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

(e purpose of this paper is to study α-ideals in a more general context, in universal algebras having a constant 0. Several characterizations are obtained for an ideal I of an algebra A to be an α-ideal. It is shown that the class of all α-ideals of an algebra A forms an algebraic lattice. Prime α-ideals and several related properties are investigated. Some properties of the spectral space of prime α-ideals equipped with the hull-kernel topology are derived.


Introduction
α-ideals were first studied by Cornish [1] for a class of distributive lattices as a generalization of annihilator ideals. Later on, the notion of α-ideals was extended to the class of 0-distributive lattices by Pawar and Khopade [2], to the class of almost distributive lattices by Rao and Rao [3], to the class of C-algebras by Rao [4], and more generally to arbitrary posets by Mokbel [5]. Jayaram [6] studied prime α-ideals in 0-distributive lattices and he topologized the class of all prime α-ideals in 0-distributive lattices. Bigard [7] has also studied α−ideals in the context of lattice-ordered groups. Instead of annulets, he used the dual lattice of carriers.
Cornish, in his paper [1], studied α−ideals in a distributive lattice L, by using the lattice A 0 (L) of all annulets of the form (x) * (where (x) * � a ∈ L: a ∧ x � 0 { }) which is a sublattice of the Boolean algebra of all annihilator ideals in L. Cornish has defined two operators α and α ← as follows: for an ideal J in L, is a filter in A 0 (L) and, conversely, for a filter F of A 0 (L), α is an ideal in L. It is observed that the map J ⟼ α ← α(J) forms a closure operator on the class of all ideals of L and α−ideals in L are defined to be those ideals which are closed with respect to this closure operator, i.e., J is an α−ideal of L if J � α ← α(J) � x ∈ L: (x) * � (y) * for some y ∈ J .
In fact, this property can be equivalently expressed in the following way: J is an α−ideal of L if and only if (x) * * ⊆ J for all x ∈ J. Cornish has observed that, by using the structure of A 0 (L), results can be transferred to give information on the ideal structure of L. e most interesting result of this type is that L is a generalized Stone lattice if and only if each prime ideal contains a unique prime α−ideal. Moreover, necessary and sufficient conditions are also given for a distributive lattice L with 0 to be disjunctive using α−ideals.
In [8], Chajda and Halas have introduced the notion of annihilators in general universal algebras having a constant 0. 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 α-ideals as a generalization of those annihilator ideals in universal algebras. We have also studied the basic topological properties of the space of α-prime ideals in universal algebras. is is a reasonable abstraction of the theory of α-ideals of distributive lattices to general universal algebras with abstract finitary operations. e results of this paper are important to extend the properties of distributive lattices to other classes

