# On State Ideals and State Relative Annihilators in De Morgan State Residuated Lattices

Authors: Francis Woumfo; Blaise Bleriot Koguep Njionou; Etienne Romuald Temgoua Alomo; Celestin Lele
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

The concept of state has been considered in commutative and noncommutative logical systems, and their properties are at the center for the development of an algebraic investigation of probabilistic models for those algebras. This article mainly focuses on the study of the lattice of state ideals in De Morgan state residuated lattices (DMSRLs). First, we prove that the lattice of all state idealsSℐL of a DMSRL L,τ is a coherent frame. Then, we characterize the DMSRL for which the lattice SℐL is a Boolean algebra. In addition, we bring in the concept of state relative annihilator of a given nonempty subset with respect to a state ideal in DMSRL and investigate various properties. We prove that state relative annihilators are a particular kind of state ideals. Finally, we investigate the notion of prime state ideal in DMSRL and establish the prime state ideal theorem.


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## Body

## 1. Introduction

It is known to all that the algebraic research on logical systems has considerable applications. In particular, it plays a meaningful role in artificial intelligence, which make computer simulate human being in dealing with fuzzy and uncertain information. In a large number of multivalued logic and fuzzy logic of algebraic systems, the residuated lattice is an important class, which was brought in by Ward and Dilworth in Ref. [1] as a generalization of ideal lattices of rings. Then, in 2018, Liviu-Constantin Holdon introduced an important variety of this structure called De Morgan residuated lattice, which comprises salient subclasses of residuated lattices such as Boolean algebras, BL-algebras, MTL-algebras, Stonean residuated lattices, and regular residuated lattices (MV-algebras, IMTL-algebras) (see Ref. [2]). Furthermore, he considered ideals and annihilators in this new structure.The concept of internal state also called state operator was introduced by Flaminio and Montagna ([3, 4]) by adding a unary operation τ, which preserves the usual properties of states to the language of MV-algebras. Since then, several authors deeply investigated this topic in other algebraic structures (see [5–8]). State residuated lattices were initiated by He et al. [9]. They introduced the concept of state operators on residuated lattices and investigated some related properties. Moreover, they inserted the notion of state filter in state residuated lattices. Following this study, Kondo and Kawaguchi recently studied generalized state operators on state residuated lattices (see [10, 11]). In 2017, Dehghani and Forouzesh [12] made a deep investigation on state filters in state residuated lattices and inducted the concept of prime state filters in state residuated lattices and proved the prime state filter theorem. Woumfo et al. [13] introduced the notion of state ideal in state residuated lattices and established that the lattice of state ideals is a complete lattice. In this article, stimulated by the previous research on the structure of De Morgan residuated lattices and by the importance of the theory of ideals and annihilators in MV-algebras, BL-algebra, and Stonean residuated lattices (see [14–17]), we analyze the notions of state ideal and state relative annihilator in a new class of state residuated lattices called De Morgan state residuated lattices.This article is divided into four sections: in the first one, we describe some preliminaries comprising the basic definitions, some rules of calculus and theorems that are needed in the sequel. Section2 is the part in which we consider the algebraic structure of the set SℐL of all state ideals in a DMSRL L,τ. It is shown that SℐL,⊆ is a coherent frame. Also, we characterize the DMSRL for which the lattice SℐL is a Boolean algebra. In Section 3, we bring in the notion of state annihilator of a nonempty set X with respect to a state ideal I in a DMSRL and analyze some of its properties. We demonstrate that state relative annihilators are a particular kind of state ideals. In the last section, we put emphasis on the notion of prime state ideals in a DMSRL. We prove the prime state ideal theorem and investigate some allied properties.


