Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 48, No. 1 (2026) pp. 47–59.

ON MINIMAL n-PRIMES SUBGROUPS OF ℓ-GROUPS

Diksha Saste¹, Mayur Kshirsagar², Nilesh Mundlik³, and

Aditi S. Phadke⁴

Abstract. This paper explores the notion of n-prime ℓ-subgroups in ℓ-groups. It derives a

characterization for minimal n-prime ℓ-subgroups and deﬁnes a topology on the set P_n(G)

of minimal n-prime ℓ-subgroups of an ℓ-group. The paper studies various properties of this

topology and establishes a necessary and suﬃcient condition for P_n(G) to be compact. It

proves that P_n(G) to be compact if and only if it is ﬁnite.

Keywords: prime ℓ-subgroup, minimal prime ℓ-subgroup, n-prime ℓ-subgroup, ℓ-group.

2010 AMS Subject Classiﬁcation: Primary 06F15 Secondary 06F20.

1. Introduction

In this paper we introduce the concept of n-prime ℓ-subgroups in ℓ-groups. The concept

of n-points in a topological space was introduced and studied by Pierce [10]. Hindman

[5] extended this idea and studied n-prime ideals in a commutative ring. Research on

the space of minimal n-prime ideals in commutative rings with unity has been done by

Hindman [5]. An ideal I in a commutative ring R is called n-prime(n ≥ 2) if S ∩ I = ∅,

whenever S is a set of n elements of R with each pairwise product lying in I. Building on

this, Anderson and Badawi [1] explored n- absorbing ideals in commutative rings, which

bear similarities to Hindman’s n-prime ideals (see Hindman[5]). Additionally, Halaˇs [4]

introduced n-prime ideals in posets, while Mundlik and Joshi [7] extended the concept of

n-prime ideals introduced in Hindman[5] for 0-distributive lattices.

In the study of lattice-ordered groups (ℓ-groups), the notion of primality arises in two

distinct contexts. On one hand, when an ℓ-group is viewed purely as a lattice, we consider

prime ideals in the lattice-theoretic sense. On the other hand, when we consider the full

algebraic structure of the ℓ-group, we encounter prime ℓ-subgroups. It is important to

note that these two notions of primality do not coincide in general.

To illustrate this distinction, consider the group Z with its usual order. Treated as

a lattice, the set (x] = {y ∈ Z | y ≤ x} forms a prime ideal for any x ∈ Z. However,

when Z is regarded as an ℓ-group, it possesses only two prime ℓ-subgroups: Z itself and

the set (0]. For a detailed discussion of this, one may refer to Darnel[3]. This example

clearly demonstrates that the set of all prime ideals in a lattice and the set of all prime

ℓ-subgroups in an ℓ-group are, in general, diﬀerent. Consequently, it becomes essential to

study the structure of prime ℓ-subgroups, particularly focusing on minimal and n-prime

ℓ-subgroups, in the broader context of ℓ-groups.

47

48

Diksha Saste, Mayur Kshirsagar, Nilesh Mundlik, and Aditi S. Phadke

P. Bhattacharjee and W. Wm. McGovern [2] have studied space of all minimal prime

subgroups of G endowed with the inverse topology. In this paper, we generalize this

concept to n-prime ℓ-subgroup in an ℓ-group G.

In section 1, we describe basic fundamental concepts and terminology related to ℓ-

groups. In section 2, we deﬁne the notion of n-prime ℓ-subgroup and minimal n-prime

ℓ-subgroup and give its characterization. In section 3, we deﬁne a toplogy on the set

P_n(G) of minimal n-prime ℓ-subgroups of an ℓ-group.

We begin with basic notions and terminologies related to ℓ-groups.

Deﬁnition 1.1 (Darnel [3]). A group (G, +) is said to be a partially ordered group (po-

group) if it is equipped with a reﬂexive, antisymmetric and transitive relation ≤ (a partial

order) which is compatible with +, that is if g ≤ h and x, y ∈ G then x+g +y ≤ x+h+y.

Deﬁnition 1.2 (Darnel [3]). Let (G, +) be a po-group. Then the subset G⁺= {g ∈ G : g ≥ 0}

is called the positive cone of G. Here 0 is identity of group G.

Observe that, the partial order is determined by G⁺, in the sense that g ≤ h if and only

if h − g ∈ G⁺.

Deﬁnition 1.3 (Darnel [3]). A po-group G is called lattice-ordered group abbreviated as

ℓ-group, if the underlying structure is lattice.

Deﬁnition 1.4 (Darnel [3]). Let L be a lattice and A be a subset of L. Then A is a

sublattice of L if for any a, b ∈ A, a ∨ b ∈ A and a ∧ b ∈ A.

Deﬁnition 1.5 (Darnel [3]). Let S be a subgroup of an ℓ-group G. Then S is called an

ℓ-subgroup of G if it is a sublattice of G.

Deﬁnition 1.6 (Darnel [3]). A subset S of a po-group G is called convex if whenever

s, t ∈ S and g ∈ G with s ≤ g ≤ t then g ∈ S.

Deﬁnition 1.7 (Darnel [3]). Let G be an ℓ-group with identity 0 and let g ∈ G.

1) The positive part of g is g₊= g ∨ 0.

2) The negative part of g is g = −g ∨ 0.

−

3) The absolute value of g is |g| = (g₊) + (g ).

−

Denote by C(G) the set of all convex ℓ-subgroups of G. Observe that if S is a convex

ℓ-subgroup of G then a ∈ S if and only if |a| ∈ S.

Deﬁnition 1.8 (Darnel [3]). A convex ℓ-subgroup P of an ℓ-group G is called prime

ℓ-subgroup if whenever 0 ≤ a, 0 ≤ b, and a ∧ b ∈ P, then either a ∈ P or b ∈ P.

Let S be a subgroup of group G. Denote by R(S) the set of all right cosets of S in G.

Proposition 1.9 (Darnel [3]). Let G be a po-group and let S be a convex subgroup of G.

Deﬁne the relation “ ≥ ” on R (S) by S + x ≥ S + y if and only if there exists s ∈ S such

that x ≥ s + y. Then “ ≥ ” is a partial order on R (S) .

