Alpha Ideals and the Space of Prime Alpha Ideals in Universal Algebras

Abstract

The purpose of this paper is to study -ideals in a more general context, in universal algebras having a constant . Several characterizations are obtained for an ideal of an algebra to be an -ideal. It is shown that the class of all -ideals of an algebra forms an algebraic lattice. Prime -ideals and several related properties are investigated. Some properties of the spectral space of prime -ideals equipped with the hull-kernel topology are derived.

1. Introduction

-ideals were first studied by Cornish [1] for a class of distributive lattices as a generalization of annihilator ideals. Later on, the notion of -ideals was extended to the class of -distributive lattices by Pawar and Khopade [2], to the class of almost distributive lattices by Rao and Rao [3], to the class of -algebras by Rao [4], and more generally to arbitrary posets by Mokbel [5]. Jayaram [6] studied prime -ideals in -distributive lattices and he topologized the class of all prime -ideals in -distributive lattices. Bigard [7] has also studied ideals in the context of lattice-ordered groups. Instead of annulets, he used the dual lattice of carriers.

Cornish, in his paper [1], studied ideals in a distributive lattice , by using the lattice of all annulets of the form (where ) which is a sublattice of the Boolean algebra of all annihilator ideals in . Cornish has defined two operators and as follows: for an ideal in ,is a filter in and, conversely, for a filter of ,is an ideal in . It is observed that the map forms a closure operator on the class of all ideals of and ideals in are defined to be those ideals which are closed with respect to this closure operator, i.e., is an ideal of if

In fact, this property can be equivalently expressed in the following way: is an ideal of if and only if for all . Cornish has observed that, by using the structure of , results can be transferred to give information on the ideal structure of . The most interesting result of this type is that is a generalized Stone lattice if and only if each prime ideal contains a unique prime ideal. Moreover, necessary and sufficient conditions are also given for a distributive lattice with to be disjunctive using ideals.

In [8], Chajda and Halas have introduced the notion of annihilators in general universal algebras having a constant . In [9], we propose a new approach to study annihilators in universal algebras by the use of commutator terms. In the present paper, we continue our study and define -ideals as a generalization of those annihilator ideals in universal algebras. We have also studied the basic topological properties of the space of -prime ideals in universal algebras. This is a reasonable abstraction of the theory of -ideals of distributive lattices to general universal algebras with abstract finitary operations. The results of this paper are important to extend the properties of distributive lattices to other classes of algebras such as groups, rings, BCK/BCI algebras, MV-algebras, and Hilbert algebras.

2. Preliminaries

This section contains some definitions and results which will be used in this paper. For the standard concepts in universal algebras, we refer to [10–12]. Throughout this paper, , where is a class of algebras of a fixed type and assume that there is an equationally definable constant in all algebras of denoted by . For a positive integer , we write to denote the tuple . The ary terms of type are formal expressions obtained in finitely many steps by the following process:(1)The variables are ary terms of type (2)If , are ary terms of type and , then is also a term of type

Definition 1 (see [13]). An -ary term is said to be an ideal term in iffor all . A nonempty subset of will be called an ideal of if for all and any ideal term in .
We denote the class of all ideals of by . It is easy to check that the intersection of any family of ideals of is an ideal. So, for a subset , there always exists a smallest ideal of containing which we call the ideal of generated by and it is denoted by . Note that if and only if for some , and where is an ary ideal term in . If , then we write instead of . In this case, if and only if for some , where is an ary ideal term in .

Definition 2 (see [13]). A class of algebras is called ideal determined if every ideal is the zero-congruence class of a unique congruence relation denoted by . In this case, the map defines an isomorphism between the lattice of ideals and the congruence lattice of .

Definition 3 (see [13, 14]). A term is said to be a commutator term in if it is an ideal term in and an ideal term in .

Definition 4 (see [13]). In an ideal determined variety, the commutator of ideals and is the zero-congruence class of the commutator congruence .
Throughout this paper, is assumed to be an ideal-determined variety and each has a -unit.

Theorem 1 (see [13, 14]). Let and be ideals of . Then

For subsets and of , denotes the product . In particular, for , is denoted by . The following lemma gives some properties of the commutator of ideals. Further details on ideals of universal algebras can be found in [15–17].