Preliminaries
is section contains some definitions and results which will be used in this paper. For the standard concepts in universal algebras, we refer to [10][11][12]. roughout this paper, A ∈ K, where K is a class of algebras of a fixed type F and assume that there is an equationally definable constant in all algebras of K denoted by 0. For a positive integer n, we write a → to denote the n−tuple 〈a 1 , a 2 , . . . , a n 〉 ∈ A n . e n−ary terms of type F are formal expressions obtained in finitely many steps by the following process: (1) e variables x 1 , . . . , x n are n−ary terms of type F (2) If m ∈ Z 0 , t 1 , . . . , t m are n−ary terms of type F and f ∈ F m , then f(t 1 , . . . , t m ) is also a term of type F Definition 1 (see [13]). An (n + m)-ary term p( x → , y → ) is said to be an ideal term in y → if p a 1 , . . . , a n , 0, 0, . . . , 0 � 0, for all a 1 , . . . , a n ∈ A. A nonempty subset I of A will be called We denote the class of all ideals of A by i(A). It is easy to check that the intersection of any family of ideals of A is an ideal. So, for a subset S ⊆ A, there always exists a smallest ideal of A containing S which we call the ideal of A generated by S and it is denoted by 〈S〉. Note that x ∈ 〈S〉 if and only if x � p(a 1 , . . . , a n , b 1 , . . . , b m ) for some a 1 , . . . , a n ∈ A, and b 1 , . . . , b m ∈ S where p( x → , y → ) is an (n + m)−ary ideal term in y → . If S � a { }, then we write 〈a〉 instead of 〈S〉. In this case, x ∈ 〈a〉 if and only if x � p(a 1 , . . . , a n , a) for some a 1 , . . . , a n ∈ A, where p( x → , y → ) is an (n + 1)−ary ideal term in y → .
Definition 2 (see [13]). A class K of algebras is called ideal determined if every ideal I is the zero-congruence class of a unique congruence relation denoted by I δ . In this case, the map I ⟼ I δ defines an isomorphism between the lattice of ideals and the congruence lattice of A.
Definition 3 (see [13,14] Definition 4 (see [13]). In an ideal determined variety, the commutator [I, J] of ideals I and J is the zero-congruence class of the commutator congruence [I δ , J δ ]. roughout this paper, K is assumed to be an idealdetermined variety and each A ∈ K has a c-unit.
Theorem 1 (see [13,14]). Let I and J be ideals of A ∈ K. en For subsets H and G of A, [H, G] denotes the product [〈H〉, 〈G〉]. In particular, for a, b ∈ A, [〈a〉, 〈b〉] is denoted by [a, b]. e following lemma gives some properties of the commutator of ideals. Further details on ideals of universal algebras can be found in [15][16][17].

) e commutator of ideals is distributive over arbitrary joins of ideals
Definition 5 (see [14]). A nonzero element u in A is said to be a formal unit if A � 〈u〉, i.e., A is generated by u 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 u ∈ A is a called a commutator unit (or c-unit for short) in A if [a, u] � 〈a〉 for all a ∈ A [9].

Example 1
(1) In a ring R with unity, the elements 1 and −1 are c-unit (2) In a bounded distributive lattice L, the largest element 1 is a c-unit (3) In a bounded Hilbert algebra A, the least element 0 is a c-unit (4) Every maximal element in an almost distributive lattice is a c-unit

Annihilators
is section contains some important results on annihilators and annihilator ideals of universal algebras taken from.
Definition 7 (see [9]). Let I ∈ i(A). For any subset S of A, we define We call (S: I) the relative annihilator of S with respect to I. For a subset S of A, the annihilator of S denoted by S * is defined to be S * � (S: (0)).
If S � a { }, then we denote S * by (a) * , i.e., It is observed that S * is an ideal of A and S * � 〈S〉 * for all S ⊆ A. Also, it can be verified that (0) * � A and A * � (0).

Lemma 2. For any subsets S and T of A,
An element a in a ring R is said to be idempotent if a 2 � a. By imitating this property to the general case of universal algebras, we define the following.
If all elements of A are idempotent, then we call A a commutator idempotent algebra (or a c-idempotent algebra for short). In other words, the commutator of ideals in A is idempotent.
Boolean rings and more generally lattices with least element 0 are the most natural examples of c-idempotent algebras. One can also verify that regular rings (a ring R is regular if for each x ∈ R there exists a ∈ R such that xax � x) are examples of c-idempotent algebras. One of the most important properties of c−idempotent algebras is that the commutators of ideals coincide with their intersection. Further, the separation axiom holds in c−idempotent algebras. Definition 9. An ideal I of A is called an annihilator ideal if I � S * for some nonempty subset S of A.
It is immediate from the definition that an ideal I of A is an annihilator ideal if and only if I * * � I. We denote the class of all annihilator ideals of A by i * (A). If A is a c−idempotent algebra, then it is proved that i * (A) forms a Boolean algebra.

α − Ideals
In this section, we study α-ideals in universal algebras in a more general context. Notation 1. We write F ⊂ ⊂ A to say that F is a finite subset of A.
We denote by i α (A) the class of all α−ideals of A.
It is easy to check that 〈0〉 and A belong to the class i α (A).

