On Generalised Quasi-ideals in Ordered Ternary Semigroups

Mohammad Yahya Abbasi; Sabahat Ali Khan; Abul Basar

doi:10.5666/KMJ.2017.57.4.545

Articles

Kyungpook Mathematical Journal 2017; 57(4): 545-558

Published online December 23, 2017

On Generalised Quasi-ideals in Ordered Ternary Semigroups

Mohammad Yahya Abbasi, Sabahat Ali Khan and Abul Basar

Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India

Received: August 7, 2016; Accepted: December 29, 2016

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Generalised Quasi-ideals
4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups
Acknowledgements
References

In this paper, we introduce generalised quasi-ideals in ordered ternary semigroups. Also, we define ordered m-right ideals, ordered (p, q)-lateral ideals and ordered n-left ideals in ordered ternary semigroups and studied the relation between them. Some intersection properties of ordered (m, (p, q), n)-quasi ideals are examined. We also characterize these notions in terms of minimal ordered (m, (p, q), n)-quasi-ideals in ordered ternary semigroups. Moreover, m-right simple, (p, q)-lateral simple, n-left simple, and (m, (p, q), n)-quasi simple ordered ternary semigroups are defined and some properties of them are studied.

Keywords: quasi ideal, ordered ternary semigroup

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Generalised Quasi-ideals
4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups
Acknowledgements
References

The idea of investigation of n-ary algebras i.e. the sets with one n-ary operation was given by Kasner’s [10]. Dornte [6] introduced the notion of n-ary groups. A ternary semigroup is a particular case of an n-ary semigroup for n=3[14]. Ternary semigroups are universal algebra with one associative operation. Different applications of ternary structures in physics are described by Kerner [12]. Sioson [15] studied the ideal theory in ternary semigroup. He also introduced the notion of regular ternary semigroups and characterized them by using the notion of quasi-ideals. Dixit and Dewan [5] studied the properties of quasi-ideals and bi-ideals in ternary semigroups.

Steinfeld [16] introduced the notion of quasi-ideal for semigroups. It is a generalization of the notion of one sided ideal. The concept of the (m, n)-quasi-ideal in semigroups was given by Lajos [13]. It is studied by many researchers in different algebraic structures [1, 3]. Dubey and Anuradha [7] introduced generalised quasi-ideals and generalised bi-ideals in ternary semigroups and characterized these notions in terms of minimal quasi-ideals and minimal bi-ideals in ternary semigroups.

Kehayopulu build-up the theory of partially ordered semigroups. In [11] he introduced the concept and notion of ordered quasi-ideals in ordered semigroups. Abbasi and Basar [2] characterized intra-regular po-Γ-semigroups through ordered quasi-Γ-ideals, ordered right Γ-ideals and ordered left Γ-ideals. Iampan [8] introduced ordered ternary semigroup and characterized the minimality and maximality of ordered lateral ideals in ordered ternary semigroups. Daddi and Pawar [4] introduced the concepts of ordered quasi-ideals, ordered bi-ideals in ordered ternary semigroups and studied their properties. Jailoka and Iampan [9] studied some results on the minimality and maximality of ordered quasi-ideals in ordered ternary semigroups.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Generalised Quasi-ideals
4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups
Acknowledgements
References

Definition 2.1.([14])

A non-empty set S with a ternary operation S × S × S → S, written as (x₁, x₂, x₃) ↦ [x₁, x₂, x₃], is called a ternary semigroup if it satisfies the following identity, for any x₁, x₂, x₃, x₄, x₅ ∈ S,

[[x1x2x3]x4x5]=[x1[x2x3x4]x5]=[[x1x2[x3x4x5]].

For non-empty subsets A,B and C of a ternary semigroup S,

[ABC]:={[abc]:a∈A,b∈B and c∈C}.

If A = {a}, then we write [{a}BC] as [aBC] and similarly if B = {b} or C = {c}, we write [AbC] and [ABc], respectively. For the sake of simplicity, we write [x₁x₂x₃] as x₁x₂x₃ and [ABC] as ABC.

Definition 2.2

A non-empty subset T of a ternary semigroup S is called a ternary subsemigroup of S if TTT ⊆ T.