ON MINIMAL n-PRIMES SUBGROUPS OF ℓ-GROUPS

49

Proposition 1.10 (Darnel [3]). Let P be a convex ℓ-subgroup of an ℓ-group G. Then the

following conditions are equivalent:

1) P is prime ℓ-subgroup

2) If g ∧ h = 0 then g ∈ P or h ∈ P

3) If g, h ∈ G⁺\ P then g ∧ h > 0

4) R (P) is totally ordered.

5) If A, B ∈ C(G) such that P ⊆ A and P ⊆ B, then either A ⊆ B or B ⊆ A.

6) If P ⊂ A and P ⊂ B, then P ⊂ A ∩ B

7) If g, h ∈ G⁺\ P, then g ∧ h ∈ G⁺\ P

Deﬁnition 1.11 (Darnel [3]). A prime ℓ-subgroup P of an ℓ-group G is called a minimal

prime ℓ-subgroup if there does not exist another prime ℓ-subgroup Q such that Q ꢀ P.

Lemma 1.12 (Darnel [3]). Let G be an ℓ-group. Then there exist a minimal prime

ℓ-subgroup of G and each prime ℓ-subgroup of G contains atleast one minimal prime ℓ-

subgroup.

Lemma 1.13 (Darnel [3]). Let G be an ℓ-group and let A, B, P ∈ C(G). Then P is a

prime ℓ-subgroup of G if and only if A ∩ B ⊆ P implies A ⊆ P or B ⊆ P.

Denote by Spec(G), the set of all prime ℓ-subgroups of an ℓ-group G. It is known

that for a prime ℓ-subgroup P, the complement G⁺\ P, is a ﬁlter on the lattice G⁺.

Furthermore, the intersection of a chain of prime ℓ-subgroups yields another prime ℓ-

subgroup. Consequently, by Zorn’s lemma, minimal prime ℓ-subgroups exist, and every

prime ℓ-subgroup contains a minimal prime ℓ-subgroup. Min(G) denotes the set of all

minimal prime ℓ-subgroups of G. For a subset A ⊆ G, the polar of A is denoted by

A^⊥= {g ∈ G : |g| ∧ |a| = 0 for all a ∈ A}. For A = {a}, we write it’s polar as a^⊥. Ob-

serve that, polars are convex ℓ-subgroups.

Deﬁnition 1.14 (Bhattacharjee and McGovern [2]). Let G be an ℓ-group. A subset F

of G⁺, maximal with respect to the property that a ∧ b > 0 for all a, b ∈ F is called an

ultraﬁlter on G⁺.

Theorem 1.15 (Bhattacharjee and McGovern [2]). Let G be an ℓ-group. For P ∈

Spec(G), the following statements are equivalent:

1) P is a minimal prime ℓ-subgroup.

2) G⁺\ P is an ultraﬁlter on G⁺.

ꢁ

ꢂ

ꢀ

3) P =

g^⊥: g ∈/ G⁺\ P .

4) P is prime and for all g ∈ P, g^⊥⊆ P.

We now obtain a characterization of minimal prime ℓ-subgroup. This characterization is

frequently used in the development of the paper.

Lemma 1.16. Let P be a prime ℓ-subgroup of an ℓ-group G. Then the following state-

ments are equivalent.

50

Diksha Saste, Mayur Kshirsagar, Nilesh Mundlik, and Aditi S. Phadke

1) P is a minimal prime ℓ-subgroup.

2) For every x ∈ P, there exists y ∈/ P such that |x| ∧ |y| = 0.

3) For any x ∈ P and I is any convex ℓ-subgroup of G not contained in P, then there

exists a ∈ I\P such that |x| ∧ |a| = 0.

Proof. 1) ⇔ 2) : follows from the Theorem 1.15.

2) ⇒ 3) : Let P be a prime ℓ-subgroup of an ℓ-group G. Suppose that, x ∈ P and I is a

convex ℓ-subgroup of G such that I ⊆ P. By the hypothesis, there exists y ∈/ P such that

|x| ∧ |y| = 0. Choose t ∈ I \ P. Setting a = |t| ∧ |y|, we have 0 ≤ |t| ∧ |y| ≤ |t|. As I is a

convex ℓ-subgroup, |t| ∧ |y| ∈ I but |t| ∧ |y| ∈/ P. Note that, |x| ∧ |a| = |x| ∧ | (|y| ∧ |t|) | =

|x| ∧ |y| ∧ |t| = 0 ∧ |t| = 0.

3) ⇒ 1) : Let P be a prime ℓ-subgroup of an ℓ-group G and let x ∈ P. In particular, if we

choose I = G then by the hypothesis there exists y ∈ G \ P such that |x| ∧ |y| = 0.

□

2. Minimal n-prime ℓ-subgroups

In this section, we deﬁne the notion of n-prime ℓ-subgroup and minimal n-prime ℓ-

subgroup and give its characterization.

Deﬁnition 2.1. A proper ℓ-subgroup P of an ℓ-group G is said to be an n-prime ℓ-

subgroup(n ≥ 2) if it can be represented as an intersection of at most (n − 1) distinct

prime ℓ-subgroups.

In the following example, we show that n-prime ℓ-subgroup need not be a prime ℓ-

subgroup.

Example 2.2. Consider the ℓ-group G = Z × Z with the component wise order and

addition. Observe that, A = {0} × Z and B = Z × {0} are both prime ℓ-subgroups of G

but A∩B = {(0, 0)} is not a prime ℓ-subgroup. However, {(0, 0)} is a 3-prime ℓ-subgroup

of G.

It can be easily observed that every n-prime ℓ-subgroup becomes an (n + 1)-prime ℓ-

subgroup and the notion 2-prime ℓ-subgroup is equivalent to the standard concept of

prime ℓ-subgroup.

Deﬁnition 2.3. An n-prime ℓ-subgroup P of an ℓ-group G is called a minimal n-prime

ℓ-subgroup if there does not exist another n-prime ℓ-subgroup Q such that Q ꢀ P.

