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

Authors: Gezahagne Mulat Addis
Journal: Journal of Mathematics
 (2021)

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## Abstract

The purpose of this paper is to studyα-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.


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## Body

## 1. 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 A0L 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,(1)αJ=x∗:x∈J,is a filter in A0L and, conversely, for a filter F of A0L,(2)α←F=x∈L:x∗∈F,is an ideal in L. It is observed that the map J⟼α←αJ 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(3)J=α←αJ=x∈L:x∗=y∗for somey∈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 A0L, results can be transferred to give information on the ideal structure of L. The 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. This is a reasonable abstraction of the theory of α-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, 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 a1,a2,…,an∈An. The n−ary terms of type F are formal expressions obtained in finitely many steps by the following process:(1)
The variablesx1,…,xn are n−ary terms of type F(2)
Ifm∈Z0, t1,…,tm are n−ary terms of type F and f∈Fm, then ft1,…,tm is also a term of type FDefinition 1 (see [13]).
Ann+m-ary term px⟶,y⟶ is said to be an ideal term in y⟶ if(4)pa1,…,an,0,0,…,0=0,for all a1,…,an∈A. A nonempty subset I of A will be called an ideal of A if pa⟶,b⟶∈I for all a⟶∈An,b⟶∈Im and any ideal term px⟶,y⟶ in y⟶.
We denote the class of all ideals ofA by iA. 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=pa1,…,an,b1,…,bm for some a1,…,an∈A, and b1,…,bm∈S where px⟶,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=pa1,…,an,a for some a1,…,an∈A, where px⟶,y⟶ is an n+1−ary ideal term in y⟶.Definition 2 (see [13]).
A classK of algebras is called ideal determined if every ideal I is the zero-congruence class of a unique congruence relation denoted by Iδ. In this case, the map I⟼Iδ defines an isomorphism between the lattice of ideals and the congruence lattice of A.Definition 3 (see [13, 14]).
A termtx⟶,y⟶,z⟶ is said to be a commutator term in y⟶,z⟶ if it is an ideal term in y⟶ and an ideal term in z⟶.Definition 4 (see [13]).
In an ideal determined variety, the commutatorI,J of ideals I and J is the zero-congruence class of the commutator congruence Iδ,Jδ.
Throughout this paper,K is assumed to be an ideal-determined variety and each A∈K has a c-unit.Theorem 1 (see [13, 14]).
LetI and J be ideals of A∈K. Then(5)I,J=ta⟶,i⟶,j⟶:a⟶∈An,i⟶∈Imandj⟶∈Jkwheretx⟶,y⟶,z⟶is a commutator term iny⟶,z⟶.For subsetsH and G of A, H,G denotes the product H,G. In particular, for a,b∈A, a,b is denoted by a,b. 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 eachI,J,K∈iA, we have the following:(1)
I,J=J,I(2)
I⊆K implies I,J⊆K,J(3)
The commutator of ideals is distributive over arbitrary joins of idealsDefinition 5 (see [14]).
A nonzero elementu 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 elementu∈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 ringR with unity, the elements 1 and −1 are c-unit(2)
In a bounded distributive latticeL, the largest element 1 is a c-unit(3)
In a bounded Hilbert algebraA, the least element 0 is a c-unit(4)
Every maximal element in an almost distributive lattice is ac-unit


## 3. Annihilators

This section contains some important results on annihilators and annihilator ideals of universal algebras taken from.Definition 7 (see [9]).
LetI∈iA. For any subset S of A, we define(6)S:I=a∈A:a,S⊆I.
We callS: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(7)S∗=S:0.
IfS=a, then we denote S∗ by a∗, i.e.,(8)a∗=b∈A:a,b=0.
It is observed thatS∗ 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 subsetsS and T of A,(1)
S∗=∩a∈Sa∗(2)
S⊆S∗∗(3)
S∗∗∗=S∗(4)
S⊆T⟹T∗⊆S∗(5)
S∪T∗=S∗∩T∗
An elementa in a ring R is said to be idempotent if a2=a. By imitating this property to the general case of universal algebras, we define the following.Definition 8.
An elementa∈A is called idempotent if(9)a,a=a.
If all elements ofA 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 element0 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 idealI 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 idealI 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.Definition 10.
An idealI of A is called a dense ideal if I∗=0; otherwise, I is called a nondense ideal.