Lemma 1 (see [14]). For each , we have the following:(1)(2) implies (3)The commutator of ideals is distributive over arbitrary joins of ideals

Definition 5 (see [14]). A nonzero element in is said to be a formal unit if , i.e., is generated by as an ideal.
A cyclic group, a ring with unity, a bounded lattice, and an almost distributive lattice with maximal elements are examples of an algebra having a formal unit [9].

Definition 6. A nonzero element is a called a commutator unit (or -unit for short) in if for all [9].

Example 1. (1)In a ring with unity, the elements and are -unit(2)In a bounded distributive lattice , the largest element is a -unit(3)In a bounded Hilbert algebra , the least element is a -unit(4)Every maximal element in an almost distributive lattice is a -unit

3. Annihilators

This section contains some important results on annihilators and annihilator ideals of universal algebras taken from.

Definition 7 (see [9]). Let . For any subset of , we defineWe call the relative annihilator of with respect to . For a subset of , the annihilator of denoted by is defined to beIf , then we denote by , i.e.,It is observed that is an ideal of and for all . Also, it can be verified that and .

Lemma 2. For any subsets and of ,(1)(2)(3)(4)(5)An element in a ring is said to be idempotent if . By imitating this property to the general case of universal algebras, we define the following.

Definition 8. An element is called idempotent ifIf all elements of are idempotent, then we call a commutator idempotent algebra (or a -idempotent algebra for short). In other words, the commutator of ideals in is idempotent.
Boolean rings and more generally lattices with least element are the most natural examples of -idempotent algebras. One can also verify that regular rings (a ring is regular if for each there exists such that ) are examples of -idempotent algebras. One of the most important properties of idempotent algebras is that the commutators of ideals coincide with their intersection. Further, the separation axiom holds in idempotent algebras.

Definition 9. An ideal of is called an annihilator ideal if for some nonempty subset of .
It is immediate from the definition that an ideal of is an annihilator ideal if and only if . We denote the class of all annihilator ideals of by . If is a idempotent algebra, then it is proved that forms a Boolean algebra.

Definition 10. An ideal of is called a dense ideal if ; otherwise, is called a nondense ideal.

4. Ideals

In this section, we study -ideals in universal algebras in a more general context.

Notation 1. We write to say that is a finite subset of .

Definition 11. An ideal of is called an ideal ifWe denote by the class of all ideals of .
It is easy to check that and belong to the class .

Remark 1. If is a distributive lattice, then this definition coincides to the definition proposed by Cornish [1].

Lemma 3. Every annihilator ideal of is an ideal.

Remark 2. The converse of Lemma 3 does not hold in general. For instance, in an infinite algebra with , every proper dense -ideal is not an annihilator ideal.

Lemma 4. In a finite algebra , every ideal is an annihilator ideal.

Theorem 2. An ideal of is an ideal if and only if

Proof. Suppose that is an -ideal. Then for all finite subsets of , which givesAlso, if , then is a finite subset of such that . So, the other inclusion holds. The converse part is straightforward.

Definition 12. For each , let us define a set as follows:In the following lemma, we give another description for the set which will be useful to prove Theorem 2.

Lemma 5. For each , we have

Proof. Let us define two sets and as follows:We show that . Since for all , it holds immediately that . To prove the other inclusion, let . Then there is and such that and , which gives . So, and therefore the equality holds.

Lemma 6. For each , we have the following:(1)(2)(3)

Theorem 3. For each , is an ideal of .

Proof. Let be an -tuple of elements from . Then, for each , there exist and finite subsets of such that and . Put and . Then is a finite subset of and such that and and . Since , for all . So, . That is, is an ideal of such that . Hence, . Now if and is an ideal term in , then . Therefore, is an ideal of .

Theorem 4. For each , is the smallest -ideal of containing .