Example 2.4. Consider the ℓ-group G = Z × Z × Z with the component wise order and

addition. Observe that, A = {0} × Z × Z, B = Z × {0} × Z, B = Z × Z × {0} are

three prime ℓ-subgroups of G. Here A ∩ B = {0} × {0} × Z, A ∩ C = {0} × Z × {0},

B ∩ C = Z × {0} × {0}. Thus A ∩ B, A ∩ C, B ∩ C are minimal 3-prime ℓ-subgroups.

Denote by P_n(G) the set of all minimal n-prime ℓ-subgroups of an ℓ-group G. If n = 2,

we observe that P₂(G) is equivalent to Min(G).

We now prove the following Lemma which is cruical as it implies that for ﬁxed n, where

n ≥ 2, if an ℓ-subgroup is minimal n-prime, then it cannot be minimal (n − 1)-prime.

ON MINIMAL n-PRIMES SUBGROUPS OF ℓ-GROUPS

51

Lemma 2.5. Let G be an ℓ-subgroup with |P₂(G)| = |Min(G)| > n−2. Then no element

of P_n(G) is an (n−1)-prime ℓ-subgroup. Consequently, every minimal n-prime ℓ-subgroup

can be represented as an intersection of exactly (n − 1) distinct prime ℓ-subgroups of G.

Proof. Let G be an ℓ-subgroup with |P₂(G)| = |Min(G)| > n−2. Suppose on the contrary

m

ꢃ

that, Q ∈ P_n(G) and it is minimal (n − 1)-prime ℓ-subgroup. Then Q =

P_i^′s are distinct prime ℓ-subgroups of G and m ≤ (n − 2). Since |Min(G)| > (n − 2),

P_i, where

i=1

there exists a minimal prime ℓ-subgroup of G say P^′such that P^′= P_i, 1 ≤ i ≤ m.

ꢄ

ꢅ

ꢄ

ꢅ

m

ꢃ

Let R = Q

P^′=

P_i

P^′. If R = Q then

P_i

⊆ P^′. As P_i∈ C(G), by the

Lemma 1.13 and minimality of P^′, we have P_k= P^′ⁱ⁼fo¹r some k, a contradiction. Thus

R ꢀ Q implies that R is an (n−1)- prime ℓ-subgroup. As every (n−1)- prime ℓ-subgroup

of G is an n-prime ℓ-subgroup, R is an n-prime ℓ-subgroup. But R ꢀ Q, a contradiction

to the fact that Q is a minimal n-prime ℓ-subgroup. Consequently, every minimal n-

prime ℓ-subgroup can be represented as an intersection of exactly (n − 1) distinct prime

i=1

ℓ-subgroups of G.

□

Lemma 2.6. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n−2 and let P be a minimal

n-prime ℓ-subgroup of G. If x₁, x₂, . . . , x_nare elements of G satisfying |x_i|∧|x_j| ∈ P, i = j

then there exists x_i(1 ≤ i ≤ n) such that |x_i| ∈ P.

Proof. Let P be a minimal n-prime ℓ-subgroup of G such that P = P₁∩P₂∩· · ·∩P_(n−1), for

P₁, P₂, . . . , P_(n−1)∈ Spec(G). Let x₁, x₂, . . . , x_nbe elements of G such that |x_i| ∧ |x_j| ∈ P

for all i = j and 1 ≤ i, j ≤ n. Then |x_i| ∧ |x_j| ∈ P_kfor every k ∈ {1, 2, . . . , (n − 1)}.

By primeness of each P_k, we have |x_i| ∈ P_kor |x_j| ∈ P_k. So there exists at most one

x_isuch that |x_i| ∈/ P_kfor every k. Since k < n, there exists some x_isuch that |x_i| ∈

P₁∩ P₂∩ · · · ∩ P_(n−1)and so, |x_i| ∈ P.

□

We now characterize the minimal n-prime ℓ-subgroups in an ℓ-group G.

Theorem 2.7. Let P be an n-prime ℓ-subgroup of an ℓ-group G with |P₂(G)| = |Min(G)| >

n − 2. Then the following statements are equivalent:

1) P is a minimal n-prime ℓ-subgroup.

2) P is an intersection of exactly n − 1 minimal prime ℓ-subgroups.

3) For each x ∈ P, there exist a₁, a₂, . . . , a_n−1∈/ P with |x| ∧ |a_i| = 0, 1 ≤ i ≤ n − 1

and |a_i| ∧ |a_j| = 0 for i = j.

Proof. 1) ⇒ 2) Let P be a minimal n-prime ℓ-subgroup of an ℓ-group G with |P₂(G)| =

n−1

ꢃ

|Min(G)| > n − 2. By the Lemma 2.5, P =

P_i, where each P_iis prime ℓ-subgroup.

i=1

Again, by the Lemma 1.12, each prime ℓ-subgroup P_icontains a minimal prime ℓ-subgroup

m

ꢃ

say P_i^′. Now, if R =

P_i^′, where m ≤ n − 1 then R is an n-prime ℓ-subgroup and clearly

i=1

R ⊆ P. As P is a minimal n-prime ℓ-subgroup, we have R = P. If m ≤ n − 2 then P

52

Diksha Saste, Mayur Kshirsagar, Nilesh Mundlik, and Aditi S. Phadke

is an (n − 1)-prime ℓ-subgroup, a contradiction to the Lemma 2.5 and so, m = n − 1.

n−1

ꢃ

Therefore, P =

P_i^′is an intersection of exactly n − 1 minimal prime ℓ-subgroups.

i=1

2) ⇒ 3) Suppose that, every n-prime ℓ-subgroup of an ℓ-group G is an intersection of

exactly n − 1 minimal prime ℓ-subgroups. Let P be an n-prime ℓ-subgroup of an ℓ-group

G.

We prove the result by the induction on n. By the Lemma 1.16, 3) holds for n = 2.