## 4.α−Ideals

In this section, we studyα-ideals in universal algebras in a more general context.Notation 1.
We writeF⊂⊂A to say that F is a finite subset of A.Definition 11.
An idealI of A is called an α−ideal if(10)F∗∗⊆Ifor eachF⊂⊂I.
We denote byiαA the class of all α−ideals of A.
It is easy to check that0 and A belong to the class iαA.Remark 1.
IfA is a distributive lattice, then this definition coincides to the definition proposed by Cornish [1].Lemma 3.
Every annihilator ideal ofA is an α−ideal.Remark 2.
The converse of Lemma3 does not hold in general. For instance, in an infinite algebra A with 0, every proper dense α-ideal is not an annihilator ideal.Lemma 4.
In a finite algebraA, every α−ideal is an annihilator ideal.Theorem 2.
An idealI of A is an α−ideal if and only if(11)I=∪F⊂⊂IF∗∗.Proof.
Suppose thatI is an α-ideal. Then F∗∗⊆I for all finite subsets F of I, which gives(12)∪F⊂⊂IF∗∗⊆I.
Also, ifa∈I, then F=a is a finite subset of I such that a∈F∗∗. So, the other inclusion holds. The converse part is straightforward.Definition 12.
For eachI∈iA, let us define a set αI as follows:(13)αI=x∈A:F∗⊆x∗for someF⊂⊂I.
In the following lemma, we give another description for the setαI which will be useful to prove Theorem 2.Lemma 5.
For eachI∈iA, we have(14)αI=∪H∈iA:F∗⊆H∗for someF⊂⊂I.Proof.
Let us define two setsK and G as follows:(15)K=x∈A:F∗⊆x∗for someF⊂⊂I,G=∪H∈iA:F∗⊆H∗for someF⊂⊂I.
We show thatK=G. Since a∗=a∗ for all a∈A, it holds immediately that K⊆G. To prove the other inclusion, let a∈G. Then there is H∈iA and F⊂⊂I such that a∈H and F∗⊆H∗, which gives H∗⊆a∗. So, a∈K and therefore the equality holds.Lemma 6.
For eachI,J∈iA, we have the following:(1)
I⊆αI(2)
I⊆J⟹αI⊆αJ(3)
ααI=αITheorem 3.
For eachI∈iA, αI is an ideal of A.Proof.
Letb⟶=b1,b2,…,bm be an m-tuple of elements from αI. Then, for each bi, there exist H1,…,Hm∈iA and finite subsets F1,…,Fm of I such that bi∈Hi and Fi∗⊆Hi∗. Put F=∪i=1mFi and H=∨i=1mHi. Then F is a finite subset of I and H∈iA such that F∗=∩i=1mFi∗ and H∗=∩i=1mHi∗ and b1,b2,…,bm∈H. Since Fi⊆F, F∗⊆Fi∗⊆Hi∗ for all i. So, F∗⊆∩i=1mHi∗=H∗. That is, H is an ideal of A such that F∗⊆H∗. Hence, H⊆αI. Now if a⟶∈An and px⟶,y⟶ is an ideal term in y⟶, then pa⟶,b⟶∈H⊆αI. Therefore, αI is an ideal of I.Theorem 4.
For eachI∈iA, αI is the smallest α-ideal of A containing I.Proof.
We first show thatαI is an α-ideal. Let us put(16)G=∪H∈iA:F∗⊆H∗for someF⊂⊂I.
LetF⊂⊂G. We show that F∗∗⊆G. If F=∅, then F∗∗=0⊆G. Suppose that F≠∅, and let F=a1,…,an. Then, for each ai, there exist H1,…,Hn∈iA and finite subsets F1,…,Fn of I such that ai∈Hi and Fi∗⊆Hi∗ for all i. Put E=∪i=1nFi and H=∨i=1nHi. Then E is a finite subset of I, and H∈iA such that a1,…,an∈H and E∗=∩i=1nFi∗ and H∗=∩i=1nHi∗. So, E∗⊆Fi∗⊆Hi∗ for all i, which gives E∗⊆H∗. Since F⊆H, we have F∗∗⊆H∗∗. It remains to show that H∗∗⊆G. If we put K=H∗∗, then K is an ideal of A such that K∗=H∗. Since E is a finite subset of I with E∗⊆K∗, we get K⊆G. Therefore G=αI is an α-ideal. It is also observed in Lemma 6 that I⊆αI. Now let J be any other α-ideal of A such that I⊆J and let a∈αI. Then there exists H∈iA 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:(17)a∈H⊆H∗∗⊆F∗∗⊆J.