## 2. Preliminaries

We summarize here some fundamental definitions and results about residuated lattices. Readers can obtain more details in Refs. [1, 18–21].A nonempty setL with four binary operations ∧,∨,⊙,⟶ and two constants 0,1 is called a bounded integral commutative residuated lattice or shortly residuated lattice if the following axioms are verified:(C1)L,∧,∨,0,1 is a bounded lattice;(C2)L,⊙,1 is a commutative monoid (with the unit element 1);(C3) For allx,y∈L, x⊙y≤z⇔x≤y⟶z.The following notations of residuated lattices will be used:L will stand for L,∧,∨,⊙,⟶,0,1, a residuated lattice and for any x∈L and n∈ℕ∗, x′:=x⟶0, x″:=x′′, x0:=1 and xn:=xn−1⊙x.In Refs. [2,9,14,21], we have the following definitions and examples:(1)
A residuated lattice satisfying the divisibility condition (div):x∧y=x⊙x⟶y is called a R ℓ-monoid.(2)
A residuated lattice satisfying the prelinearity condition (pre):x⟶y∨y⟶x=1 is called a MTL-algebra.(3)
A residuated lattice satisfying prelinearity and divisibility conditions is called a BL-algebra.(4)
A residuated lattice satisfying the double negation condition (dn):x″=x is called a regular residuated lattice.(5)
A BL-algebra satisfying the double negation condition is called an MV-algebra.(6)
A MTL-algebra satisfying the double negation condition is called an IMTL-algebra.(7)
A residuated lattice satisfying the Boolean propertiesx∨x′=1 and x∧x′=0 is called a Boolean algebra;(8)
A residuated lattice satisfying the Stone propertyx′∨x″=1 is called a Stonean residuated lattice;(9)
A residuated lattice satisfying the De Morgan propertyx∧y′=x′∨y′ is called a De Morgan residuated lattice.We shall notice from (see [2]) that Boolean algebras, BL-algebras, MTL-algebras, Stonean residuated lattices, and regular residuated lattices (MV-algebras, IMTL-algebras) are particular important subclasses of De Morgan residuated lattices.Example 1.
LetL=0,a,b,c,d,e,f,g,1 endowed with the Hasse diagram and Caley tables (Figure 1):
Then,L=L,∧,∨,⊙,⟶,0,1 is a nonregular residuated lattice. One can easily check that L is a De Morgan residuated lattice, which is Stonean.Figure 1
A De Morgan residuated lattice which is Stonean.Example 2.
LetL=0,p,a,b,c,d,e,f,q,1 with his Hasse diagram and Cayley tables of ⊙ and ⟶ be the following (Figure 2):
Then,L=L,∧,∨,⊙,⟶,0,1 is a residuated lattice.
One can readily verify thatL is a De Morgan residuated, which is regular. But L is not a BL-algebra, L is not a MV-algebra, L is not a Boolean algebra, L is not a Stonean residuated lattice, L is not a MTL-algebra, and L is not a IMTL-algebra.
The following basic arithmetic of residuated lattices will be used for anyx,y,z∈L (see [19,22]):(RL1):1⟶x=x, x⟶x=1, x⟶1=1, 0⟶x=1;(RL2):x≤y⇔x⟶y=1;(RL3):x⟶y=y⟶x=1⇔x=y;(RL4): ifx≤y , then y⟶z≤x⟶z , z⟶x≤z⟶y , x⊙z≤y⊙z and y′≤x′;(RL5):x⊙x⟶y≤y; x⊙x⟶y≤x∧y;(RL6):x⊙y≤x∧y≤x,y≤x∨y; x≤y⟶x ; x⊙y≤x⟶y,y⟶x;(RL7):x⊙y″=x″⊙y″,x∨y′=x′∧y′ and x∧y′≥x′∨y′;(RL8):0′=1, 1′=0;(RL9):x≤x″≤x′⟶x;(RL10):x⟶y≤y′⟶x′;(RL11):x′′′=x′, x⊙y′=x⟶y′=y⟶x′=x″⟶y′;(RL12):x⊙x′=0 , x⊙y=0⇔x≤y′ ; x⊙0=0;(RL13):x′⟶y≤x′⊙y′′;(RL14):x⟶x∧y=x⟶y;(RL15):x⊙y=x⊙x⟶x⊙y.
LetL be a residuated lattice, and we set x⊘y=x′⟶y, for every x,y∈L.Figure 2
A De Morgan residuated lattice which is not Stonean.Definition 1 (see [19]).
An ideal ofL is a nonempty subset I of L, which satisfies the following conditions for every x,y∈L:(I1) Ify∈I and x≤y, then x∈I;(I2) Ifx,y∈I, then x⊘y∈I.
By the fact that⊘ is neither commutative nor associative in residuated lattices and in order to get an operation with properties closed to addition properties, Buşneag et al. defined a commutative and associative operation ⊕ in residuated lattices as follows: x⊕y=x′⊙y′′, for every x,y∈L.