Assume that, if P is an intersection of exactly n − 2 minimal prime ℓ-subgroups then

for each x ∈ P there exists a₁, a₂, . . . , a_n−2∈ P with |x| ∧ |a_i| = 0 and |a_i| ∧ |a_j| = 0,

for i = j. Now, suppose that P is an intersection of exactly (n − 1) minimal prime ℓ-

n−1

ꢃ

subgroups. Then P =

P_i, where each P_iis a minimal prime ℓ-subgroups. Let x ∈ P.

i=1

n−2

ꢃ

Consider, Q =

P_i, implies that x ∈ Q. By the induction assumption, there exist

i=1

a₁, a₂, . . . , a_n−2∈/ Q such that |x| ∧ |a_i| = 0 and |a_i| ∧ |a_j| = 0, for i = j. Now, for ﬁx

k ∈ {1, 2, . . . , n − 2}, |a_i| ∧ |a_j| ∈ P_k, for all i = j. Therefore, there exists at most one a_i

such that |a_i| ∈ P_k. Otherwise, we have |a_i|, |a_i| ∈ P_kimplies that |a_i|∧|a_i| = 0 ∈ Q, a

1

2

1

2

ꢆ

ꢉ

n−2

ꢃ

ꢇ

ꢈ

ꢊ

ꢋ

contradiction. Without loss of generality, we may assume that |a_i| ∈

P_j

\ P_i. Now

j=1

j=i

we have following two cases:

(A) : |a_i| ∈ P_n−1, i = 1, 2, . . . , (n − 2). Since P_n−1in a minimal prime, by the Lemma

1.16 there exists c₁, c₂, . . . , c_n−2∈/ P_n−1such that |a_i| ∧ |c_i| = 0. Also, as x ∈ P_n−1

there exists f ∈ P_n−1such that |x| ∧ |f| = 0 follows from the case n = 2. Consider

a_n−1= |f| ∧ |c₁| ∧ |c₂| ∧ . . . ∧ |c_n−2|. Observe that |x| ∧ |a_n−1| = 0 and |a_i| ∧ |a_n−1| = 0 for

each i, i = 1, 2, . . . , (n − 2). Thus in this case 3) holds.

(B) : Now, assume that P_n−1does not contain at most one |a_i|, say |a_k| ∈ P_n−1

.

Otherwise, |a_k| ∧ |a_j| = 0 ∈ P_n−1for i = j, but |a_j|, |a_k| ∈ P_n−1, a contradiction. As

P_k= P_n−1, there exists d such that d ∈ P_n−1\P_k. Consider, a^′= |a_k| ∧ |d|. Clearly,

k

|a^′| = |a_k| ∧ |d| ≤ |d| ∈ P_n−1and hence |a^′| ∈ P_n−1. Also, |a_k| ∈/ P_kand |d| ∈/ P_kimplies

k

that |a^′| ∈/ P_k. Thus |a^′| ∈ P_n−1\P_k. Replacing a_kby a^′in the set {a₁, a₂, . . . , a_n−2} ,

k

this case follows from (A).

3) ⇒ 1) : Suppose on the contrary that, Q is an n-prime ℓ-subgroup with Q ꢀ P. Choose

x ∈ P\Q. By the hypothesis, there exist a₁, a₂, . . . , a_n−1∈/ P such that |x| ∧ |a_i| = 0 and

|a_i| ∧ |a_j| = 0 for i = j, and i, j ∈ {1, 2, . . . , n − 1}. But Q is an n-prime ℓ-subgroup.

Therefore by the Lemma 2.6, |x| ∈ Q or |a_i| ∈ Q ⊆ P, a contradiction. Thus, P is a

minimal n-prime ℓ-subgroup of G.

□

ON MINIMAL n-PRIMES SUBGROUPS OF ℓ-GROUPS

53

3. Space of minimal n-prime ℓ-subgroups

In this section we deﬁne a toplogy on the set P_n(G) of all minimal n-prime ℓ-subgroups

of an ℓ-group. Here we study the open and closed subsets in P_n(G). We prove few

results related to compactness, Hausdorﬀ property and regularity of this topological space.

We further consider the product of distinct minimal prime ℓ-subgroups and study its

properties. Finally we prove the characterization that P_n(G) is compact if and only if

P_n(G) is ﬁnite.

For a ∈ G, we deﬁne the subset h_n(a) = {P ∈ P_n(G) : a ∈ P} of P_n(G).

Lemma 3.1. Let G be an ℓ-group and let a ∈ G. Then h_n(a) = h_n(|a|).

Proof. Let P be a minimal prime n-subgroup of an ℓ-group G such that P ∈ h_n(a). P is

a convex ℓ-subgroup of G and thus 0 ∈ P. Then a ∨ 0 = a₊∈ P and −a ∨ 0 = a ∈ P.

−

This implies that, a₊∨ a = |a| ∈ P and so, P ∈ h_n(|a|). Conversely, let P ∈ h_n(|a|).

Then |a| ∈ P. Since 0 ≤⁻a ≤ |a| ∈ P or 0 ≤ −a ≤ |a| ∈ P. By using convexity of P,

we have a ∈ P or −a ∈ P. Consequently, we have a ∈ P implies that P ∈ h_n(a). Thus,

h_n(a) = h_n(|a|).

□

Proposition 3.2. Let G be an ℓ-group and let a, b ∈ G. Then h_n(|a|) ∩ h_n(|b|) = h_n(|a| ∨

|b|).

Proof. Let P be a minimal prime n-subgroup of an ℓ-group G. Suppose that, P ∈ h_n(|a|)∩

h_n(|b|). Then |a| ∈ P and |b| ∈ P, implies that |a| ∨ |b| ∈ P. Thus P ∈ h_n(|a| ∨ |b|).

Conversely, if P ∈ h_n(|a| ∨ |b|) then |a| ∨ |b| ∈ P. As |a| ≤ |a| ∨ |b| and |b| ≤ |a| ∨ |b|

implies that |a| ∈ P and |b| ∈ P. Thus P ∈ h_n(|a|) ∩ h_n(|b|) and so, h_n(|a|) ∩ h_n(|b|) =

h_n(|a| ∨ |b|).

□

Deﬁnition 3.3 (Munkres [8]). A subbasis S for a topology on X is a collection of subsets

of X whose union equals X. The topology generated by the subbasis S is deﬁned to be the

collection τ of all unions of ﬁnite intersections of elements of S.