Hence,αI⊆J and therefore αI is the smallest α-ideal containing I.Theorem 5.
An idealI of A is an α−ideal if and only if I=αI.Proof.
Suppose thatI is an α-ideal. Since the other inclusion is trivial, we proceed to show that αI⊆I. Let a∈αI. Then there exists H∈iA and F⊂⊂I such that a∈H and F∗⊆H∗. We have the following:(18)a∈H⊆H∗∗⊆F∗∗⊆I.
Therefore,αI⊆I and hence the equality holds. The converse part is straightforward.
From Lemma6, it can be deduced that the map α:iA⟶iA:I⟼αI is a closure operator on iA and α−ideals of A are precisely the closed elements of iA 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 thatI is an α-ideal and let F⊂⊂A and E⊆A such that F∗=E∗. Then 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 F∗∗⊆I. Put E=F∗∗. Then E⊆A such that F∗=E∗. As F⊂⊂I, we get E⊆I. Hence, the result holds.Theorem 7.
LetA 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 thatP is nondense. Then there exists a nonzero a∈P∗ which implies that P=a∗. Thus P is an annihilator ideal and hence it follows from Lemma 3 that P is an α-ideal.Theorem 8.
The collectioniαA,⊆ forms a complete lattice.Proof.
It is enough to show thatiαA is closed under arbitrary intersection. Let Jλλ∈Δ be an indexed family of α−ideals in A. Put J=∩λ∈ΔJλ. If Δ is empty, then J=A and hence it is an α−ideal. Assume that Δ is nonempty. Then 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 anyI,J∈iαA, their supremum is given by(19)I⊔J=αI∨J.
For a nonempty setX, remember from [12] that a family ℭ of subsets of X will be called a closure system in PX 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.Theorem 9.
The collectioniαA,⊆ is an algebraic lattice.Proof.
It is enough to show that every chain iniαA has an upper bound in iαA. Let Jλλ∈Δ be a chain in iαA. Put J=∪λ∈ΔJλ. We first show that J is an ideal. Let px⟶,y⟶ be an n+m−ary ideal term in y⟶, a1,…,an∈A, and b1,…,bm∈J. Then, there exist λ1,…,λm∈Δ such that each bi∈Jλi. Since the family Jλλ∈Δ is a chain, we can find λ∈Δ with b1,…,bm∈Jλ. With Jλ being an ideal of A, it holds that(20)pa1,…,an,b1,…,bm∈Jλ⊆J.
Therefore,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=a1,…,an. Then, there exist λ1,…,λn∈Δ such that each ai∈Jλi. Since the family Jλλ∈Δ is a chain, we can find λ∈Δ with a1,…,an∈Jλ, i.e., F⊂⊂Jλ. With Jλ being an α−ideal of A, it follows that F∗∗⊆Jλ⊆J. Therefore, J is an α−ideal. This completes the proof.
It can be deduced from the above theorem that the closure operatorJ⟼αJ is in fact an algebraic closure operator. Moreover, the compact elements in iαA are those finitely generated α−ideals of A.Definition 13 (see [14]).