For any positive integers m and n with m ≤ n and any elements x₁, x₂, x₃………x_{2_n} and x_{2_n+1} of a ternary semigroup [15], we can write

[x1x2x3………x2n+1]=[x1,x2,x3..[[xmxm+1xm+2]xm+3xm+4]…x2n+1].

Example 2.3.([5])

Let S = {−i, 0, i}. Then S is a ternary semigroup under the multiplication over complex number while S is not a semigroup under complex number multiplication.

Definition 2.4.([8])

A ternary semigroup S is called a partially ordered ternary semigroup if there exits a partially ordered relation ≤ such that for any a, b, x, y ∈ S, a ≤ b ⇒ axy ≤ bxy, xay ≤ xby, and xya ≤ xyb.

Example 2.5

Let

S={ (a00bc0defg0hi00j):a,b,c,d,e,f,g,h,i,j∈ℕ∪{0}},

where ℕ ∪{0} is an ordered ternary semigroup under the ordinary multiplication of numbers with partial ordered relation ≤_ℕ is ”less than or equal to”. Now we define partial order relation ≤_S on S by, for any A, B ∈ S

A≤S B if and only if aij≤ℕ bij, for all i and j.

Then it is easy to verify that S is an ordered ternary semigroup under usual multiplication of matrices over ℕ ∪{0} with partial order relation ≤_S.

For a subset H of S, we denote (H]:= {s ∈ S | s ≤ h for some h ∈ H}. If H = {a}, we also write ({a}] as (a].

Definition 2.6.([8])

A ternary subsemigroup T of S is called an ordered ternary subsemigroup of S if (T] ⊆ T.

Theorem 2.7.([4])

Let S be an ordered ternary semigroup, then the following hold:

A ⊆ (A], for all A ⊆ S.
If A ⊆ B ⊆ S, then (A] ⊆ (B].
((A]] = (A], for all A⊆ S.
(A](B](C] ⊆ (ABC], for all A,B,C ⊆ S.

Definition 2.8.([8])

An element z of S is called a zero element if

zxy = xzy = xyz = z for all x, y ∈ S, and
z ≤ x for all x ∈ S.

If z ∈ S is a zero element, it is denoted by 0.

Definition 2.9.([4])

An element a of S is called regular if there exists an element x in S such that a ≤ axa. S is called regular ordered ternary semigroup if every element of S is regular.

Theorem 2.10.([4])

Let T be an ordered ternary subsemigroup of S. Then T is regular if and only if a ∈ (aT a], for all a ∈ T.

Definition 2.11.([4])

A non-empty subset I of S is called an ordered right (resp, ordered left, ordered lateral) ideal if

ISS ⊆ I (resp., SSI ⊆ I, SIS ⊆ I), and
(I] ⊆ I.

Example 2.12

In Example 2.5, let

R={ (a00b00000000c00d):a,b,c,d∈ℕ∪{0}} s.t. R⊆S.

Then R is an ordered right ideal of S.

A non-empty subset I of S is called an ordered ideal of S if I is an ordered left, an ordered right and an ordered lateral ideal of S.

Example 2.13

In Example 2.5, let

I={ (a00eb00fc00gd00h):a,b,c,d,e,f,g,h∈ℕ∪{0}} s.t. I⊆S.

Then I is an ordered ideal of S.

Definition 2.14.([4])

A non-empty subset Q of S is called an ordered quasi-ideal of S if

(SSQ] ∩ (SQS] ∩ (QSS] ⊆ Q,
(SSQ] ∩ (SSQSS] ∩ (QSS] ⊆ Q, and
(Q] ⊆ Q.

Example 2.15

In Example 2.5, let

Q={ (000a000b00000000):a,b∈ℕ∪{0}} s.t. Q⊆S.

Then Q is an ordered quasi ideal of S which is not an ordered ideal of S.

We can easily prove that {0} is the smallest ordered quasi-ideal of S with a zero element and it is called a zero ordered quasi-ideal of S. Moreover, 0 ∈ Q for all ordered quasi-ideal Q of S.

Definition 2.16.([4])

A non-empty subset B of S is called an ordered bi-ideal of S if,

BSBSB ⊆ B,
For a ∈ B, b ∈ S such that b ≤ a implies b ∈ B. i.e. (B] = B.