We deﬁne S = {P_n(G) \ h_n(|a|) : a ∈ G} the collection of a set complement of h_n(|a|),

for a ∈ G. S is subset of P_n(G). Note that, if P ∈ P_n(G) then there exists some

a ∈ G such that |a| ∈/ P. Thus P ∈ P_n(G) \ h_n(|a|). Therefore the collection S =

{P_n(G) \ h_n(|a|) : a ∈ G} forms a subbase for a topology on P_n(G). Thus any basic

open set in this topology is a ﬁnite intersections of members of S. Also, every open set of

this topology is a union of ﬁnite intersections of members of S.

The following lemma provides a detailed description of the basis elements for this topology.

Lemma 3.4. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n−2. A basis for the open

ꢌ

ꢍ

ⁿ⁻¹P_n(G)\h_n(|a_i|) : a_i∈ G .

ꢃ

sets in P_n(G) is formed by the collection

i=1

m

ꢃ

Proof. Let P ∈

(P_n(G)\h_n(|b_i|)) for some m. As P is a minimal n-prime ℓ-subgroup, it

i=1

can be expressed as the intersection of n−1 minimal prime ℓ-subgroups, say, P₁, P₂, . . . , P_(n−1)

.

54

Diksha Saste, Mayur Kshirsagar, Nilesh Mundlik, and Aditi S. Phadke

n−1

ꢃ

Thus P =

P_j. Since |b_i| ∈/ P for each i = 1, 2, . . . , m, there exists some minimal prime

j=1

ℓ-subgroup P_jsuch that |b_i| ∈/ P_j. Now, for each i ∈ {1, 2, . . . , n − 1}, deﬁne the set

B_i= {|b_j| : |b_j| ∈/ P_i}. Clearly, each B_iis ﬁnite and nonempty for some i. Without loss

of generality, assume that B₁= ∅ and let a₁= ∧B₁.

ꢌ

∧B_i, if B_i= ∅;

Now let us deﬁne a_i=

where 2 ≤ i ≤ n − 1.

a₁,

if B_i= ∅,

m

n−1

m

ꢎ

We show that,

h_n(|b_j|) ⊆

h_n(|a_i|). Let Q ∈

h_n(|b_j|) be a minimal n-prime

j=1

i=1

j=1

ℓ-subgroup. Then |b_j| ∈ Q for some 1 ≤ j ≤ m. By the deﬁnition of a_i’s, the expression

of some |a_k| must contains the element |b_j|. Hence Q ∈ h_n(|a_k|) for 1 ≤ k ≤ n − 1.

n−1

ꢎ

Now, we show that P ∈/

h_n(|a_i|). Suppose on the contrary that, P ∈

h_n(|a_i|),

i=1

then P ∈ h_n(|a_i|) for some i ∈ {1, 2, . . . , n − 1}. Let B_i= {|b₁|, |b₂|, . . . , |b_k|} then P_idoes

ꢏ

not contain |b₁|, |b₂|, . . . , |b_k|. As P_iis prime, we have |a_i| =

B_i∈/ P_i. Thus P ∈/ h_n(|a_i|),

ꢌ

ꢍ

a contradiction. Therefore, P ∈ ⁿ⁻¹P_n(G) \ h_n(|a_i|). Thus

ⁿ⁻¹P_n(G) \ h_n(|a_i|) : a_i∈ G

ꢃ

i=1

forms a basis for the topology generated by S on P_n(G).

□

n−1

ꢎ

From Lemma 3.4, the closed sets in P_n(G) are of the form

h_n(|a_i|), a_i∈ G.

i=1

Lemma 3.5. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n − 2. The space P_n(G)

has a subbasis of clopen sets.

Proof. Since S = {P_n(G) \ h_n(|a|) : a ∈ G} forms a subbase of topology on P_n(G), the

set P_n(G) \ h_n(|a|) is open. Now, we prove that h_n(|a|) is open for any a ∈ G. Clearly, if

h_n(|a|) = ∅ or h_n(|a|) = P_n(G) then h_n(|a|) is open. Let P ∈ h_n(|a|), for a ∈ G. Then by

the Theorem 2.7, there exist a₁, a₂, . . . , a_n−1∈/ P such that |a| ∧ |a_i| = 0 and |a_i| ∧ |a_j| =

n−1

ꢎ

0, i = j. Clearly, P ∈/ h_n(|a_i|) for each i = 1, 2, . . . , n − 1. Thus P ∈/

h_n(|a_i|). Now,

i=1

if R is any minimal n-prime ℓ-subgroup then |a| ∧ |a_i| = 0 = |a_i| ∧ |a_j| ∈ R. Therefore, by

ꢄ

ꢅ

n−1

ꢎ

the Lemma 2.6, |a| ∈ R or |a_i| ∈ R for some i. Thus h_n(|a|)

h_n(|a_i|)

= P_n(G)

i=1

implies that h_n(|a|) is open. Therefore, P_n(G) has a subbasis of clopen sets.

□

Corollary 3.6. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n − 2. The collection

ꢌ

ꢍ

ⁿ⁻¹P_n(G)\h_n(|a_i|) : a_i∈ G

forms a basis of clopen sets for P_n(G).

ꢃ

i=1

Deﬁnition 3.7. (Munkres [8]) A topological space is said to be zero-dimensional if it is

a non-empty T₁space with a base consisting of clopen sets.

ON MINIMAL n-PRIMES SUBGROUPS OF ℓ-GROUPS

55

Deﬁnition 3.8. (Munkres [8]) A topological space X is said to be a completely regular

space if every closed set A in X and a point x₀∈ X, x₀∈/ A, then there exists a continuous

function f : X → [0, 1], such that f(x₀) = 0 and f(A) = 1.

Corollary 3.9. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n − 2. P_n(G) is zero-

dimensional, completely regular and Hausdorﬀ.