Proof. We first show that is an -ideal. Let us putLet . We show that . If , then . Suppose that , and let . Then, for each , there exist and finite subsets of such that and for all . Put and . Then is a finite subset of , and such that and and . So, for all , which gives . Since , we have . It remains to show that . If we put , then is an ideal of such that . Since is a finite subset of with , we get . Therefore is an -ideal. It is also observed in Lemma 6 that . Now let be any other -ideal of such that and let . Then there exists and such that and . So, . Also, since and is an -ideal, we get . Now consider the following:Hence, and therefore is the smallest -ideal containing .

Theorem 5. An ideal of is an ideal if and only if .

Proof. Suppose that is an -ideal. Since the other inclusion is trivial, we proceed to show that . Let . Then there exists and such that and . We have the following:Therefore, and hence the equality holds. The converse part is straightforward.
From Lemma 6, it can be deduced that the map is a closure operator on and ideals of are precisely the closed elements of with respect to this closure operator.

Theorem 6. An ideal I of A is an α-ideal if and only if for each finite subset F of A and any subset E of A, F^∗ = E^∗ and F ⊆ I together imply E ⊆ I.

Proof. Suppose that is an -ideal and let and such that . Then . If , then and hence . So . Conversely, suppose that the above condition is satisfied. Let . We show that . Put . Then such that . As , we get . Hence, the result holds.

Theorem 7. Let be a idempotent algebra and be a prime ideal of . If is nondense, then it is an ideal.

Proof. Suppose that is nondense. Then there exists a nonzero which implies that . Thus is an annihilator ideal and hence it follows from Lemma 3 that is an -ideal.

Theorem 8. The collection forms a complete lattice.

Proof. It is enough to show that is closed under arbitrary intersection. Let be an indexed family of ideals in . Put . If is empty, then and hence it is an ideal. Assume that is nonempty. Then is an ideal of . Moreover, if is a finite subset of , then it can be easily observed that and therefore it is an ideal.
Note also that, for any , their supremum is given byFor a nonempty set , remember from [12] that a family of subsets of will be called a closure system in if it forms a complete lattice together with the usual inclusion order. Furthermore, a closure system is an algebraic lattice if and only if every chain in has supremum in .

Theorem 9. The collection is an algebraic lattice.

Proof. It is enough to show that every chain in has an upper bound in . Let be a chain in . Put . We first show that is an ideal. Let be an ary ideal term in , , and . Then, there exist such that each . Since the family is a chain, we can find with . With being an ideal of , it holds thatTherefore, is an ideal of . It remains to show that is an ideal. Let be a finite subset of . If , then . Assume that is nonempty and let . Then, there exist such that each . Since the family is a chain, we can find with , i.e., . With being an ideal of , it follows that . Therefore, is an ideal. This completes the proof.
It can be deduced from the above theorem that the closure operator is in fact an algebraic closure operator. Moreover, the compact elements in are those finitely generated ideals of .

Definition 13 (see [14]). A subset of is called an system if, for all , .
In the following lemma, we give another description for systems for our purpose.

Lemma 7. A nonempty subset of is an system if and only if, for any nonempty subsets and of ,

Proof. Suppose that is an -system. Let be nonempty subsets of M. Choose and . Then . Since , , which gives . The converse part is straightforward.

Definition 14. For an system of , let us define a set as follows:

Lemma 8. For each system of ,

Theorem 10. For each system of , the set is an ideal of .

Proof. Let be a -tuple of elements from . Then for each there exist and nonempty subsets of such that and for all . If we put , then such that . On the other hand, let us define a sequence as follows:Since is an -system, each is a nonempty subset of such that . Moreover,Now consider the following:Therefore, . Now, if and is an ideal term in , we get . So, . Therefore is an ideal.

Theorem 11. For each system of , the set is an ideal of .

Proof. Let be a finite subset of . Then there exist and nonempty subsets of such that and for all . If we let , then such that . On the other hand, let us define a sequence as follows:Since is an -system, each is a nonempty subset of such that . Moreover,Now consider the following:That is, , where is a nonempty subset of . So, , which gives . Thus, and hence . Since , we have . So, . Therefore, is an -ideal.

Definition 15. For each prime ideal of , let us define as follows:If we let , then is an -system of such that . So, it follows from Theorems 10 and 11 that is an -ideal of . Moreover, if is minimal prime ideal, then is an -ideal containing .