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

Lemma 3. Every annihilator ideal of
e converse of Lemma 3 does not hold in general. For instance, in an infinite algebra A with 0, every proper dense α-ideal is not an annihilator ideal.

Theorem 2. An ideal I of A is an α−ideal if and only if
Proof. Suppose that I is an α-ideal. en F * * ⊆ I for all finite subsets F of I, which gives Also, if a ∈ I, then F � a { } is a finite subset of I such that a ∈ F * * . So, the other inclusion holds. e converse part is straightforward. □ Definition 12. For each I ∈ i(A), let us define a set α(I) as follows: In the following lemma, we give another description for the set α(I) which will be useful to prove eorem 2.
Proof. Let us define two sets K and G as follows: We show that K � G. Since (a) * � 〈a〉 * for all a ∈ A, it holds immediately that K ⊆ G. To prove the other inclusion, let a ∈ G.
en there is H ∈ i(A) and F ⊂ ⊂ I such that a ∈ H and F * ⊆ H * , which gives H * ⊆ (a) * . So, a ∈ K and therefore the equality holds.

Journal of Mathematics
It is also observed in Lemma 6 that I ⊆ α(I). Now let J be any other α-ideal of A such that I ⊆ J and let a ∈ α(I). en there exists H ∈ i(A) and F ⊂ ⊂ I such that a ∈ H and F * ⊆ H * . So, H * * ⊆ F * * . Also, since F ⊆ I ⊆ J and J is an α-ideal, we get F * * ⊆ J. Now consider the following: Hence, α(I) ⊆ J and therefore α(I) is the smallest α-ideal containing I.

Theorem 5. An ideal I of A is an α−ideal if and only if
Proof. Suppose that I is an α-ideal. Since the other inclusion is trivial, we proceed to show that α(I) ⊆ I. Let a ∈ α(I).
en there exists H ∈ i(A) and F ⊂ ⊂ I such that a ∈ H and F * ⊆ H * . We have the following:

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
en F * * � E * * . If F ⊆ I, then F * * ⊆ I and hence E * * ⊆ I. So E ⊆ I. Conversely, suppose that the above condition is satisfied. Let F ⊂ ⊂ I. We show that en E ⊆ A such that F * � E * . As F ⊂ ⊂ I, we get E ⊆ I. Hence, the result holds. □ Theorem 7. Let A be a c−idempotent algebra and P be a prime ideal of A. If P is nondense, then it is an α−ideal.
Proof. Suppose that P is nondense. en there exists a nonzero a ∈ P * which implies that P � (a) * . us P is an annihilator ideal and hence it follows from Lemma 3 that P is an α-ideal. Proof. It is enough to show that i α (A) is closed under arbitrary intersection. Let J λ λ∈Δ be an indexed family of α−ideals in A. Put J � ∩ λ∈Δ J λ . If Δ is empty, then J � A and hence it is an α−ideal. Assume that Δ is nonempty. en J is an ideal of A. Moreover, if F is a finite subset of J, then it can be easily observed that F * * ⊆ J and therefore it is an α−ideal.
Note also that, for any I, J ∈ i α (A), their supremum is given by For a nonempty set X, remember from [12] that a family C of subsets of X will be called a closure system in P(X) if it forms a complete lattice together with the usual inclusion order. Furthermore, a closure system C is an algebraic lattice if and only if every chain in C has supremum in C.
Proof. It is enough to show that every chain in i α (A) has an upper bound in i α (A). Let J λ λ∈Δ be a chain in i α (A). Put J � ∪ λ∈Δ J λ . We first show that J is an ideal. Let p( x → , y → ) be an (n + m)−ary ideal term in y → , a 1 , . . . , a n ∈ A, and b 1 , . . . , b m ∈ J.
en, there exist λ 1 , . . . , λ m ∈ Δ such that each b i ∈ J λ i . Since the family J λ λ∈Δ is a chain, we can find erefore, J is an ideal of A. It remains to show that J is an α−ideal. Let F be a finite subset of F. If F � ∅, then F * * � (0) ⊆ J. Assume that F is nonempty and let F � a 1 , . . . , a n }.
is completes the proof. It can be deduced from the above theorem that the closure operator J ⟼ α(J) is in fact an algebraic closure operator. Moreover, the compact elements in i α (A) are those finitely generated α−ideals of A.
Journal of Mathematics Definition 13 (see [14]). A subset M of A is called an In the following lemma, we give another description for m−systems for our purpose.