A subsetM of A is called an m−system if, for all a,b∈M, a,b∩M≠∅.
In the following lemma, we give another description form−systems for our purpose.Lemma 7.
A nonempty subsetM of A is an m−system if and only if, for any nonempty subsets S1 and S2 of M,(21)M∩S1,S2≠∅.Proof.
Suppose thatM is an m-system. Let S1,S2 be nonempty subsets of M. Choose a∈S1 and b∈S2. Then a,b⊆S1,S2. Since a,b∈M, M∩a,b≠∅, which gives M∩S1,S2≠∅. The converse part is straightforward.Definition 14.
For anm−system M of A, let us define a set 0M as follows:(22)0M=x∈A:x,S=0for someS⊆M.Lemma 8.
For eachm−system M of A,(23)0M=∪H∈iA:H,S=0for someS⊆M.Theorem 10.
For eachm−system M of A, the set 0M is an ideal of A.Proof.
Letb⟶=b1,b2,…,bk be a k-tuple of elements from 0M. Then for each bi there exist H1,…,Hk∈iA and nonempty subsets S1,…,Sk of M such that bi∈Hi and Hi,Si=0 for all i. If we put H=∨i=1kHi, then H∈iA such that b1,b2,…,bk∈H. On the other hand, let us define a sequence M1,M2,…Mk as follows:(24)M1=S1andfor1≤j<k,Mj+1=Mj,Sj+1∩M.
SinceM is an m-system, each Mj is a nonempty subset of M such that Mj⊆Sj. Moreover,(25)Mk⊆Mk−1⊆⋯⊆M2⊆M1.
Now consider the following:(26)H,Mk=Vi=1kHi,Mk=∨i=1kHi,Mk⊆∨i=1kHi,Si=0.
Therefore,H⊆0M. Now, if a⟶∈An and px⟶,y⟶ is an ideal term in y⟶, we get pa⟶,b⟶∈H. So, pa⟶,b⟶∈0M. Therefore 0M is an ideal.Theorem 11.
For eachm−system M of A, the set 0M is an α−ideal of A.Proof.
LetF=a1,a2,…,an be a finite subset of 0M. Then there exist H1,…,Hn∈iA and nonempty subsets S1,…,Sn of M such that ai∈Hi and Hi,Si=0 for all i. If we let H=∨i=1mHi, then H∈iA such that a1,a2,…,am∈H. On the other hand, let us define a sequence M1,M2,…,Mn as follows:(27)M1=S1andfor1≤k<n,Mk+1=Mk,Sk+1∩M.
Sincem− is an M-system, each Mk is a nonempty subset of M such that Mk⊆Sk. Moreover,(28)Mn⊆Mn−1⊆⋯⊆M2⊆M1.
Now consider the following:(29)H,Mn=Vi=1nHi,Mn=Vi=1nHi,Mn⊆Vi=1nHi,Si=0.
That is,H,Mn=0, where Mn is a nonempty subset of M. So, H⊆Mn∗, which gives H∗∗⊆Mn∗∗∗=Mn∗. Thus, H∗∗,Mn=0 and hence H∗∗⊆0M. Since F⊆H, we have F∗∗⊆H∗∗. So, F∗∗⊆0M. Therefore, 0M is an α-ideal.Definition 15.
For each prime idealP of A, let us define 0′P as follows:(30)0′P=a∈A:a,S=0for someS⊆AwithS∩P=∅.
If we letM=A−P, then M is an m-system of A such that 0′P=0M. So, it follows from Theorems 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.
LetA be a c−idempotent algebra. Then the following are equivalent:(1)
Every idealI of A is an α-ideal(2)
Every prime idealP of A is an α-idealProof.
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 αI⊆I. Suppose not. Then there exists x∈αI 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 αI⊆αP=P. This is a contradiction. Thus, αI⊆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 Theorem12 is necessary and sufficient condition for A to be disjunctive.