Example 2.17

Consider S = L₄(ℕ∪{0}), be the set of all strictly lower triangular 4× 4 matrices over ℕ∪{0}. As we know that ℕ ∪{0} is an ordered ternary semigroup under the ordinary multiplication of numbers with partial ordered relation ≤_ℕ is ”less than or equal to”. Then S is an ordered ternary semigroup under the usual multiplication of matrices over ℕ ∪ {0} with partial order relation ≤_S, as defined in the Example 2.5. Let

B4={ (00000000a0000b00):a,b∈ℕ∪{0}}.

Clearly B₄ is a ternary subsemigroup of S. We have that B₄SB₄SB₄ ⊆ B₄ and (B₄] ⊆ B₄. But (B₄SS] ∩ (SB₄S ∪ SSB₄SS] ∩ (SSB₄] =

{ (00000000a000bc00):a,b,c∈ℕ∪{0}}⊈B4.

Therefore B₄ is an ordered bi-ideal of S which is not an ordered quasi-ideal of S.

3. Generalised Quasi-ideals

Go to

Abstract
1. Introduction
2. Preliminaries
3. Generalised Quasi-ideals
4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups
Acknowledgements
References

In this section, we define ordered (m, (p, q), n)-quasi-ideal of an ordered ternary semigroup and establish some of their elementary properties.

Definition 3.1

A ternary subsemigroup Q of S is called a generalised quasi-ideal or an ordered (m, (p, q), n)-quasi-ideal of S if

(Q(SS)^m] ∩ ((S^pQS^q ∪ S^pSQSS^q)] ∩ ((SS)ⁿQ] ⊆ Q, where m, n, p, q are positive integers greater than zero and p + q = even,
(Q] ⊆ Q.

Example 3.2

All the ordered quasi ideals of the Examples 2.12, 2.13 and 2.15 are ordered (m, (p, q), n) quasi ideals of S.

Remark 3.3

Every ordered quasi-ideal of S is an ordered (1, (1, 1), 1)-quasi-ideal of S. But an ordered (m, (p, q), n)-quasi-ideal need not be an ordered quasi-ideal of S.

Example 3.4

Let

S={ (00000a0000bc000def00ghij0):a,b,c,d,e,f,g,h,i,j∈ℕ∪{0}}.

As we know that ℕ ∪{0} is an ordered ternary semigroup under ordinary multiplication of numbers with partial ordered relation ≤_ℕ is ”less than or equal to”. Then S is an ordered ternary semigroup under the usual multiplication of matrices over ℕ ∪ {0} with partial order relation ≤_S, as defined in the Example 2.5. Let

Qgen={ (0000000000a00000b00000c00):a,b,c∈ℕ∪{0}}.

Then it is easy to see that Q_gen is a ternary subsemigroup of S and Q_gen is an ordered (2, (2, 2), 2) quasi ideal of S. Now (Q_genSS] ∩ (SQ_genS ∪ SSQ_genSS] ∩ (SSQ_gen] =

{ (00000000000000000000a0000):a∈ℕ∪{0}}⊈Qgen.

Therefore Q_gen is not an ordered (1, (1, 1), 1) quasi-ideal ideal of S. Although Q_gen is an ordered (1, (1, 1), 1) bi-ideal of S.

Lemma 3.5

Let {T_i | i ∈ I} be the arbitrary collection of ordered ternary subsemigroups of S such that∩i∈ITi≠∅. Then∩i∈ITiis an ordered ternary subsemigroup of S.

Proof

Let T_i be an ordered ternary subsemigroup of S for all i ∈ I such that ∩i∈ITi≠∅ and let t1,t2,t3∈∩i∈ITi for all i ∈ I. As T_i is an ordered ternary subsemigroup of S for all i ∈ I, we have t₁t₂t₃ ∈ T_i for all i ∈ I. Therefore t1t2t3∈∩i∈ITi.

Now suppose that x∈(∩i∈ITi]. Then x ≤ a, for some a∈∩i∈ITi. Now a ∈ T_i, for all i ∈ I, it implies x ∈ (T_i] = T_i, for all i ∈ I. Thus we have x∈∩i∈ITi, which shows that (∩i∈ITi]⊆∩i∈ITi. Hence ∩i∈ITi is an ordered ternary subsemigroup of S.

