# Onα-Multiplier on Almost Distributive Lattices

Authors: Ying Wang; Abid Mahboob; Abdul Rauf Khan; Zafar Ullah; Zahid Karim; Mamoona Karim
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we initiate the concept ofα-multiplier on almost distributive lattices. We prove some useful results by using the notion of α-multiplier and generalize the idea of multiplier on almost distributive lattices.


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## Body

## 1. Introduction

A lattice is an advanced abstract structure that has been studied in abstract algebra during last few decades. Birkhoff introduced the concept of lattice theory in 1940 [1]. Lattice is generalization of Boolean and fuzzy algebras. Latter on Gratzer and Schmidt worked together and showed their interest in the development of lattice theory [2]. In 1955, Helgason introduced the concept of multiplier in Banach Algebra [3]. The idea of multiplier in lattice was given by Larsen [4] in 1971, and Cornish extended this concept of multiplier in distributive lattice [5].In 1981, the idea of ADLs was initiated by Swamy and Rao [6]. An almost distributive lattice satisfies all the properties of distributive lattice except commutativity of ∧ and ∨ and right distributivity of ∨ over ∧. Recently, Kim has introduced the idea of a multiplier in ADLs [7] and discussed some fundamental properties of this notion. For detailed study of the subject, we refer to readers [8–10].Now, we have generalized certain properties ofα-multiplier. The notion of α-multiplier for an almost distributive lattice is introduced, and some related properties are investigated. Moreover, we introduced principle α-multiplier and isotone α-multiplier on almost distributive lattices.


## 2. Preliminaries

Definition 1.
(see [6]). An algebra G,∧,∨,0 is said to be an almost distributive lattice if it satisfies the following:(i)
e1∨0=e1(ii)
0∧e1=0(iii)
e1∨h1∧ω1=e1∧ω1∨h1∧ω1(iv)
e1∧h1∨ω1=e1∧h1∨e1∧ω1(v)
e1∨h1∧ω1=e1∨h1∧e1∨ω1(vi)
e1∨h1∧h1=h1(vii)
e1∨h1∧e1=e1(viii)
e1∨h1∧e1=e1, ∀e1, h1, ω1∈GLemma 1.
(see [6]). Let G be an almost distributive lattice. For any e1, h1∈G, we have(i)
e1∨e1=e1(ii)
e1∨e1=e1(iii)
e1∧h1∨h1=h1(iv)
e1∧e1∨h1=e1(v)
e1∨h1∧e1=e1(vi)
e1∨h1=e1⇔e1∧h1=h1(vii)
e1∨h1=h1⇔e1∧h1=e1Definition 2.
(see [6]). For any e1, h1∈G, we say that e1≤h1 if e1∧h1=e1 or equivalently, e1∨h1=h1.Lemma 2.
(see [6]). Let G be an almost distributive lattice. For any e1, h1, ω1, h3∈G, then the following identities hold:(i)
e1∧h1≤h1 and e1≤e1∨h1(ii)
e1∧h1=h1∧e1 whenever e1≤h1(iii)
e1∨h1∨ω1∧h3=e1∨h1∨ω1∧h3(iv)
e1≤h1⇒e1∧ω1≤h1∧ω1,ω1∧e1≤ω1∧h1 and ω1∨e1≤ω1∨h1Definition 3.
(see [6]). Let G be a lattice, and 0 is known as a zero element of a lattice G if 0∧e1=0,∀e1∈G.Lemma 3.
(see [6]). Let G be an almost distributive lattice. If G has 0, then for any e1, h1∈G, the following identities hold:(i)
e1∨0=e1 and 0∨e1=e1(ii)
e1∧0=0(iii)
e1∧h1=0 if and only if h1∧e1=0Definition 4.
(see [6]). Let I be a nonempty subset of G which is called an ideal of G if h3∨ω3∈I and h3∧e3∈I whenever h3, ω3∈I and e3∈G.
IfI is an ideal of G and e3, h3∈G, then e3∧h3∈I if and only if h3∧e3∈I.Lemma 4.
(see [6]). For any e1, h1∈G, we have(i)
e1∧h1∨h1=h1(ii)
e1∨e1∧h1=e1=e1∧e1∨h1(iii)
e1∨h1∧e1=e1=e1∨h1∧e1Definition 5.
(see [7]). Let G be an almost distributive lattice and ζ1andζ2 be two self maps. We define ζ1∨ζ2:G⟶G by ζ1∨ζ2e1=ζ1e1∨ζ2e1,∀e1∈G.Definition 6.
(see [7]). Let G1 and G2 be two almost distributive lattices. Then, G1×G2 is also an ADL with respect to the pointwise operation given by e1,h1∧ω1,x=e1∧ω1,h1∧xande1,h1∨ω1,x=e1∨ω1,h1∨x,∀e1,h1∈G1, ω1,x∈G2.Definition 7.
(see [7]). Let G be an almost distributive lattice and ζ be a multiplier of G. Define a set fixζG by fixζG=e1∈G:ζe1=e1.Definition 8.
(see [7]). Let G,∧,∨,0 be an almost distributive lattice. For any u∈G, define Γu=e1,h1∈G×G|fue1=fuh1, where fu is a principle multiplier induced by u∈G.