## 5. The Space of Primeα-Ideals

Definition 16.
By a primeα-ideal of A, we mean an α-ideal P of A satisfying the condition(31)I,J⊆P⟹I⊆PorJ⊆P,for all I,J∈iA.Theorem 13.
LetI be an α-ideal of A and M be an m-system of A such that I∩M=∅. Then there exists a prime α-ideal P of A such that I⊆P and P∩M=∅.Proof.
Let us put(32)L=J∈iαA:I⊆JandJ∩M=∅.
ClearlyL 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. Then there exist a,b∈A−P such that a,b⊆P. If we put J=P⊔αa and K=P⊔αb, 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:(33)x,y⊆J,K=P⊔αa,P⊔αb=P,P⊔P,αa⊔P,αb⊔αa,αb⊆P.
SinceP∩M=∅, it follows that x,y=∅. This is a contradiction. Therefore, P is prime.Corollary 1.
LetI 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.
SinceA is c-idempotent, every singleton a is an m-system. So, the proof follows from Theorem 13.Theorem 14.
LetA be a c-idempotent algebra. Then, for any α-ideal I of A, we have(34)I=∩P:Pis a primeα−ideal ofAandI⊆P.Proof.
Let us put(35)H=∩P:Pis a primeα−ideal ofAandI⊆P.
Clearly,I⊆H. To prove the other inclusion, let x∉I. Then, by Corollary 1, there exists a prime α-ideal P of A such that I⊆P and x∉P, i.e., x∉H. Thus, H⊆I and hence the equality holds.Theorem 15.
LetA be a c-idempotent algebra. Then, for any α-ideal I of A, we have(36)I∗=∩P:Pis a primeα−ideal ofAandI⊈P.Proof.
Let us put(37)G=∩P:Pis a primeα−ideal ofAandI⊈P.
Letx∈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. Then x∉P and hence x∉G. Therefore, 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:(38)PαA=P:Pis a primeα−ideal ofA.
For each subsetS of A, let(39)DαS=P∈PαA:S⊈P,VαS=P∈PαA:S⊆P.
For eacha∈A, we write Dαa instead of Dαa. For any subset S of A, one can verify that(40)DααS=DαS=DαS.Lemma 9.
For anyα-ideal I of A, we have(41)DαI=∪x∈IDαx=∪x∈IDαx∗∗.Proof.
The equalityDαI=∪x∈IDαx holds trivially. For the other equality, it is enough to show that Dαx=Dαx∗∗ 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 10.
The following conditions hold for allI,J∈iA:(1)
I⊆J⟹DαI⊆DαJ(2)
DαI∩J=DαI∩DαJ=DαI,J(3)
DαI∨J=DαI∪DαJ(4)
Dα∨λ∈ΔIλ=∪λ∈ΔDαIλLemma 11.
The following conditions are equivalent for any idealsI and J of A:(1)
DαI=DαJ(2)
VαI=VαJ(3)
αI=αJTheorem 16.
The familyTα=DαI:Iis anα−ideal ofA forms a topology on PαA.Proof.
Clearly, both∅ and PαA belong to the family Tα. Moreover, it follows from (2) and (4) of Lemma 10 that Tα is closed under finite intersection and arbitrary unions and thus it is a topology on Pα.Definition 17.
We call the topologyTα the α-prime spectrum of A and sometimes it may be denoted as SpecαA.Lemma 12.
The subfamilyℬα=Dαx:x∈A constitutes a base for the topology Tα.Proof.
LetI be n-ideal of A and P∈DαI. Then there exists x∈I such that x∉P, i.e., P∈Dαx⊆DαI. Thus ℬα is a base for Tα.Theorem 17.
EachDαx is a compact open subset of PαA.Proof.
LetDαxλλ∈Δ be a basic open cover for Dαx, i.e.,(42)Dαx⊆∪λ∈ΔDαxλ.
LetI be an ideal of A generated by xλ:λ∈Δ. Then Dαx⊆DαI=DααI. Our aim is to show that x∈αI. Suppose on the contrary that x∉αI. Then there exists a prime α-ideal P of A such that αI⊆P and x∉P, i.e., P∈Dαx but P∉DααI, which is a contradiction. Thus, x∉αI. There exists a finite subset F of I such that F∗⊆x∗. Since I is generated by xλ:λ∈Δ, there exist k∈N and λ1,…,λk∈Δ such that F⊆xλ1,…,xλk. If we put E=xλ1,…,xλk, then E is a finite subset of I such that E∗⊆F∗⊆x∗. Now we show that(43)Dαx⊆∪i=1kDαxλi.