Theorem 3.6

Let S be an ordered ternary semigroup and Q_i be an ordered (m, (p, q), n)-quasi ideal of S such that∩i∈IQi≠∅. Then∩i∈IQiis an ordered (m, (p, q), n)-quasi ideal of S.

Proof

Let {Q_i | i ∈ I} be a family of ordered (m, (p, q), n)-quasi ideal of S. Clearly Q=∩i∈IQi is an ordered ternary subsemigroup of S by the Lemma 3.5. We claim that Q is an ordered (m, (p, q), n)-quasi ideal of S. Now

(Q(SS)m]∩(SpQSq∪SpSQSSq]∩((SS)nQ]=(∩i∈IQi(SS)m]∩(Sp∩i∈IQiSq∪SpS∩i∈IQiSSq]∩((SS)n∩i∈IQi]⊆(Qi(SS)m]∩(SpQiSq∪SpSQiSSq]∩((SS)nQi], for all i∈I.⊆Qi, for all i∈I.

Therefore (Q(SS)m]∩(SpQSq∪SpSQSSq]∩((SS)nQ]⊆∩i∈IQi. Consequently Q is an ordered (m, (p, q), n)-quasi ideal of S.

Definition 3.7

Let S be an ordered ternary semigroup. Then a ternary subsemigroup

R of S is called an ordered m-right ideal of S if R(SS)^m ⊆ R and (R] = R,
M of S is called an ordered (p, q)-lateral ideal of S if (S^pMS^q ∪S^pSMSS^q) ⊆ M and (M] = M,
L of S is called an ordered n-left ideal of S if (SS)ⁿL ⊆ L and (L] = L. where m, n, p, q are positive integers and p + q is an even positive integer.

Theorem 3.8

Every ordered m-right, ordered (p, q)-lateral and ordered n-left ideal of S is an ordered (m, (p, q), n)-quasi ideal of S. But converse need not be true.

Proof

Proof is straight forward. Conversely, take an ordered ternary semigroup S given in the Example 2.5. Let

H={ (000a00bc00000000):a,b,c∈ℕ∪{0}}.

Then H is an ordered (3, (2, 2), 3)-quasi ideal of S. But it is not an ordered 3-right ideal, an ordered (2, 2)-lateral ideal and an ordered 3-left ideal of S.

Theorem 3.9

Let S be an ordered ternary semigroup. Then the following statements hold:

Let R_i be an ordered m-right ideal of S such that∩i∈IRi≠∅. Then∩i∈IRiis an ordered m-right ideal of S.
Let M_i be an ordered (p, q)-lateral ideal of S such that∩i∈IMi≠∅. Then∩i∈IMiis an ordered (p, q)-lateral ideal of S.
Let L_i be an ordered n-left ideal of S such that∩i∈ILi≠∅. Then∩i∈ILiis an ordered n-left ideal of S.

Proof

Analogous to the proof of the Theorem 3.6.

Theorem 3.10

Let R be an ordered m-right ideal, M be an ordered (p, q)-lateral ideal and L be an ordered n-left ideal of S. Then R ∩ M ∩ L is an ordered (m, (p, q), n)-quasi-ideal of S.

Proof

Suppose that Q = R ∩ M ∩ L. By the Theorem 3.8, every ordered m-right, ordered (p, q)-lateral and ordered n-left ideal of S are ordered (m, (p, q), n)-quasiideals of S. Therefore R, M and L are ordered (m, (p, q), n)-quasi-ideals of S. If R ∩ M ∩ L is non-empty. Then by the Theorem 3.6, we have Q = R ∩ M ∩ L is an ordered (m, (p, q), n)-quasi-ideal of S.

Theorem 3.11

Let A be any non-empty subset of S. Then

(A(SS)^m] is an ordered m-right ideal of S,
(S^pAS^q ∪ S^pSASS^q] is an ordered (p, q)-lateral ideal of S,
((SS)ⁿA] is an ordered n-left ideal of S,
(A(SS)^m] ∩ (S^pAS^q∪S^pSASS^q] ∩ ((SS)ⁿA] is an ordered (m, (p, q), n)-quasi ideal of S.