(24)
Since M is an m-system, each M j is a nonempty subset of M such that M j ⊆ S j . Moreover, Now consider the following:  Proof. Let F � a 1 , a 2 , . . . , a n be a finite subset of 0(M Since m− is an M-system, each M k is a nonempty subset of M such that M k ⊆ S k . Moreover, Now consider the following: If we let M � A − P, then M is an m-system of A such that 0 ′ (P) � 0(M). So, it follows from eorems 10 and 11 that 0 ′ (P) is an α-ideal of A. Moreover, if P is minimal prime ideal, then 0 ′ (P) is an α-ideal containing P.

Theorem 12. Let
A be a c−idempotent algebra. en the following are equivalent:

(1) Every ideal I of A is an α-ideal (2) Every prime ideal P of A is an α-ideal
Proof. (1) ⟹ (2) is obvious. We proceed to show (2) ⟹ (1). Assume (2). Let I be an ideal of A. It is enough if we show that α(I) ⊆ I. Suppose not. en there exists x ∈ α(I) but x ∉ I. Since A is c−idempotent, we can apply Zorn's lemma to obtain a prime ideal P of A such that I ⊆ P and x ∉ P. So α(I) ⊆ α(P) � P. is is a contradiction. us, α(I) ⊆ I and hence I is an α−ideal. □ Remark 3. If the algebra we are working on is a distributive lattice, then one (and hence all) of the conditions of eorem 12 is necessary and sufficient condition for A to be disjunctive. Definition 16. By a prime α-ideal of A, we mean an α-ideal P of A satisfying the condition

The Space of Prime α-Ideals
for all I, J ∈ i(A).

Journal of Mathematics
Theorem 13. Let I be an α-ideal of A and M be an m-system of A such that I ∩ M � ∅. en there exists a prime α-ideal P of A such that I ⊆ P and P ∩ M � ∅.
Proof. Let us put Clearly L is a nonempty family of α-ideals of A satisfying the hypothesis of Zorn's lemma. So L has a maximal element, say P. Our aim is to show that P is prime. Suppose not. en there exist a, b ∈ A − P such that [a, b] ⊆ P. If we put J � P ⊔ α(〈a〉) and K � P ⊔ α(〈b〉), then J and K are α-ideals of A properly containing P. By the maximality of P in L, both J and K do not belong to L. So J ∩ M ≠ ∅ and K ∩ M ≠ ∅. Choose x, y ∈ M such that x ∈ J and y ∈ K. Consider the following: is is a contradiction. erefore, P is prime.