Note that in Ref. [2], it was shown that a nonempty subset I of a residuated lattice L is an ideal of L if and only if the following conditions hold for any x,y∈L:(I3) Ify∈I and x≤y, then x∈I;(I4) Ifx,y∈I, then x⊕y∈I.
The set of all ideals of a residuated latticeL will be denoted by ℐL.
For a nonempty subsetX of a residuated lattice L, the ideal generated by X is X:=a∈L:a≤x1⊕x2⊕,….,⊕xn,for somen∈ℕ∗,xi∈X,for1≤i≤n [14].
Here are some properties of the operation⊕ (see [13,14]). Let L be a residuated lattice. For any x,y,z,t∈L, we have(P1):x⊕y=x′⟶y″=y′⟶x″;(P2):x⊕x′=1, x⊕0=x″, x⊕1=1;(P3):x⊕y=y⊕x, x,y≤x⊕y;(P4):x⊕y⊕z=x⊕y⊕z;(P5): Ifx≤y, then x⊕z≤y⊕z;(P6): Ifx≤y and z≤t, then x⊕z≤y⊕t.
LetL be a residuated lattice. For any x∈L and n∈ℕ, we define 0x=0, 1x=x, and nx=n−1x⊕x, for n≥2.
Then, the following relations hold for anyx,y∈L and m,n≥2:(P7):m≤n⇒mx≤nx. In particular, x≤nx;(P8):x≤y⇒mx≤my;(P9):nx⊕y=nx⊕ny;(P10):x⊕ny≤nx⊕y;(P11):x⊘y≤x⊕y.Lemma 1.
In any De Morgan residuated latticeL, the following relations hold for any x,y,z∈L AND m,n≥2[2]:(P12):x∧y″=x″∧y″;(P13):x⊕y∧z=x⊕y∧x⊕z;(P14):x∧ny≤nx∧y;(P15):mx∧ny≤mnx⊙y≤mnx∧y;(P16):x∧y⊕z≤x∧y⊕x∧z.
Now, we give some necessary results for the sequel about lattices and frames. It is worth noting that the main references for frames theory are the following books [23, 24].
A latticeL,∧,∨ is called Brouwerian if it satisfies the equality x∧∨k∈Kyk=∨k∈Kx∧yk (whenever the arbitrary joins exist), for any x,yk∈L, k∈K.Definition 2.
We call frame a complete latticeL that satisfies the infinite distributive law x∧∨A=∨x∧a:a∈A, for all x∈L and A⊆L[25].Remark 1.
(1)
Every Brouwerian latticeL,∧,∨ is distributive;(2)
A frame is a complete Brouwerian lattice.
An elementx of a complete lattice L is called compact if for all A⊆L, a≤∨A implies that a≤∨H for some finite H⊆A (see [9,26]). We will denote by CL the set of all compact elements of a complete lattice L.Proposition 1.
LetL be a frame and x∈L [27]. Then, x∈CL if for all A⊆L, x=∨A implies that x=∨H for some finite H⊆A.Definition 3.
A frameL is called coherent if the following conditions hold [27]:(i)
CL is a sublattice of L; that is, for all x,y∈L, if x,y∈CL, then x∧y,x∨y∈CL;(ii)
For allx∈L, x=∨k∈Kxk,withxk∈CL.
Recall that ifL is a lattice with 0 its bottom element, and x∈L, then y∈L is said to be a pseudocomplement of x if x∧y=0 and for every z∈L, x∧z=0 implies z≤y. L is called pseudocomplemented if every element has a pseudocomplement. Every frame is pseudocomplemented (see [28]).
For everyx,y∈L, we call a relative pseudocomplement of x with respect to y, the greatest element (if it exists) z∈L such that x∧z≤y.
In what follows, we recall some results about state residuated lattices, which will be used in the sequel.Definition 4.
(see [9]). A map τ:L⟶L is said to be a state operator on L if the following conditions hold for any x,y∈L [25]:(SO1):τ0=0;(SO2):x⟶y=1 implies τx⟶τy=1;(SO3):τx⟶y=τx⟶τx∧y;(SO4):τx⊙y=τx⊙τx⟶x⊙y;(SO5):ττx⊙τy=τx⊙τy;(SO6):ττx⟶τy=τx⟶τy;(SO7):ττx∨τy=τx∨τy;(SO8):ττx∧τy=τx∧τy.