Proof

(1) It is easy to show that (A(SS)^m] is a ternary subsemigroup and ((A(SS)^m]] = (A(SS)^m]. Now

(A(SS)m](SS)m⊆(A(SS)m]((SS)m]⊆(A(SS)m(SS)m]=(A(SSSS)m]⊆(A(SS)m].

Therefore (A(SS)^m] is an ordered m-right ideal of S.

(2), (3) and (4) can be proved analogously to (1).

Theorem 3.12

Let A be an ordered ternary subsemigroup of S. Then

(A ∪ A(SS)^m] is an ordered m-right ideal of S containing A,
(A ∪ S^pAS^q ∪ S^pSASS^q] is an ordered (p, q)-lateral ideal of S containing A,
(A ∪ (SS)ⁿA] is an ordered n-left ideal of S containing A,
((A(SS)^m] ∩ (S^pAS^q∪S^pSASS^q] ∩ ((SS)ⁿA]) ∪ (A] is an ordered (m, (p, q), n)- quasi ideal of S containing A.

Proof

Proof is analogous to the Theorem 3.11.

Theorem 3.13

Let Q be an ordered (m, (p, q), n)-quasi ideal of S. Then

R= (Q ∪ Q(SS)^m] is an ordered m-right ideal of S,
M= (Q ∪ S^pQS^q ∪ S^pSQSS^q] is an ordered (p, q)-lateral ideal of S,
L= (Q ∪ (SS)ⁿQ] is an ordered n-left ideal of S.

Proof

Proof is analogous to the Theorem 3.11.

An ordered (m, (p, q), n)-quasi ideal Q has the (m, (p, q), n) intersection property if Q is the intersection of an ordered m-right ideal, an ordered (p, q)-lateral and an ordered n-left ideal of S.

Remark 3.14

Every ordered m-right ideal, ordered (p, q)-lateral ideal and ordered n-left ideal have the intersection property.

Theorem 3.15

Let S be an ordered ternary semigroup and Q be an ordered (m, (p, q), n)-quasi ideal of S. Then the following statements are equivalent:

Q has the (m, (p, q), n) intersection property;
(Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ] = Q;
(Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q;
(Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ] ⊆ Q;
(Q ∪ Q(SS)^m] ∩ (S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ] ⊆ Q.

Proof

(1) ⇒ (2) : Let Q has the (m, (p, q), n) intersection property. It is obvious that Q ⊆ (Q∪Q(SS)^m]∩(Q∪S^pQS^q∪S^pSQSS^q]∩(Q∪(SS)ⁿQ]....(i). Now to prove (2) we will show that (Q∪Q(SS)^m]∩(Q∪S^pQS^q ∪S^pSQSS^q]∩(Q∪(SS)ⁿQ] ⊆ Q. As it is known that Q has (m, (p, q), n) intersection property, it implies there exist an ordered m-right ideal R, an ordered (p, q)-lateral ideal M and an ordered n-left ideal L of S s.t. R ∩ M ∩ L = Q. Then Q ⊆ R, Q ⊆ M and Q ⊆ L. Also we have that ((SS)ⁿQ] ⊆ ((SS)ⁿL] ⊆ L and in the similar way (S^pQS^q ∪ S^pSQSS^q] ⊆ M and (Q(SS)^m] ⊆ R which implies Q ∪((SS)ⁿQ] = (Q∪(SS)ⁿQ] ⊆ L, Q ∪(S^pQS^q ∪ S^pSQSS^q] = (Q∪ S^pQS^q ∪ S^pSQSS^q] ⊆ M and Q ∪(Q(SS)^m] = (Q∪Q(SS)^m] ⊆ R. Hence we have (Q∪Q(SS)^m]∩(Q∪S^pQS^q∪S^pSQSS^q]∩(Q∪(SS)ⁿQ] ⊆ L∩M∩R = Q....(ii). From (i) and (ii), we have (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ] = Q.

(2) ⇒ (1) : Consider (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ] = Q. By the Theorem 3.13, (Q ∪ Q(SS)^m] is an ordered m-right ideal of S, (Q ∪ S^pQS^q ∪ S^pSQSS^q] is an ordered (p, q)-lateral ideal of S and (Q ∪ (SS)ⁿQ] is an ordered n-left ideal of S. Let R= (Q∪Q(SS)^m], M=(Q∪ S^pQS^q ∪ S^pSQSS^q] and L=(Q ∪ (SS)ⁿQ]. Now (Q(SS)^m] ∩ (S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q, as Q is an ordered (m, (p, q), n)-quasi ideal of S. We have

L∩M∩R=(Q∪Q(SS)m]∩(Q∪SpQSq∪SpSQSSq]∩(Q∪(SS)nQ]=Q∪(Q(SS)m]∩(SpQSq∪SpSQSSq]∩((SS)nQ]⊆Q∪Q=Q.