For any primeα-ideal P of A, if x∉P, then x∗∗⊈P and hence E∗∗⊈P, i.e.,(44)Dαx⊆DαE=∪i=1kDαxλi.
This completes the proof.Corollary 2.
IfA has a dense element, then PαA is compact.Proof.
IfA has a dense element say d, then Dαd=PαA. It follows from Theorem 17 that PαA is compact.Theorem 18.
If every finitely generated ideal inA is a principal ideal, then ℬα is the set of all compact open subsets of PαA.Proof.
Suppose that every finitely generated ideal inA is a principal ideal. It is proved in Theorem 17 that every element in ℬα is compact and open. Let H be a compact open subset of PαA. Then there is some α-ideal I of A such that(45)H=DαI=∪x∈IDαx.
SinceH is compact, there exist x1,…,xm∈I such that(46)H=∪i=1mDαxi=Dαxi:1≤i≤m.
It follows from our assumption that there is somey∈A such that H=Dαy. Hence proved.Corollary 3.
If every finitely generated ideal inA is a principal ideal and PαA is compact, then A has dense elements.Theorem 19.
There is a lattice isomorphism between the lattice of allα-ideals of A and the lattice of all open subsets of PαA.Proof.
One can easily verify that the mapI⟼DαI is an isomorphism between the lattice of all α-ideals of A and the lattice of open subsets of PαA.
We denote the closure, interior, and exterior of a subfamilyℱ of PαA by clℱ, intℱ, and extℱ, respectively. The following theorem shows that the topology Tα on PαA is the hull-kernel topology.Theorem 20.
For any subfamilyℱ of PαA, clℱ=Vα∩H∈ℱH.Proof.
Suppose thatP∈clℱ. Then every neighborhood of P meets ℱ. If we put J=∩H∈ℱH, 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. Then P∈DαJ and hence there is some H∈ℱ such that J⊈H, which is a contradiction. Thus, P∈VαJ. Conversely, suppose that P∈VαJ and let I be any α-ideal of A with I⊈P. We show that DαI∩ℱ≠∅. If not, then I⊆H for all H∈ℱ and hence I⊆P, which is a contradiction. Therefore, P∈clℱ and the equality holds.Theorem 21.
For anyα-ideal I of A,(47)extclDαI=DαI∗.Proof.
By Theorem20, we have(48)clDαI=Vα∩P∈DαAP=Vα∩I⊈PP=Q∈PαA:∩I⊈PP⊆Q=Q∈PαA:I∗⊆Qby Theorem22VαI∗.
SoPαA−clDαI=DαI∗. Now consider the following:(49)extclDαI=intPαA−clDαI=intDαI∗=DαI∗.
An open subset of a topological spaceX 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 anα-ideal I of A, DαI is an open domain if and only if I=I∗∗.Proof.
It follows from the above theorem thatclDαI=VαI∗. Now consider the following:(50)intclDαI=intVαI∗=intPαA−DαI∗=PαA−clDαI∗PαA−VαI∗∗=DαI∗∗.
IfI=I∗∗, then intclDαI=DαI∗∗=DαI and hence DαI is an open domain. Conversely, if intclDαI=DαI, then DαI∗∗=DαI and it follows from Lemma 13 that I=I∗∗.Corollary 4.
For anyα-ideal I of A, DαI is semiregular. In particular, if A has a dense element, then PαA is semiregular.Theorem 23.
If every finitely generated ideal inA is principal, then the following conditions are equivalent:(1)
For eachx∈A, there exists y∈A such that x∗∗=y∗(2)
Exterior of every compact open subset ofPαA is compactProof.
Suppose that (1) holds. LetY be a compact open subset of PαA. By Theorem 18, Y=Dαx for some x∈A. Now extY=extDαx=Dαx∗. By (1), there exists y∈A such that x∗=y∗∗. Thus, extY=Dαx∗=Dαy∗∗=Dαy. Again, by Theorem 18, extY is compact. Conversely assume (2). Let x∈A. Then Dαx and its exterior are compact, i.e., Dαx∗ is compact and open. By Theorem 18, there exists y∈A such that Dαx∗=Dαy=Dαy∗∗. It follows from Lemma 11 that x∗=y∗∗ and this completes the proof.