The pairL,τ is called a state residuated lattice.
We shall notice that for any residuated latticeL, the identity map idL is a state operator on L, which is an endomorphism, but in general, a state operator τ is not an endomorphism.Definition 5.
LetL,τ be a state residuated lattice [9, 13]. An ideal I of L is said to be a state ideal of L,τ if τI⊆I (i.e., for all x∈L, x∈I⇒τx∈I). Similarly, a filter F of L is called a state filter if τF⊆F.
SℐL will stand for the set of all state ideals of L,τ. It is obvious that 0,L∈SℐL⊆ℐL.
For computational issues, we will use the following properties (see [9, 13]).
LetL,τ be a state residuated lattice. Then, for any x,y∈L, for all n≥1, we have(SO9):τ1=1;(SO10):x≤y implies τx≤τy;(SO11):τx′=τx′;(SO12):τx⊙y≥τx⊙τy and if x⊙y=0, then τx⊙y=τx⊙τy;(SO13):τx⊙y′≥τx⊙τy′ and if x≤y, then τx⊙y′=τx⊙τy′;(SO14):τx⟶y≤τx⟶τy. Particularly, if x,y are comparable,  then τx⟶y=τx⟶τy;(SO15): Ifτ is faithful, then x<y implies τx<τy;(SO16):τ2x=ττx=τx;(SO17):τL=Fixτ, where Fixτ=x∈L:τx=x;(SO18):τL is a subalgebra of L;(SO19):kerτ=x∈L:τx=1 is a state filter of L,τ;(SO20):cokerτ=x∈L:τx=0 is a state ideal of L,τ;(SO21):τx″=τx″;(SO22):τx⊕y≤τx⊕τy;(SO23): Ifx,y∈τL, then x⊕y∈τL.(SO24):τnx≤nτx.
For any nonempty subsetX of L, we denote by Xτ the state ideal of L,τ generated by X,=; that is, Xτ is the smallest state ideal of L,τ containing X, and for an element a∈L, aτ:=aτ is called the principal state ideal of L,τ. If I∈SℐL and a∉I, we denote by I,aτ:=I∪aτ.
The next theorem gives the concrete description of the state ideal generated by a nonempty subset of a state residuated lattice.Theorem 1 (see [13]).
LetX be a nonempty subset of L, I,I1,I2∈SℐL and a∈L\I. Then,(1)
Xτ=x∈L:x≤n1x1⊕τx1⊕…⊕nkxk⊕τxk,for somek∈ℕ∗,xi∈X,ni∈ℕ∗,for1≤i≤k;(2)
aτ=x∈L:x≤na⊕τa,for somen∈ℕ∗;(3)
I,aτ=x∈L:x≤i⊕na⊕τa,for somei∈Iand∈ℕ∗;(4)
I1∨I2:=I1∪I2τ=x∈L:x≤i1⊕i2,withi1∈I1andi2∈I2.Lemma 2 (see [13]).
LetL,τ be a state residuated lattice. For all a,b∈L, we have(5)
a≤b⇒aτ⊆bτ;(6)
τaτ⊆aτ;(7)
a⊕τaτ=aτ;(8)
a⊕τa∧b⊕τbτ⊆aτ∩bτ;(9)
aτ∨bτ=a∨bτ=a⊕bτ.Proposition 2.
LetL,τ be a state residuated lattice [13]. Then,
SℐL,⊆ is a bounded complete lattice with the bottom element 0 and the top element L.
Now, we introduce the concept of De Morgan state residuated lattice.Definition 6.
A state residuated latticeL,τ is called De Morgan if L is a De Morgan residuated lattice. More precisely, a De Morgan state residuated lattice is a De Morgan residuated lattice endowed with a state operator.
The following remarks give the relationship between the above definition and the notion of state-morphism operator studied on universal algebras in Ref. [29], also with the notion of generalized state in Ref. [18].Remark 2.
LetL be a De Morgan residuated lattice.(1)
If a mapτ:L⟶L is an idempotent De Morgan-endomorphism (i.e., τ is an endomorphism of L such that τ2=τ), then τ is a state operator on L and it is said to be a state-morphism operator. Therefore, the couple L,τ is called a De Morgan state-morphism residuated lattice. So, by taking a state-morphism operator τ (which is a particular type of state operator), our theorems can be extended to the general setting of universal algebras as in Ref. [29].(2)
Letτ be a state operator on L. From Definition 1.9, we have(SO1)τ0=0;(SO3)τx⟶y=τx⟶τx∧y, for any x,y∈L.