(2) ⇒ (3) : Consider Q = (Q∪Q(SS)^m] ∩ (Q∪S^pQS^q∪S^pSQSS^q] ∩ (Q∪(SS)ⁿQ]. As we know ((SS)ⁿQ] ⊆ (Q ∪ (SS)ⁿQ], we have (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ]. Hence (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q.

(3) ⇒ (2): Let (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q. Then Q ⊆ (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ]. Now we have to show that (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q. For this suppose that x ∈ (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ]. Then we have to show that x ∈ Q. Now(Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q. We have x ∈ Q. Therefore (Q ∪ Q(SS)^m] ∩ (Q ∪ S^pQS^q ∪ S^pSQSS^q] ∩ (Q ∪ (SS)ⁿQ]=Q.

The proofs for (2) ⇒ (4), (2)⇒ (5) and (4) ⇒(2), (5) ⇒ (2) are analogous to the proofs of (2)⇒(3) and (3) ⇒ (2), respectively.

Theorem 3.16

Every regular ordered ternary semigroup S has the intersection property of ordered (m, (p, q), n)-quasi-ideals for any positive integer m, p, q, n and p + q is even.

Proof

Let S be a regular ordered ternary semigroup and Q be an ordered (m, (p, q), n)-quasi-ideal of S. Then by the Theorem 3.13, R = (Q ∪ Q(SS)^m], M=(Q ∪ S^pQS^q ∪ S^pSQSS^q] and L=(Q ∪ (SS)ⁿQ] are an ordered m-right, an ordered (p, q)-lateral and an ordered n-left ideal of S respectively. Clearly Q ⊆ R,Q ⊆ M and Q ⊆ L implies Q ⊆ R ∩ M ∩ L. As S is regular, we have Q ⊆ (Q(SS)^m], Q ⊆ (S^pQS^p ∪ S^pSQSS^q] and Q ⊆ ((SS)ⁿQ]. Therefore R = (Q(SS)^m], M = (S^pQS^q ∪ S^pSQSS^q] and L = ((SS)ⁿQ]. Hence we have R ∩ M ∩ L = (Q(SS)^m] ∩ ((S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q. It implies Q = R ∩ M ∩ L. Therefore Q has the (m, (p, q), n) intersection property.

4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups

Go to

Abstract
1. Introduction
2. Preliminaries
3. Generalised Quasi-ideals
4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups
Acknowledgements
References

In this section, we introduce the concept of a minimal ordered (m, (p, q), n)- quasi-ideal, a minimal ordered m-right ideal, a minimal ordered (p, q)-lateral ideal and a minimal ordered n-left ideal in ordered ternary semigroups and study the relationship between them. Also m-right simple, (p, q)-lateral simple, n-left simple and (m, (p, q), n)-quasi-simple ordered ternary semigroups are defined and some properties of them are investigated.

Definition 4.1

An ordered m-right ideal R of S is called minimal ordered m-right ideal if it does not properly contain any ordered m-right ideal of S.

Definition 4.2

An ordered (p, q)-lateral ideal M of S is called minimal ordered (p, q)-lateral ideal if it does not properly contain any ordered (p, q)-lateral ideal of S.

Definition 4.3

An ordered n-left ideal L of S is called minimal ordered n-left ideal if it does not properly contain any ordered n-left ideal of S.

Definition 4.4

An ordered (m, (p, q), n)-quasi ideal Q of S is called minimal ordered (m, (p, q), n)-quasi ideal if it does not properly contain any ordered (m, (p, q), n)-quasi ideal of S.

Theorem 4.5

Let S be an ordered ternary semigroup and Q be an ordered (m, (p, q), n)-quasi-ideal of S. Then Q is minimal if and only if Q is the intersection of some minimal ordered m-right ideal R, minimal ordered (p, q)-lateral ideal M and minimal ordered n-left ideal L of S.