Proof. It is well known that, every Hausdorﬀ space is T₁and every zero-dimensional space

is completely regular. By the Lemma 3.5, P_n(G) has a basis of clopen sets. Thus it is

enough to prove that P_n(G) is Hausdorﬀ space. Let P, Q be distinct elements of P_n(G)

and let a ∈ P \ Q. Then P ∈ h_n(|a|). By Theorem 2.7, there exist a₁, a₂, . . . , a_n−1∈/ P

n−1

ꢎ

such that |a| ∧ |a_i| = 0 and |a_i| ∧ |a_j| = 0. Then P ∈/

h_n(|a_i|) and Q ∈/ h_n(|a|). Since

i=1

n−1

ꢎ

|a_i| ∧ |a_j| = 0 ∈ Q, by Lemma 2.6, |a_k| ∈ Q for some k and so, Q ∈

h_n(|a_i|). As

i=1

ꢄ

ꢅ

n−1

ꢎ

h_n(|a|)

h_n(|a_i|)

= P_n(G) implies that P_n(G) is Hausdorﬀ. Therefore, P_n(G) is

i=1

zero-dimensional and completely regular.

□

Deﬁnition 3.10. Denote π_n(G) = P₂(G) × · · · × P₂(G) \ ∆_n(G) where

ꢐ

ꢑꢒ

ꢓ

(n−1)-times

ꢔ

ꢘ

ꢕ

ꢖ

ꢕ

ꢙ

∆_n(G) =

(P₁, P₂, . . . , P_n−1) ∈ P₂(G) × · · · × P₂(G) : P_i= P_jfor some i = j

ꢐ

ꢑꢒ

ꢓ

ꢕ

ꢗ

ꢕ

ꢚ

(n−1)-times

From Pawar and Thakare [9] (see Mundlik, Joshi and Halas [6]), the collection B =

{P₂(G) \ h(|x|) : x ∈ G} forms a basis for open sets in P₂(G). Thus B × · · · × B forms

ꢐ

ꢑꢒ

ꢓ

(n−1)-times

a basis for open sets for the product topology on P₂(G) × · · · × P₂(G). The subsequent

ꢐ

ꢑꢒ

ꢓ

(n−1)-times

Lemma gives us the basis for open sets for a subspace topology on π_n(G).

Lemma 3.11. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n − 2. The collection

ꢌ

ꢍ

n−1

ꢛ

(P₂(G) \ h₂(|a_i|)) : |a_i| ∧ |a_j| = 0 for i = j

i=1

forms a basis for open sets in π_n(G).

n−1

ꢛ

Proof. Let (P₁, P₂, . . . , P_n−1) ∈

(P₂(G) \ h₂(|b_i|)) . Since P₁, P₂, . . . , P_n−1are min-

i=1

imal prime ℓ-subgroups, therefore for each i, j ∈ {1, 2, . . . , n − 1}, there exists x_ij∈

P_i\ P_j, for i = j. By Theorem 2.6, there exists y_i∈/ P_isuch that |x_ij| ∧ |y_i| = 0. Let

r_j∈ P_j\ P_iand |y_i| ∧ |r_j| = |x_ji|(say) for i = j. Clearly, |x_ji| ∈ P_j\P_i. This gives

that |x_ij| ∧ |x_ji| = |x_ij| ∧ |y_i| ∧ |r_j| = 0. Now, setting |a_i| = |b_i| ∧ |x_1i| ∧ |x_2i| ∧ . . . ∧

|x_(i−1)i| ∧ |x_(i+1)i| ∧ . . . ∧ |x_(n−1)i| for each i ∈ {1, 2, . . . , n − 1}, gives that each |a_i| ∈/ P_i

56

Diksha Saste, Mayur Kshirsagar, Nilesh Mundlik, and Aditi S. Phadke

n−1

ꢛ

and so P_i∈ P₂(G) \ h₂(|a_i|). Note that (P₁, P₂, . . . , P_n−1) ∈

(P₂(G) \ h₂(|a_i|)) ⊆

i=1

n−1

ꢛ

(P₂(G) \ h₂(|b_i|)), where |a_i| ∧ |a_j| = 0, i = j .

□

i=1

Deﬁnition 3.12. (Munkres [8]) Let X and Y be topological spaces. A function f : X → Y

is a local homeomorphism, if for every point x ∈ X, there exists an open set U containing

x such that the image f(U) is open in Y and the restriction f|_U: U → f(U) is a

homeomorphism.

Theorem 3.13. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n − 2. The map

n−1

ꢃ

φ : π_n(G) → P_n(G) deﬁned by φ(P₁, P₂, . . . , P_n−1) =

P_iis a local homeomorphism.

i=1

Proof. Clearly, φ is well-deﬁned. Now, we prove that φ is continuous.

n−1

ꢎ

By the Corollary 3.6, let

h_n(|a_i|) be an open set in P_n(G).

i=1

ꢄ

ꢅ

n−1

ꢎ

We claim that φ⁻¹

h_n(|a_i|)

=

h₂(|a_r|) × h₂(|a_r|) × · · · × h₂(|a_r|) .

ꢐ

ꢑꢒ

ꢓ

i=1

1≤r≤n−1

(n−1)−times

ꢄ

ꢅ

n−1

ꢎ

ꢃ

ꢎ

Let (P₁, P₂, . . . , P_n−1) ∈ φ⁻¹

h_n(|a_i|) . Then φ(P₁, P₂, . . . , P_n−1) =

P_i∈

h_n(|a_i|).

i=1

n−1

ꢃ

Thus

P_i∈ h_n(|a_r|) for some r ∈ {1, 2, . . . , n − 1}. Hence |a_r| ∈ P_ifor i = {1, 2, . . . , n − 1} ,

i=1

n−1

ꢃ

implies that |a_r| ∈

P_i⊆ P_i.

i=1

Therefore (P₁, P₂, . . . , P_n−1) ∈ h₂(|a_r|) × h₂(|a_r|) × · · · × h₂(|a_r|) ⊆

ꢎ

h₂(|a_r|) × h₂(|a_r|) × · · · × h₂(|a_r|). Thus |a_r| ∈ P_ifor i ∈ {1, 2, . . . , n − 1} , implies