Thus, from Ref. [18], Proposition 8.1 (iii) and Definition 8.1 (1), τ is a generalized Bosbach state of type 1. Moreover, from (SO10), we have x≤y implies τx≤τy, for any x,y∈L. Therefore, any state operator τ on a De Morgan residuated lattice L can be seen as an ordering-preserving generalized Bosbach state of type 1.
A state operatorτ is called cofaithful if cokerτ=0 and uncofaithful otherwise.Example 3.
SetL=0;a;b;1 with 0<a<b<1. Then, L is De Morgan residuated lattice that is a BL-algebra but not an MV-algebra with the operations (Figure 3):
Let define the unary operatorτ on L by(1)τx=0,if x=0,a,if x=a;1,if x∈b,1.
One can easily check thatτ is a state operator on L. Hence, the couple L,τ is a DMSRL. In addition, cokerτ=0 and τ verified the following properties: τx⟶y=τx⟶τy and τx⊙y=τx⊙τy, for any x,y∈L. Therefore, τ is a cofaithful De Morgan state-morphism operator. Furthermore, the state ideals of L,τ are 0 and L.Figure 3
Operations of the De Morgan residuated lattice of exemple3.Example 4.
LetL1 and L2 be two nontrivial De Morgan residuated lattices and h:L1⟶L2 a homomorphism. We define the map τh:L1×L2L1×L2x,y↦τhx,y=x,hx. Then, one can check that τh is an idempotent endomorphism on L1×L2. So, τh is a state-morphism operator and the couple L1×L2,τh is a De Morgan state-morphism residuated lattice. In addition, cokerτh=0×L2. Hence, τh is a uncofaithful state operator.Remark 3.
(1)
For every De Morgan residuated latticeL, L,idL is a De Morgan state residuated lattice. That is, a De Morgan residuated lattice L can be seen as a De Morgan state residuated lattice. One can see that all ideals of L is a state ideal of L,idL.(2)
LetL be a De Morgan residuated lattice, S a sub-De Morgan residuated lattice of L and τ:L⟶L be a state operator on L such that τS⊆S. Then, the restriction of τ on S is a state operator on S. We can see that cokerτ/S=S∩cokerτ is a state ideal of S.This means that if a state operator preserves a substructure, then its restriction to this substructure is a state operator.(3)
LetLi:i∈I be a nonempty family of De Morgan residuated lattices and τi:i∈I a family of state operators such that τi is a state operator on Li for each i∈I. We define the map τ:∏i∈ILi⟶∏i∈ILixii∈I↦τixii∈I..
Then,τ is a state operator on ∏i∈ILi. We can check that cokerτ=∏i∈Icokerτi.
So, if we have state operators on structures, then we can define a state operator on the product and compute its cokernel easily.Lemma 3.
IfL,τ is a De Morgan state residuated lattice, then for all a,b∈L, we have a⊕τa∧b⊕τbτ=aτ∩bτ.Proof.
Leta,b∈L. Then, from Lemma 2 (8), we have a⊕τa∧b⊕τbτ⊆aτ∩bτ. Now, let x∈aτ∩bτ. Then, x∈aτ and x∈bτ. Hence, from Theorem 1 (2), there exists m,n≥1 such that, x≤ma⊕τa and x≤nb⊕τb. Therefore,(2)x≤ma⊕τa∧nb⊕τb≤P15mna⊕τa∧b⊕τb≤P3mna⊕τa∧b⊕τb⊕τa⊕τa∧b⊕τb
That is,x∈a⊕τa∧b⊕τbτ. Hence, aτ∩bτ⊆a⊕τa∧b⊕τbτ. Thus, a⊕τa∧b⊕τbτ=aτ∩bτ.


## 3. The Lattice of All State Ideals of a DMSRL

In this section, we focus on the study of the algebraic structure of the setSℐL of all state ideals of a De Morgan state residuated lattice L,τ.From now on, unless otherwise specified,L,τ will always denote a De Morgan state residuated lattice L,∨,∧,⊙,⟶,0,1; that is, L is a De Morgan residuated lattice and τ is a state operator on L.Proposition 3.