Theorem 12. Let be a idempotent algebra. Then the following are equivalent:(1)Every ideal of is an -ideal(2)Every prime ideal of is an -ideal

Proof. is obvious. We proceed to show . Assume . Let be an ideal of . It is enough if we show that . Suppose not. Then there exists but . Since is idempotent, we can apply Zorn’s lemma to obtain a prime ideal of such that and . So . This is a contradiction. Thus, and hence is an ideal.

Remark 3. If the algebra we are working on is a distributive lattice, then one (and hence all) of the conditions of Theorem 12 is necessary and sufficient condition for to be disjunctive.

5. The Space of Prime -Ideals

Definition 16. By a prime -ideal of , we mean an -ideal of satisfying the conditionfor all .

Theorem 13. Let be an -ideal of and be an -system of such that . Then there exists a prime -ideal of such that and .

Proof. Let us putClearly is a nonempty family of -ideals of satisfying the hypothesis of Zorn’s lemma. So has a maximal element, say . Our aim is to show that is prime. Suppose not. Then there exist such that . If we put and , then and are -ideals of properly containing . By the maximality of in , both and do not belong to . So and . Choose such that and . Consider the following:Since , it follows that . This is a contradiction. Therefore, is prime.

Corollary 1. Let be an -ideal of and such that . If is -idempotent, then there exists a prime -ideal of such that and .

Proof. Since is -idempotent, every singleton is an -system. So, the proof follows from Theorem 13.

Theorem 14. Let be a -idempotent algebra. Then, for any -ideal of , we have

Proof. Let us putClearly, . To prove the other inclusion, let . Then, by Corollary 1, there exists a prime -ideal of such that and , i.e., . Thus, and hence the equality holds.

Theorem 15. Let be a -idempotent algebra. Then, for any -ideal of , we have

Proof. Let us putLet and be a prime -ideal of with . If , then . Since is prime and it follows that , i.e., , and hence . To prove the other inclusion, let and be any prime -ideal of with . If , then . Choose a nonzero . By Corollary 1, there exists a prime -ideal of such that , which gives . Then and hence . Therefore, and the equality holds.
In the rest of our work, is assumed to be -idempotent.
Let us now give the following notations:For each subset of , letFor each , we write instead of . For any subset of , one can verify that

Lemma 9. For any -ideal of , we have

Proof. The equality holds trivially. For the other equality, it is enough to show that for all . For any prime -ideal of , one can verify that if and only if .

Lemma 10. The following conditions hold for all :(1)(2)(3)(4)

Lemma 11. The following conditions are equivalent for any ideals and of :(1)(2)(3)

Theorem 16. The family forms a topology on .

Proof. Clearly, both and belong to the family . Moreover, it follows from (2) and (4) of Lemma 10 that is closed under finite intersection and arbitrary unions and thus it is a topology on .

Definition 17. We call the topology the -prime spectrum of and sometimes it may be denoted as .

Lemma 12. The subfamily constitutes a base for the topology .

Proof. Let be n-ideal of and . Then there exists such that , i.e., . Thus is a base for .

Theorem 17. Each is a compact open subset of .

Proof. Let be a basic open cover for , i.e.,Let be an ideal of generated by . Then . Our aim is to show that . Suppose on the contrary that . Then there exists a prime -ideal of such that and , i.e., but , which is a contradiction. Thus, . There exists a finite subset of such that . Since is generated by , there exist and such that . If we put , then is a finite subset of such that . Now we show thatFor any prime -ideal of , if , then and hence , i.e.,This completes the proof.

Corollary 2. If has a dense element, then is compact.

Proof. If has a dense element say , then . It follows from Theorem 17 that is compact.

Theorem 18. If every finitely generated ideal in is a principal ideal, then is the set of all compact open subsets of .

Proof. Suppose that every finitely generated ideal in is a principal ideal. It is proved in Theorem 17 that every element in is compact and open. Let be a compact open subset of . Then there is some -ideal of such thatSince is compact, there exist such thatIt follows from our assumption that there is some such that . Hence proved.

Corollary 3. If every finitely generated ideal in is a principal ideal and is compact, then has dense elements.