IfA 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. This can be proved by applying Zorn’s lemma.Theorem 24.
IfA has a formal unit, then PαA is a T1-space if and only if every α-prime ideal of A is maximal.Proof.
Suppose thatPαA is T1. 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. As PαA is a T1-space, there exist basic open sets Dαa and Dαb such that P∈Dαa, M∈Dαb, P∉Dαb, and M∉Dαa, i.e., a∈M−P and b∈P−M=∅. This is a contradiction. Thus, P is maximal. Conversely, suppose that every prime α-ideal of A is maximal. Let P and Q be two distinct elements of PαA. Then, by assumption, P⊈Q and Q⊈P. This implies that there exist a,b∈A such that a∈P−Q and b∈Q−P. Then we have P∈Dαb−Dαa and Q∈Dαa−Dαb. Thus, PαA is a T1-space.Theorem 25.
IfPαA is a Hausdorff space containing more than one element, then there exist a,b∈A such that(51)PαA=Dαa∪Dαb∪Vαa,b.Proof.
Suppose thatPαA is Hausdorff. Let P and Q be distinct elements in PαA. Then there exist two basic open sets, say Dαa and Dαb, such that P∈Dαa, Q∈Dαb, and Dαa∩Dαb=∅. For any K∈PαA, as Dαa∩Dαb=∅, the conditions a∉K and b∉K are not possible.Case 1.
Ifa∈K and b∉K, then K∈Dαb.Case 2.
Ifb∈K and a∉K, then K∈Dαa.Case 3.
Ifa,b∈K, then K∈Vαa,b.
In all cases, it holds thatK∈Dαa∪Dαb∪Vαa,b. This completes the proof.Theorem 26.
The following are equivalent:(1)
PαA is Hausdorff(2)
For any two distinct elementsP and Q of PαA, there exist a,b∈A such that a∉P,b∉Q and there does not exist any element K of PαA such that a∉K and b∉KProof.
Suppose thatPαA is Hausdorff and let P and Q be distinct elements of PαA. Then there exist two basic open sets Dαa and Dαa such that P∈Dαa,Q∈Dαb and Dαa∩Dαb=∅. Now suppose on the contrary that there is K∈PαA such that a∉K and b∉K. Then K∈Dαa∩Dαb, which is absurd. Therefore, there does not exist any element K of PαA such that a∉K and b∉K. Conversely, assume (2). Let P and Q be two distinct elements in PαA. Then, by assumption, there exist a,b∈A such that a∉P and b∉Q and there does not exist any element K in PαA such that a∉K and b∉K, i.e., P∈Dαa,b∈Dαb and Dαa∩Dαb=∅. Thus, PαA is Hausdorff.Theorem 27.
PαA is regular if and only if, for any P∈PαA and a∉P (a∈A), there exist an ideal I of A and b∈A such that P∈Dαb⊆VαI⊆Dαa.Proof.
Suppose thatPαA is regular. Let P∈PαA and a∉P. Then, P∉Vαa. As PαA is regular, there exist α-ideals I and J of A such that P∈DαI, Vαa⊆DαJ, and DαI∩DαJ=∅. Then, VαJ⊆Dαa and DαJ⊆VαI. From P∈DαI, we have I⊈P. So we can choose b∈I−P. Now we show that DαJ⊆Vαb. If K∈DαJ⊆VαI, then I⊆K and hence b∈K. Thus, K∈Vαb and DαJ⊆Vαb. Taking their complement, we get Dαb⊆VαJ. Thus, for any P∈PαA and a∉P, there exist an ideal J of A and b∈A such that(52)P∈Dαb⊆VαJ⊆Dαa.
Conversely, suppose that the condition of the theorem is satisfied. To show thatPαA is regular, let P∈PαA and I be an α-ideal of A such that P∉VαI. Then I⊈P. Choose a∈I and a∉P. By assumption, there exist an α-ideal J of A and b∈A such that(53)P∈Dαb⊆VαJ⊆Dαa.
If we putG=Dαb and ℋ=DαJ, then G and ℋ are disjoint open sets in PαA such that P∈G and VαI⊆ℋ. Thus, PαA is regular.


## 6. Conclusion

The study of the space ofα-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.



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*Source: 4045182-2021-06-09.xml*