ꢐ

ꢑꢒ

ꢓ

1≤r≤n−1

(n−1)−times

n−1

ꢃ

ꢎ

that |a_r| ∈

P_i. This means that

P_i∈ h_n(|a_r|) ⊆

h_n(|a_i|). Thus (P₁, P₂, . . . , P_n−1) ∈

i=1

ꢄ

ꢅ

n−1

ꢎ

φ⁻¹

h_n(|a_i|)

and so, φ is continuous.

i=1

Now, we prove that φ is an open map. Let ⁿ⁻¹P₂(G) \ h₂(|a_i|) be a basic open set in

ꢛ

i=1

ꢄ

ꢅ

n

ꢛ

π_n(G), where |a_i|∧|a_j| = 0 for i = j ∈ {1, 2, . . . , n−1}. We claim that φ

P_n(G)\ⁿ⁻¹h_n(|a_i|). Let Q ∈ φ

P₂(G) \ h₂(|a_i|)

=

i=1

ꢄ

ꢅ

n

ꢎ

ꢛ

P₂(G) \ h₂(|a_i|) . Then there exists (P₁, P₂, . . . , P_n−1) ∈

i=1

ON MINIMAL n-PRIMES SUBGROUPS OF ℓ-GROUPS

57

ⁿ⁻¹P₂(G) \ h₂(|a_i|) such that

ꢛ

i=1

n−1

P_i= Q. Since (P₁, P₂, . . . , P_n−1) ∈ ⁿ⁻¹P₂(G) \ h₂(|a_i|), we have

ꢃ

ꢛ

φ (P₁, P₂, . . . , P_n−1) =

i=1

n−1

P_i= Q, implies that Q ∈ P_n(G) \ ⁿ⁻¹h_n(|a_i|). Let

ꢃ

ꢎ

|a_i| ∈/ P_ifor each i. Hence |a_i| ∈/

i=1

ꢄ

ꢅ

n−1

ꢎ

R ∈ P_n(G) \

h_n(|a_i|) . Then |a_i| ∈/ R for each i. By the Theorem 1.15, we have

i=1

n−1

ꢃ

R =

P_i, where P_i’s are minimal prime ℓ subgroups. Let B_i= {|a_j| : |a_j| ∈/ P_i} . Since

i=1

|a_i| ∧ |a_j| = 0 for i = j, we have |B_i| = 1 for each i. Thus there is one to one correspon-

dence between |a₁|, |a₂|, . . . , |a_n−1| and prime ℓ-subgroups P₁, P₂, . . . , P_n−1such that each

ꢄ

ꢅ

n

ꢛ

prime subgroup misses precisely one element. Hence R ∈ φ

(P₂(G) \ h₂(|a_i|))

and

i=1

so, φ is an open map. By Lemma 2.5, φ is onto.

Let (P₁, P₂, . . . , P_n−1) ∈ π_n(G) be any element. By Lemma 3.11, there exists a basic

open set U = ⁿ⁻¹P₂(G)\h₂(|a_i|) of (P₁, P₂, . . . , P_n−1), where |a_i| ∧ |a_j| = 0 for i = j.

ꢛ

i=1

We show φ|_Uis one-to-one. Let (Q₁, Q₂, . . . , Q_n−1) , (R₁, R₂, . . . , R_n−1) ∈ U be such that

n−1

ꢃ

φ (Q₁, Q₂, . . . , Q_n−1) = φ (R₁, R₂, . . . , R_n−1), that is,

Q_i=

R_i. First, we claim that

i=1

{Q₁, Q₂, . . . , Q_n−1} = {R₁, R₂, . . . , R_n−1}. Suppose not; then there exists R_ksuch that

R_k∈/ {Q₁, Q₂, . . . , Q_n−1}. Then R_k= Q_ifor each i. Let |b_i| ∈ Q_i\ R_kfor each i and

n−1

ꢃ

put b = |b₁| ∧ |b₂| ∧ . . . ∧ |b_n−1|. Then |b| ∈

Q_i=

R_i, implies that |b| ∈ R_k, a

i=1

contradiction. Suppose that, (Q₁, Q₂, . . . , Q_n−1) = (R₁, R₂, . . . , R_n−1). Then there exists

some i such that Q_i= R_i. It is also clear that there exists some k such that Q_i= R_kfor

some i = k. Since (Q₁, Q₂, . . . , Q_n−1) , (R₁, R₂, . . . , R_n−1) ∈ U = ⁿ⁻¹P₂(G)\h₂(|a_i|), we

ꢛ

i=1

have |a_i| ∈/ Q_iand |a_k| ∈/ R_k= Q_ifor i = k. But |a_i| ∧ |a_k| = 0 ∈ Q_i, a contradiction.

Hence φ|_Uis one-to-one.

□

Deﬁnition 3.14. (Munkres [8]) Let X and Y be topological spaces. A surjective map

f : X → Y is said to be a perfect map, if it is continuous, closed and, for each y ∈ Y , the

set f⁻¹(y) is compact.

Although the proof of the following theorem closely follows the proof presented in Mundlik

and Joshi [7], we include it here for the sake of completeness.

Theorem 3.15. Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n − 2. Then the

following statements are equivalent for n > 2.

58

Diksha Saste, Mayur Kshirsagar, Nilesh Mundlik, and Aditi S. Phadke

1) P_n(G) is compact.

2) π_n(G) is compact.

3) P (G) is ﬁnite.

4) P²_n(G) is ﬁnite.

Proof. 1) ⇒ 2) Let G be an ℓ-group with |P₂(G)| = |Min(G)| > n − 2. By the Theorem

n−1

ꢃ

3.13, the map φ : π_n(G) → P_n(G) deﬁned by φ(P₁, P₂, . . . , P_n−1) =

P_iis (n − 1)! to

i=1

one, onto and continuous. We prove that φ is a closed map. Let C be a closed set in

π_n(G). Choose y ∈ P_n(G) \ φ(C). Since φ is (n − 1)! to one, there exist distinct points

x₁, x₂, . . . , x_(n−1)!∈ π_n(G)\C with φ(x_i) = y. Since C is closed in π_n(G), for each x_ithere

is a neighbourhood U_isuch that U_i∩C = ∅. Observe that, ⁿ⁻¹P₂(G) is a Hausdorﬀ space,

and π_n(G) is a subspace of ⁿ⁻¹P₂(G), implies that π_n(G) is a Hausdorﬀ space. Hence for

ꢛ

i=1

ꢛ

i=1

x_i= x_jthere exists disjoint neighbourhoods V_iand V_jof x_iand x_j(respectively). Now,

local homeomorphism property of φ, implies that φ(V_i) is open in P_n(G) and contains y

(n−1)!