Corollary 1. Let I be an α-ideal of A and a ∈ A such that a ∉ I. If
A is c-idempotent, then there exists a prime α-ideal P of A such that I ⊆ P and a ∉ P.
Proof. Since A is c-idempotent, every singleton a { } is an m-system. So, the proof follows from eorem 13.  Proof. Let us put G � ∩ P: P is a prime α − ideal of A and I ⊈ P .
Let x ∈ A and P be a prime α-ideal of A with I ⊈ P. If x ∈ I * , then [x, I] � (0) ⊆ P. Since P is prime and I ⊈ P it follows that x ∈ P, i.e., x ∈ G, and hence I * ⊆ G. To prove the other inclusion, let x ∈ A and P be any prime α-ideal of A with I ⊈ P. If x ∉ I * , then [x, I] ≠ (0). Choose a nonzero y ∈ [x, I]. By Corollary 1, there exists a prime α-ideal P of A such that y ∉ P, which gives [x, I] ⊈ P. en x ∉ P and hence x ∉ G. erefore, G ⊆ I * and the equality holds.
In the rest of our work, A is assumed to be c-idempotent.
Let us now give the following notations: For each subset S of A, let For each a ∈ A, we write D α (a) instead of D α ( a { }). For any subset S of A, one can verify that (40) □ Lemma 9. For any α-ideal I of A, we have Proof. e equality D α (I) � ∪ x∈I D α (x) holds trivially. For the other equality, it is enough to show that D α (x) � D α ((x) * * ) for all x ∈ A. For any prime α-ideal P of A, one can verify that x ∈ P if and only if (x) * * ⊆ P.
Lemma 11. e following conditions are equivalent for any ideals I and J of A: e family T α � D α (I): I is an α− ideal of A} forms a topology on P α (A).
Proof. Clearly, both ∅ and P α (A) belong to the family T α . Moreover, it follows from (2) and (4) of Lemma 10 that T α is closed under finite intersection and arbitrary unions and thus it is a topology on P α .
Proof. Let I be n-ideal of A and P ∈ D α (I). en there exists x ∈ I such that x ∉ P, i.e., P ∈ D α (x) ⊆ D α (I). us B α is a base for T α .

Theorem 17. Each D α (x) is a compact open subset of P α (A).
Proof. Let D α (x λ ) λ∈Δ be a basic open cover for D α (x), i.e., Let I be an ideal of A generated by x λ : λ ∈ Δ . en D α (x) ⊆ D α (I) � D α (α(I)). Our aim is to show that x ∈ α(I). Suppose on the contrary that x ∉ α(I). en there exists a prime α-ideal P of A such that α(I) ⊆ P and x ∉ P, i.e., P ∈ D α (x) but P ∉ D α (α(I)), which is a contradiction.
us, x ∉ α(I). ere exists a finite subset F of I such that F * ⊆ (x) * . Since I is generated by x λ : λ ∈ Δ , there exist k ∈ N and λ 1 , For any prime α-ideal P of A, if x ∉ P, then (x) * * ⊈ P and hence E * * ⊈ P, i.e., is completes the proof.

□
Since H is compact, there exist x 1 , . . . , x m ∈ I such that It follows from our assumption that there is some y ∈ A such that H � D α (y). Hence proved. □ Corollary 3. If every finitely generated ideal in A is a principal ideal and P α (A) is compact, then A has dense elements.