Proof

Assume that Q is minimal ordered (m, (p, q), n)-quasi ideal of S. Then

(Q(SS)m]∩(SpQSq∪SpSQSSq]∩((SS)nQ]⊆Q.

By the Theorem 3.11, (Q(SS)^m],(S^pQS^q ∪ S^pSQSS^q], ((SS)ⁿQ] are an ordered m-right, an ordered (p, q)-lateral and an ordered n-left ideal of S and by Theorem 3.10, intersection of an ordered m-right, an ordered (p, q)-lateral and an ordered n-left ideal is an ordered (m, (p, q), n)-quasi ideal of S. As Q is minimal, we have

(Q(SS)m]∩(SpQSq∪SpSQSSq]∩((SS)nQ]=Q.

To show that ((SS)ⁿQ] is an minimal ordered n-left ideal of S. Let L be an ordered n-left ideal of S contained in ((SS)ⁿQ]. Then ((SS)ⁿL] ⊆ (L] = L ⊆ ((SS)ⁿQ]. Thus, (Q(SS)^m] ∩ (S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿL] ⊆ (Q(SS)^m] ∩ (S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿQ] ⊆ Q

Now (Q(SS)^m] ∩ (S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿL] is an ordered (m, (p, q), n)-quasi ideal of S and Q is a minimal ordered (m, (p, q), n)-quasi ideal of S. We have (Q(SS)^m] ∩ (S^pQS^q ∪ S^pSQSS^q] ∩ ((SS)ⁿL] = Q. Then Q ⊆ ((SS)ⁿL] and we have ((SS)ⁿQ] ⊆ ((SS)ⁿ((SS)ⁿL]] ⊆ ((SS)ⁿ(SS)ⁿL] ⊆ ((SS)ⁿL] ⊆ L. It implies L = ((SS)ⁿQ]. Therefore ((SS)ⁿQ] is a minimal ordered n-left ideal of S. Similarly other cases can be proved.

Conversely, suppose Q= L ∩ M ∩ R, where L,M and R are minimal ordered n-left, minimal ordered (p, q)-lateral and minimal ordered m-right ideals of S, respectively. Then Q ⊆ L,Q ⊆ M and Q ⊆ R. By the Theorem 3.10, Q will be an ordered (m, (p, q), n)-quasi ideal of S. Now we have to show that Q is minimal. For this let Q′ be an ordered (m, (p, q), n)-quasi ideal of S contained in Q. By the Theorem 3.11, (Q′ (SS)^m], (S^pQ′S^q ∪ S^pSQ′SS^q], ((SS)ⁿQ′] are an ordered m-right, an ordered (p, q)-lateral and an ordered n-left ideal of S, respectively. Now,

((SS)nQ′]⊆((SS)nQ]⊆((SS)nL]⊆L.

But L is minimal, it implies ((SS)ⁿQ′] = L. Similarly (Q′ (SS)^m] = R and (S^pQ′S^q ∪ S^pSQ′SS^q] = M. As Q′ is an ordered (m, (p, q), n)-quasi ideal of S. We have

Q=L∩M∩R=((SS)nQ′]∩(SpQ′Sq∪SpSQ′SSq]∩(Q′(SS)m]⊆Q′.

It implies Q = Q′.

Therefore Q is a minimal ordered (m, (p, q), n)-quasi ideal of S.

Theorem 4.6

Let S be an ordered ternary semigroup. Then the following holds:

An ordered m-right ideal R is minimal if and only if (a(SS)^m] = R for all a ∈ R;
An ordered (p, q)-lateral ideal M is minimal if and only if (S^paS^q ∪ S^pSaSS^q] = M for all a ∈ M;
An ordered n-left ideal L is minimal if and only if ((SS)ⁿa] = L for all a ∈ L;
An ordered (m, (p, q), n)-quasi-ideal Q is minimal if and only if (a(SS)^m] ∩ (S^paS^q ∪ S^pSaSS^q] ∩ ((SS)ⁿa] = Q for all a ∈ Q.