SℐL,⊆ is a Brouwerian lattice.Proof.
LetK be an index set, I∈SℐL, and Ikk∈K be a family of state ideals of L,τ. We will show that I∧∨k∈KIk=∨k∈KI∧Ik. That is,
I∩∪k∈KIkτ=∪k∈KI∩Ikτ. Clearly, ∪k∈KI∩Ikτ⊆I∩∪k∈KIkτ.
Letx∈I∩∪k∈KIkτ. Then, x∈I and x∈∪k∈KIkτ. It follows that there exist k1,k2,…,km∈K, xkj∈Ikj, 1≤j≤, such that x≤xk1⊕xk2⊕….⊕xkm. Then, x=x∧xk1⊕xk2⊕….⊕xkm≤P16x∧xk1⊕x∧xk2⊕….⊕x∧xkm. Since I,Ikj∈SℐL, we have x∧xkj∈I∩Ikj, for every 1≤j≤m. We deduce that x∈∪k∈KI∩Ikτ. Hence, I∩∪k∈KIkτ⊆∪k∈KI∩Ikτ, that is, I∧∨k∈KIk=∨k∈KI∧Ik. Therefore, SℐL,⊆ is a Brouwerian lattice.Theorem 2.
The latticeSℐL,⊆ is a frame.Proof.
From Proposition2, SℐL,⊆ is a complete lattice. From Proposition 3, SℐL,⊆ is a Brouwerian lattice. Combining them, we have by Remark 1 (2) that SℐL,⊆ is a frame.
In the following result, we establish a concrete description of the right adjoint of the map:(3)I1∩−:SℐL⟶SℐLI↦I1∩−I=I1∩I
Now, for anyI1,I2∈SℐL, we put I1⟶I2=x∈L:I1∩xτ⊆I2.Proposition 4.
In the frameSℐL,⊆, for any I1,I2,I∈SℐL, we have(1)
I1⟶I2∈SℐL(2)
I1∩I⊆I2⇔I⊆I1⟶I2, that is, I1⟶I2=supI∈SℐL:I1∩I⊆I2 and I1⟶−:SℐL⟶SℐLI↦I1⟶−I=I1⟶I is the right adjoint of I1∩−:SℐL⟶SℐL;(3)
I1⟶I2=x∈L:i∧nx⊕τx∈I2,for alli∈I1andn∈ℕ∗.Proof.
(1)
We will show thatI1⟶I2 is a state ideal of L,τ. We have I1⟶I2≠∅. Indeed, 0τ=0 and I1∩0τ=0⊆I2. Hence, 0∈I1⟶I2.Now, letx,y∈I1⟶I2, then I1∩xτ⊆I2 and I1∩yτ⊆I2. We obtain I1∩xτ∨I1∩yτ⊆I2. From Theorem 2, it follows that I1∩xτ∨yτ⊆I2, which implies (by Lemma 3 (9)) that I1∩x⊕yτ⊆I2. Thus, x⊕y∈I1⟶I2.Assumex∈I1⟶I2 and y≤x. Then, I1∩xτ⊆I2 and yτ⊆Lemma1.129xτ. It follows that I1∩yτ⊆I1∩xτ⊆I2. Hence, y∈I1⟶I2. Finally, let x∈I1⟶I2; then, I1∩xτ⊆I2. Since I1∩τxτ⊆I1∩xτ⊆I2, we obtain that τx∈I1⟶I2. Therefore, I1⟶I2 is a state ideal of L,τ. That is, I1⟶I2∈SℐL.(2)
Now, we show thatI1∩I⊆I2⇔I⊆I1⟶I2, for any I,I1,I2∈SℐL. Assume I1∩I⊆I2 and x∈I, we obtain I1∩xτ⊆I1∩I⊆I2 (since τx∈I)). It follows that x∈I1⟶I2. That is, I⊆I1⟶I2. Conversely, let I⊆I1⟶I2 and x∈I1∩I. Then, we have x∈I⊆I1⟶I2. That is, I1∩xτ⊆I2. Since x∈I1∩xτ⊆I2, we deduce that x∈I2, which implies I1∩I⊆I2. Therefore, I1⟶I2=supI∈SℐL:I1∩I⊆I2 and I1⟶−:SℐL⟶SℐLI↦I1⟶−I=I1⟶I is the right adjoint of I1∩−:SℐL⟶SℐL.(3)
SetA=x∈L:i∧nx⊕τx∈I2,for alli∈I1andn∈ℕ∗. First, let x∈I1⟶I2. Then, I1∩xτ⊆I2. For n≥1, and i∈I1, we have nx⊕τx∈xτ (since x,τx∈xτ)). It follows that i∧nx⊕τx∈I1∩xτ, which implies that i∧nx⊕τx∈I2. Hence, x∈A.