## 3.α-Multiplier on Almost Distributive Lattices

Definition 9.
LetG be an almost distributive lattice. A function ζ:G⟶G is called α-multiplier if ζe1∧h1=ζe1∧αh1∀e1, h1∈G, where α is a mapping on G.Example 1.
LetG be an almost distributive lattice with 0∈G. A function ζ defined by ζe1=0∀e1∈G is called zero α-multiplier.Proof.
LetG be an almost distributive lattice with 0∈G; then, we have to prove that ζ is a zero α-multiplier.
Lete1,h1∈G, ζe1∧h1=ζe1∧αh1=0∧αh1=0. Hence, ζ is a zero α-multiplier.Lemma 5.
Letζ be α-multiplier of G. If α:G⟶G is homomorphism, then following conditions hold:(i)
ζe1≤αe1(ii)
ζe1∧ζh1≤ζe1∧h1, ∀e1,h1∈GProof.
(i)
Sinceζe1=ζe1∧e1=ζe1∧αe1, it implies that ζe1≤αe1.(ii)
Lete1, h1∈G. We have to show that ζe1∧ζh1≤ζe1∧h1.(1)ζe1≤αe1,ζh1≤αh1,ζe1∧ζh1≤ζe1∧αh1=ζe1∧h1. It implies ζe1∧ζh1≤ζe1∧h1.Definition 10.
LetG be an almost distributive lattice. A function Ke is defined by Keg=e∧ag,∀g∈G, where α:G⟶G is a homomorphism, then Ke is α−multiplier of G, and such α−multiplier of G is called a principle α−multiplier of G.Definition 11.
LetG,∧,∨,0 be an almost distributive lattice and K be α-multiplier on G. For any u∈G, define Γu=e1,h1∈G×G|kue1=kuh1, where Ku is a principle α-multiplier induced by u∈G.Lemma 6.
LetG be an almost distributive lattice. A function Ke3 is defined by Ke3a=e3∧αa, where α:G⟶G is a homomorphism, then Ke3 is α-multiplier of G, and such α-multiplier of G is called a principle α-multiplier of G.Proof.
Leth3, ω3, e3∈G, and K be an α-multiplier; then, we have to prove that Ke3 is an α-multiplier.(2)Keh3=e∧αh3,∀h3∈G,Keh3∧ω3=e3∧αh3∧ω3=e3∧αh3∧αω3=e3∧αh3∧αω3=Ke3h3∧αω3. This implies that Ke3 is an α-multiplier.Definition 12.
LetG be an almost distributive lattice and ζ be α−multiplier on G, where α is a mapping on G. If for e1≤h1 implies ζe1≤ζh1, then ζ is an isotone α−multiplier.Proposition 1.
LetG be an almost distributive lattice. If α:G⟶G is an increasing homomorphism, then ζah2=a∧αh2, and ∀h2∈G is an isotone α-multiplier of G.Proof.
LetG be an ADL and h2,ω2∈G, with h2≤ω2 such that αh2≤αω2, then we have ζah2∧ω2=a∧αh2∧αω2=a∧αh2∧a∧αω2=ζah2∧ζaω2. It implies ζah2≤ζaω2. Hence, ζa is an isotone α-multiplier.Lemma 7.
LetG be an almost distributive lattice and ζ be an α-multiplier of G and α be an increasing homomorphism on G. If h3≤ω3 and ζω3=αω3, then ζh3=αh3.Proof.
Leth3, ω3∈G for h3≤ω3. Since α is an increasing homomorphism, so(3)αh3≤αω3,ζω3=αω3.
By using equation (3), we have ζh3=ζω3∧h3=ζω3∧αh3=αω3∧αh3 since α is an increasing homomorphism. It implies that ζh3=αh3.Theorem 1.
LetG be an almost distributive lattice and ζ be an α-multiplier of G and α be an increasing homomorphism on G. Then, ζ is an isotone α-multiplier.Proof.
Supposeh3, ω3∈G. By using Lemma 5 (i), we have(4)ζω3≤αω3,(5)forh3≤ω3,αh3≤αω3.
Sinceh3≤ω3, therefore, we have ζh3=ζω3∧h3=ζω3∧αh3. By equations (4) and (5), we have ζh3≤ζω3∧αω3=ζω3. It implies ζh3≤ζω3. Hence, ζ is an isotoneα-multiplier.Proposition 2.
LetG be an almost distributive lattice, ζ be an α-multiplier of G, and α be homomorphism on G. Then, ζω1∨e1=ζω1∨ζe1, ∀ω1, e1∈G.Proof.