Proof

(2) Suppose that an ordered (p, q)-lateral ideal M is minimal. Let a ∈ M. Then (S^pSaSS^q ∪ S^paS^q] ⊆ (S^pSMSS^q ∪ S^pMS^q] ⊆ M. By the Theorem 3.11(2), we have (S^pSaSS^q ∪S^paS^q] is an ordered (p, q)-lateral ideal of S. As M is minimal ordered (p, q)-lateral ideal of S. We have (S^pSaSS^q ∪ S^paS^q] = M.

Conversely, suppose that (S^pSaSS^q ∪ S^paS^q] = M for all a ∈ M. Let M′ be any ordered (p, q)-lateral ideal of S contained in M. Let m ∈ M′. Then m ∈ M. By assumption, we have (S^pSmSS^q ∪ S^pmS^q] = M for all m ∈ M. M = (S^pSmSS^q ∪ S^pmS^q] ⊆ (S^pSM′SS^q ∪ S^pM′S^q] ⊆ M′. It implies M ⊆ M′. Thus, M= M′. Hence, M is minimal ordered (p, q)-lateral ideal of S.

Analogously we can prove (1), (3) and (4).

Definition 4.7

Let S be an ordered ternary semigroup. Then S is called an m-right simple if S is a unique ordered m-right ideal of S.

Definition 4.8

Let S be an ordered ternary semigroup. Then S is called an (p, q)-lateral simple if S is a unique ordered (p, q)-lateral ideal of S.

Definition 4.9

Let S be an ordered ternary semigroup. Then S is called an n-left simple if S is a unique ordered n-left ideal of S.

Definition 4.10

Let S be an ordered ternary semigroup. Then S is called an (m, (p, q), n)-quasi simple if S is a unique ordered (m, (p, q), n)-quasi ideal of S.

Theorem 4.11

Let S be an ordered ternary semigroup. The following statements hold true:

S is an m-right simple if and only if (a(SS)^m] = S for all a ∈ S;
S is an (p, q)-lateral simple if and only if (S^paS^q ∪ S^pSaSS^q] = S for all a ∈ S;
S is an n-left simple if and only if ((SS)ⁿa] = S for all a ∈ S;
S is an (m, (p, q), n)-quasi simple if and only if (a(SS)^m] ∩ (S^paS^q ∪ S^pSaSS^q] ∩ ((SS)ⁿa] = S for all a ∈ S.

Proof

(1) Assume that S is an m-right simple, we have that S is a minimal ordered m-right ideal of S. By the Theorem 4.6(1), (a(SS)^m] = S for all a ∈ S.

Conversely, suppose that (a(SS)^m] = S for all a ∈ S. By the Theorem 4.6(1), S is a minimal ordered m-right ideal of S, and therefore S is an m-right simple.

(2), (3) and (4) can be proved analogously to (1).

Theorem 4.12

Let S be an ordered ternary semigroup. The following statements hold true:

If an ordered m-right ideal R of S is an m-right simple, then R is a minimal ordered m-right ideal of S;
If an ordered (p, q)-lateral ideal M of S is an (p, q)-lateral simple, then M is a minimal ordered (p, q)-lateral ideal of S;
If an ordered n-left ideal L of S is an n-left simple, then L is a minimal ordered n-left ideal of S;
If an ordered (m, (p, q), n)-quasi ideal Q of S is an (m, (p, q), n)-quasi simple, then Q is a minimal ordered (m, (p, q), n)-quasi ideal of S.

Proof

(1) Let R be an m-right simple. By the Theorem 4.11(1), we have (a(RR)^m] = R for all a ∈ R. For every a ∈ R, we have R = (a(RR)^m] ⊆ (a(SS)^m] ⊆ (R(SS)^m] ⊆ R. Then (a(SS)^m]= R for all a ∈ R. By the Theorem 4.6(1), we have R is minimal.

(2), (3) and (4) can be proved analogously to (1).

Acknowledgements

Go to

Abstract
1. Introduction
2. Preliminaries
3. Generalised Quasi-ideals
4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups
Acknowledgements
References

The third author is thankful to the National Board of Higher Mathematics, Department of Atomic Energy, Government of India for the financial assistance provided through Post-Doctoral Fellowship under Grant No: 2/40(30)/2015/R&D-II/9473.

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Generalised Quasi-ideals
4. Generalised Minimal Quasi-ideals and (m, (p, q), n)-Quasi Simple Ordered Ternary Semigroups
Acknowledgements
References

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