ꢃ

for each i ∈ {1, 2, . . . , (n − 1)!}. Thus

φ(V_i) is a neighbourhood of y.

i=1

ꢆ

ꢉ

(n−1)!

ꢃ

ꢈ

ꢋ

We now demonstrate that

φ(V_i)

φ(C) = ∅. Assume, for the sake of con-

i=1

ꢉ

ꢆ

(n−1)!

ꢃ

ꢈ

ꢋ

tradiction, that z ∈

φ(V_i)

φ(C). Therefore for each i, 1 ≤ i ≤ (n − 1)!, there

i=1

exists v_i∈ V_iand c ∈ C such that φ(v_i) = z = φ(c), a contradicting the fact that φ is

ꢆ

ꢉ

(n−1)!

ꢃ

ꢈ

ꢋ

an (n − 1)! to one map. Therefore

φ(V_i)

φ(C)

= ∅, implies that

φ(V_i) ⊆

i=1

ꢁ

ꢂ

P_n(G)\φ(C). This establishes that φ is a closed map. Since φ⁻¹(y) = x₁, x₂, . . . , x_(n−1)!

is a ﬁnite set for each y ∈ P_n(G), it is compact in π_n(G). Therefore φ is a perfect map.

Hence φ⁻¹(P_n(G)) = π_n(G) is compact by Munkres[[8], Ex 12, Page 172].

2) ⇒ 3) Assume that π_n(G) is compact. As π_n(G) ⊆ ⁿ⁻¹P₂(G) and

P₂(G) is

ꢛ

ꢜ

n−1

i=1

Housdorﬀ implies that π_n(G) is closed. Hence, ∆_n(G) is open in P₂(G) × · · · × P₂(G).

ꢐ

ꢑꢒ

ꢓ

Now, we prove that P₂(G) is discrete. Consider an arbitrary point P ∈⁽ⁿP⁻₂¹(⁾G^−t)ⁱ.^meC^shoose

(P, P, Q₁, Q₂, . . . , Q_n−3) ∈ ∆_n(G), where all Q₁, Q₂, . . . , Q_n−3are distinct and Q_i= P.

Since ∆_n(G) is open, there exists a neighbourhood U × U × U₁× U₂× · · · × U_n−3of

(P, P, Q₁, Q₂, . . . , Q_n−3) such that U × U × U₁× U₂× · · · × U_n−3⊆ ∆_n(G), where U

is a neighbourhood of P. As each Q_i= P using the Hausdorﬀ property of P₂(G),

we can ﬁnd neighbourhoods U_iof Q_iand W_iof P such that U_i∩ W_i= ∅. Then

ON MINIMAL n-PRIMES SUBGROUPS OF ℓ-GROUPS

59

n−3

ꢃ

U^′=

W_iis a neighbourhood of P which is disjoint from each U_i, i = 1, 2, . . . , n − 3.

i=1

We claim that U^′= {P}. Suppose that U^′contains another point, say R(= P) then

(P, R, Q₁, Q₂, . . . , Q_n−3) ∈ (U^′× U^′× U₁× U₂× · · · × U_n−3) ⊆ ∆_n(G), a contradiction.

Therefore {P} is open in P₂(G) making P₂(G) is discrete and hence π_n(G) is discrete. As

it is compact, π_n(G) is ﬁnite. Thus P₂(G) is ﬁnite.

3) ⇒ 4) If P₂(G) is ﬁnite then there are only ﬁnitely many choices of subsets of P₂(G) of

size n − 1, hence only ﬁnitely many possible intersections. Consequently P_n(G) is ﬁnite.

4) ⇒ 1) If P_n(G) is ﬁnite. It is well known that every ﬁnite set of topological space is

compact. Consequently P_n(G) is compact.

□

4. Acknowledgement

The authors are thankful to the referee for valuable comments.

REFERENCES

[1] D. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm Algebra 9(5) (2011) ,

1646-1672.

[2] P. Bhattacharjee and W. McGovern, Lamron ℓ-groups, Quaestiones Mathematicae, 41:1,(2018), 81-98.

[3] M. Darnel, Theory of Lattice-ordered Groups, Monographs and Textbooks in Pure and Applied Mathematics,

Vol. 187, Marcel Dekker, New York, 1995.

[4] R. Halaˇs, On extensions of ideals in posets, Discrete Math, 308(21) (2008), 4972-4977.

[5] N. Hindman, Minimal n-prime ideal spaces, Math. Annalen. 199 (1972), 97-114.

[6] N. Mundlik and V. Joshi, R. Halaˇs, The hull-kernel topology on prime ideals in posets, Soft Comput. 21(7)

(2017),1653-1665.

[7] N. Mundlik and V. Joshi, Space of Minimal n-Prime Ideals, Order 36 (2019),225-232.

[8] J. Munkres, Topology , 2nd edn. Prentice Hall, Upper Saddle River (2000).

[9] Y. S. Pawar and N. K. Thakare, The space of minimal prime ideals in a 0-distributive semilattices, Period.

Math. Hungar. 13(4) (1982), 309-319.

[10] R. Pierce, Modules over commutative regular rings, Mem Amer. Math. Soc., No. 70 (1967).

(Received, August 28, 2025)

(Revised, November 10, 2025)

^1,3,4Department of Mathematics,

Modern Education Society’s, Nowrosjee Wadia College,

Pune-411001, India.

²Department of Mathematics,

Fergusson College (Autonomous),

Pune-411004, India.

Email¹: sastediksha2021@gmail.com

Email²: mayur.kshirsagar@fergusson.edu

Email³: mundliknilesh@gmail.com

Email⁴: phadkeaditi@gmail.com