Letω1, e1∈G and ζ be an α-multiplier of G; then, we have to show that ζω1∨e1=ζω1∨ζe1. By Definition 1, we have ζω1∨ζe1=ζω1∨e1∧ω1∨ζω1∨e1∧e1=ζω1∨e1∧αω1∨ζω1∨e1∧αe1. By Definition 1, we have ζω1∨ζe1=ζω1∨e1∧αω1∨αe1=ζω1∨e1∧αω1∨e1=ζω1∨e1. It implies that ζω1∨ζe1=ζω1∨e1.Proposition 3.
LetG be an almost distributive lattice and ζ1 and ζ2 be two α-multipliers of G. Then, ζ1∨ζ2 is also an α-multiplier of G.Proof.
LetG be an ADL and ζ1 and ζ2 be α-multiplier such that(6)ζ1h3∧ω3=ζ1h3∧αω3,ζ2h3∧ω3=ζ2h3∧αω3.
Leth3, ω3∈G, ζ1, and ζ2 be α-multipliers of G. Now, by Definition 5, we have ζ1∨ζ2h3∧ω3=ζ1h3∧ω3∨ζ2h3∧ω3=ζ1h3∧αω3∨ζ2h3∧αω3. By Definitions 1 and 5, we have ζ1∨ζ2h3∧ω3=ζ1h3∨ζ2h3∧αω3=ζ1∨ζ2h3∧αω3 which along with equation (6) implies that (ζ1∨ζ2h3∧ω3=ζ1∨ζ2h3∧αω3. Hence, ζ1∨ζ2 is called α-multiplier of G.Proposition 4.
LetG1 and G2 be two almost distributive lattices with 0. A function ζ:G1×G2⟶G1×G2 defined by ζs,ω3=0,αω3∀s,ω3∈G1×G2 and α is a homomorphism. Then, ζ is α-multiplier of G1×G2 with pointwise operation.Proof.
LetG1 and G2 be two ADLs with 0. We define a mapping ζ:G1×G2⟶G1×G2 by(7)ζs1,t1=0,αt1.
Then, we have to show thatζ is an α-multiplier with pointwise operation such that(8)ζs1,t1∧s2,t2=ζs1,t1∧αs2,αt2.
Lets1,t1,s2,t2∈G1×G2. By Definition 6, ζs1,t1∧s2,t2=ζs1∧s2,t1∧t2. By using equation (7) and Definition 1, we have ζs1,t1∧s2,t2=0,αt1∧αt1=0∧αs2,αt1∧αt2. By Definition 6 and equation (7), we have ζs1,t1∧s2,t2=0,αt1∧αs2,αt2=ζs1,t1∧αs2,αt2. This implies that ζ is an α-multiplier with pointwise operation on G1×G2.Theorem 2.
LetG be an almost distributive lattice and BG be the set of all α-multipliers of G. Then, BG under binary operations ∨ and ∧ is an almost distributive lattice, where for any ζ1, ζ2∈BG, h3∈G.(9)ζ1∧ζ2h3=ζ1h3∧ζ2h3,ζ1∨ζ2h3=ζ1h3∨ζ2h3.Proof.
Letζ1, ζ2∈BG. Then, by equation (9), we have (ζ1∧ζ2h3∧ω3=ζ1h3∧ω3∧ζ2h3∧ω3=ζ1h3∧αω3∧ζ2h3∧αω3=ζ1h3∧ζ2h3∧αω3=ζ1∧ζ2h3∧αω3. This implies that ζ1∧ζ2 is an α-multiplier. Let ζ1, ζ2∈BG, and along with equation (9), we have ζ1∨ζ2h3∧ω3=ζ1h3∧ω3∨ζ2h3∧ω3=ζ1h3∧αω3∨ζ2h3∧αω3=ζ1h3∨ζ2h3∧αω3=ζ1∨ζ2h3∧αω3. This implies that (ζ1∨ζ2) is an α-multiplier. Hence, (BG, ∨, ∧) is closed under ∨, ∧. Hence, (BG, ∨, ∧) is an ADL.Theorem 3.
LetG be an almost distributive lattice and BG be the set of all α-multiplier on G. Then, set of all principal α-multiplier PGζx|x∈G is distributive lattice with the following operation ζh2∨ζω3=ζh2∨ω3 and ζh2∧ζω3=ζh2∧ω3  for all h2, ω3∈G.Proof.
Leth2, ω3∈G. Then, ζh2∨ζω3e1=ζh2e1∨ζω3e1=h2∧αe1∨ω3∧αe1=h2∨ω3∧αe1=ζh2∨ω3e1. For some e1∈G, it implies that ζh2∨ζω3=ζh2∨ω3∈PG. Also ζh2∧ζω3e1=ζh2e1∧ζω3e1=h2∧αe1∧ω3∧αe1=h2∧ω3∧αe1=ζh2∧ω3e1. For any e1∈G, it implies ζh2∧ζω3=ζh2∧ω3∈PG. Hence, (PG, ∨, ∧) is closed, and so PG is sub almost distributive lattice. Moreover, for any e1∈G, ζh2∧ω3e1=h2∧ω3∧αe1=ω3∧h2∧αe1=ζω3∧ζh2. Thus, ζh2∧ζω3=ζω3∧ζh2. Hence, PG is a distributive lattice.


## 4. Conclusion

In this paper, we have generalized the idea of multiplier toα-multiplier in almost distributive lattices and investigated some properties of ADLs. We have also explored some results by using the notion of principal α-multiplier and isotone α-multiplier. This generalized concept played a vital role in exploring different properties of almost distributive lattices.



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