Theorem 19. ere is a lattice isomorphism between the lattice of all α-ideals of A and the lattice of all open subsets of P α (A).
Proof. One can easily verify that the map I ⟼ D α (I) is an isomorphism between the lattice of all α-ideals of A and the lattice of open subsets of P α (A).
We denote the closure, interior, and exterior of a subfamily F of P α (A) by cl(F), int(F), and ext(F), respectively. e following theorem shows that the topology T α on P α (A) is the hull-kernel topology. Proof. Suppose that P ∈ cl(F). en every neighborhood of P meets F. If we put J � ∩ H∈F H, then J is an α-ideal of A such that J ⊆ H for all H ∈ F. Our claim is to show that J ⊆ P. Suppose not. en P ∈ D α (J) and hence there is some H ∈ F such that J ⊈ H, which is a contradiction. us, P ∈ V α (J). Conversely, suppose that P ∈ V α (J) and let I be any α-ideal of A with I ⊈ P. We show that D α (I) ∩ F ≠ ∅. If not, then I ⊆ H for all H ∈ F and hence I ⊆ P, which is a contradiction. erefore, P ∈ cl(F) and the equality holds. (47) Proof. By eorem 20, we have So P α (A) − cl(D α (I)) � D α (I * ). Now consider the following:
If A has formal unit, then it was proved in [14] that every proper ideal I of A is contained in a maximal ideal M of A.
is can be proved by applying Zorn's lemma. Proof. Suppose that P α (A) is T 1 . Let P be an α-prime ideal of A, which is not maximal. Since A has a formal unit, there exists a maximal ideal M of A such that P⊊M.
i.e., a ∈ M − P and b ∈ P − M � ∅. is is a contradiction. us, P is maximal. Conversely, suppose that every prime α-ideal of A is maximal. Let P and Q be two distinct elements of P α (A).
en, by assumption, P ⊈ Q and Q ⊈ P. is implies that there exist a, b ∈ A such that a ∈ P − Q and b ∈ Q − P.
en we have P ∈ D α (b) − D α (a) and Q ∈ D α (a) − D α (b). us, P α (A) is a T 1 -space. □ Theorem 25. If P α (A) is a Hausdorff space containing more than one element, then there exist a, b ∈ A such that Proof. Suppose that P α (A) is Hausdorff. Let P and Q be distinct elements in P α (A). en there exist two basic open sets, say D α (a) and D α (b), such that P ∈ D α (a), Q ∈ D α (b), Case 2. If b ∈ K and a ∉ K, then K ∈ D α (a).
Theorem 26. e following are equivalent: (1) P α (A) is Hausdorff (2) For any two distinct elements P and Q of P α (A), there exist a, b ∈ A such that a ∉ P, b ∉ Q and there does not exist any element K of P α (A) such that a ∉ K and b ∉ K Proof. Suppose that P α (A) is Hausdorff and let P and Q be distinct elements of P α (A). en there exist two basic open sets D α (a) and D α (a) such that P ∈ D α (a), Q ∈ D α (b) and D α (a) ∩ D α (b) � ∅. Now suppose on the contrary that there is K ∈ P α (A) such that a ∉ K and b ∉ K. en K ∈ D α (a) ∩ D α (b), which is absurd. erefore, there does not exist any element K of P α (A) such that a ∉ K and b ∉ K. Conversely, assume (2). Let P and Q be two distinct elements in P α (A).
en, by assumption, there exist a, b ∈ A such that a ∉ P and b ∉ Q and there does not exist any element K in P α (A) such that a ∉ K and b ∉ K, i.e., P ∈ D α (a), b ∈ D α (b) and D α (a) ∩ D α (b) � ∅. us, P α (A) is Hausdorff. □ Theorem 27. P α (A) is regular if and only if, for any P ∈ P α (A) and a ∉ P (a ∈ A), there exist an ideal I of A and b ∈ A such that P ∈ D α (b) ⊆ V α (I) ⊆ D α (a).
Proof. Suppose that P α (A) is regular. Let P ∈ P α (A) and a ∉ P. en, P ∉ V α (a). As P α (A) is regular, there exist α-ideals I and J of A such that P ∈ D α (I), V α (a) ⊆ D α (J), and D α (I) ∩ D α (J) � ∅.
en, V α (J) ⊆ D α (a) and D α (J) ⊆ V α (I). From P ∈ D α (I), we have I ⊈ P. So we can choose b ∈ I − P. Now we show that D α (J) ⊆ V α (b). If K ∈ D α (J) ⊆ V α (I), then I ⊆ K and hence b ∈ K.
us, for any P ∈ P α (A) and a ∉ P, there exist an ideal J of A and b ∈ A such that Conversely, suppose that the condition of the theorem is satisfied. To show that P α (A) is regular, let P ∈ P α (A) and 8 Journal of Mathematics I be an α-ideal of A such that P ∉ V α (I). en I ⊈ P. Choose a ∈ I and a ∉ P. By assumption, there exist an α-ideal J of A and b ∈ A such that If we put G � D α (b) and H � D α (J), then G and H are disjoint open sets in P α (A) such that P ∈ G and V α (I) ⊆ H. us, P α (A) is regular.

Conclusion
e study of the space of α-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 c-idempotent algebra A induced by α-ideals.

Data Availability
No data were used to support the results of this study.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.