = is weakly primary to in if and only if each ⩾ is weakly primary to .
This shows that each ⩾ is weakly prime to . The converse part is obvious.

Theorem 14. If { } is a family of a elements weakly primary to in , then ⋁
is weakly primary to .
Proof. It follows from the fact that ⩽ ⋁ and from Theorem 13.
Theorem 15. Let , ∈ . Then, 0 ̸ = is nonweakly primary to if and only if each ⩽ is nonweakly primary to .
This lemma leads us to the following two obvious corollaries.

Corollary 16. If { } is a family of elements nonweakly primary to in , then ⋀
is nonweakly primary to .
Corollary 17. If ⋁ is nonweakly primary to in , then each is nonweakly primary to .
Theorem 18. If y is compact and 0 ̸ = is nonweakly primary to a, then either is nonweakly primary to or is nonweakly primary to for some ∈ + . Definition 19. Completely meet semiprimary elements are defined as follows.
An element is said to be completely meet semiprimary element if √ is a completely meet prime element.
Example 20. Every element < is not a completely meet prime element. But note that √ = . Thus, each < is a completely meet semiprimary element (see Figure 2).
Theorem 21. Suppose that is a completely meet semiprimary element of and let { } ⊆ . If each 0 ̸ = is nonweakly primary to in , then ⋁ is nonweakly primary to a.

Theorem 22.
If is a completely meet semiprimary element, then is weakly primary element and is nonweakly primary to .
Proof. By Theorem 21, it follows that is nonweakly primary element to . Let and be compact elements such that, 0 ̸ = ⩽ . Then by Theorem 15, is nonweakly primary to . Therefore, by Theorem 18, either is nonweakly primary to or is nonweakly primary to for some ∈ + . Thus, ⩽ or ⩽ for some ∈ + . This shows that is a weakly primary element.