Theorem 19. There is a lattice isomorphism between the lattice of all -ideals of and the lattice of all open subsets of .

Proof. One can easily verify that the map is an isomorphism between the lattice of all -ideals of and the lattice of open subsets of .
We denote the closure, interior, and exterior of a subfamily of by , , and , respectively. The following theorem shows that the topology on is the hull-kernel topology.

Theorem 20. For any subfamily of , .

Proof. Suppose that . Then every neighborhood of meets . If we put , then is an -ideal of such that for all . Our claim is to show that . Suppose not. Then and hence there is some such that , which is a contradiction. Thus, . Conversely, suppose that and let be any -ideal of with . We show that . If not, then for all and hence , which is a contradiction. Therefore, and the equality holds.

Theorem 21. For any -ideal of ,

Proof. By Theorem 20, we haveSo . Now consider the following:An open subset of a topological space is called an open domain if it is identical with the interior of its closure and it is called semiregular if it is a union of open domains.

Theorem 22. For an -ideal of , is an open domain if and only if .

Proof. It follows from the above theorem that . Now consider the following:If , then and hence is an open domain. Conversely, if , then and it follows from Lemma 13 that .

Corollary 4. For any -ideal of , is semiregular. In particular, if has a dense element, then is semiregular.

Theorem 23. If every finitely generated ideal in is principal, then the following conditions are equivalent:(1)For each , there exists such that (2)Exterior of every compact open subset of is compact

Proof. Suppose that (1) holds. Let be a compact open subset of . By Theorem 18, for some . Now . By (1), there exists such that . Thus, . Again, by Theorem 18, is compact. Conversely assume (2). Let . Then and its exterior are compact, i.e., is compact and open. By Theorem 18, there exists such that . It follows from Lemma 11 that and this completes the proof.
If has formal unit, then it was proved in [14] that every proper ideal of is contained in a maximal ideal of . This can be proved by applying Zorn’s lemma.

Theorem 24. If has a formal unit, then is a -space if and only if every -prime ideal of is maximal.

Proof. Suppose that is . Let be an -prime ideal of , which is not maximal. Since has a formal unit, there exists a maximal ideal of such that . As is a -space, there exist basic open sets and such that , , , and , i.e., and . This is a contradiction. Thus, is maximal. Conversely, suppose that every prime -ideal of is maximal. Let and be two distinct elements of . Then, by assumption, and . This implies that there exist such that and . Then we have and . Thus, is a -space.

Theorem 25. If is a Hausdorff space containing more than one element, then there exist such that

Proof. Suppose that is Hausdorff. Let and be distinct elements in . Then there exist two basic open sets, say and , such that , , and . For any , as , the conditions and are not possible.

Case 1. If and , then .

Case 2. If and , then .

Case 3. If , then .
In all cases, it holds that . This completes the proof.

Theorem 26. The following are equivalent:(1) is Hausdorff(2)For any two distinct elements and of , there exist such that and there does not exist any element of such that and

Proof. Suppose that is Hausdorff and let and be distinct elements of . Then there exist two basic open sets and such that and . Now suppose on the contrary that there is such that and . Then , which is absurd. Therefore, there does not exist any element of such that and . Conversely, assume (2). Let and be two distinct elements in . Then, by assumption, there exist such that and and there does not exist any element in such that and , i.e., and . Thus, is Hausdorff.

Theorem 27. is regular if and only if, for any and (), there exist an ideal of and such that .

Proof. Suppose that is regular. Let and . Then, . As is regular, there exist -ideals and of such that , , and . Then, and . From , we have . So we can choose . Now we show that . If , then and hence . Thus, and . Taking their complement, we get . Thus, for any and , there exist an ideal of and such thatConversely, suppose that the condition of the theorem is satisfied. To show that is regular, let and be an -ideal of such that . Then . Choose and . By assumption, there exist an -ideal of and such thatIf we put and , then and are disjoint open sets in such that and . Thus, is regular.

6. Conclusion

The study of the space of -prime ideals of universal algebras is initiated in the paper and it needs more investigation. It is observed by the author that it is promising to obtain a reticulation of a -idempotent algebra induced by -ideals.

Data Availability

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Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.