Conversely, letx∈A and t∈I1∩xτ. Then, there exists n∈ℕ∗ such that t≤nx⊕τx. Hence, t=t∧nx⊕τx∈I2. That is, I1∩xτ⊆I2. Hence, x∈I1⟶I2. Therefore, I1⟶I2=x∈L:i∧nx⊕τx∈I2,for alli∈I1andn∈ℕ∗.
For everyI∈SℐL, we put I′=I⟶0=x∈L:xτ∩I=0. Then, from Proposition 1 (3), we have the following corollary.Corollary 1.
For allI∈SℐL, we have I′=x∈L:i∧nx⊕τx=0,for alli∈Iandn∈ℕ∗.Theorem 3.
LetI∈SℐL. Then, CSℐL=xτ:x∈L.Proof.
⇒). Assume I∈CSℐL. Set H=xτ:x∈L. Since I=∨x∈Ixτ, there are xi1≤i≤n such that I=x1τ∨x2τ∨…∨xnτ (Proposition 1). By Lemma 2 (9), we have I=x1⊕x2⊕…⊕xnτ. Thus, I∈H, that is, CSℐL⊆H.
⇐). Let I∈H. Then, there exists x∈L such that I=xτ. Assume Ikk∈K⊆SℐL and I=xτ⊆∨k∈KIk. Then, x∈∨k∈KIk=∪k∈KIkτ. It follows that there exist kj∈K, xkj∈Ikj, for all 1≤k≤m such that x≤xk1⊕xk2⊕…⊕xkm. That is, x∈Ik1∪Ik2∪…∪Ikmτ=Ik1∨Ik2∨…∨Ikm. Hence, I=xτ⊆Ik1∨Ik2∨…∨Ikm. Hence, I∈CSℐL, that is, H⊆CSℐL. Therefore, CSℐL=xτ:x∈L.Remark 4.
Theorem3 means that a state ideal I is a compact element of the frame SℐL if and only if it is principal.Theorem 4.
The latticeSℐL,⊆ is a coherent frame.Proof.
(i)
From Theorem2, we have that SℐL,⊆ is a frame.(ii)
From Theorem3, we obtain that CSℐL=xτ:x∈L. By Lemma 3, we have xτ∧yτ=x⊕τx∧y⊕τyτ, and by Lemma 2 (9), we have xτ∨yτ=x⊕yτ for all x,y∈L. Therefore, CSℐL,⊆ is a sublattice of SℐL,⊆.(iii)
For anyI∈SℐL, we have I=∨x∈Ixτ.
(i), (ii), and (iii) combined with Definition3 imply SℐL,⊆ is a coherent frame.
We have immediately the following results.Corollary 2.
The latticeSℐL,⊆ is pseudocomplemented. Clearly, for all I∈SℐL, we have that I′=I⟶0=x∈L:I∩xτ=0 is the pseudocomplement of I.
According to Definition3 and Corollary 2, we have the following result in a De Morgan residuated lattice L.Corollary 3.
(1)
ℐL,⊆ is a coherent frame;(2)
ℐL,⊆ is a pseudocomplemented lattice.
We recall that a Heyting algebra ([30]) is a lattice L,∨,∧ with 0 such that for every x,y∈L, there exists an element x⟶y∈L (call the pseudocomplement of x with respect to y) such that for every z∈L,x∧z≤y⇔z≤x⟶y (i.e., x⟶y=supz∈L:x∧z≤y).Remark 5.
From Proposition4, SℐL,∨,∧,′,0 is a Heyting algebra, where for I∈SℐL, I′=I⟶0=x∈L:xτ∩I=0.Proposition 5.
For anya,b∈L, we have(1)
aτ′=x∈L:a⊕τa∧x⊕τx=0;(2)
aτ′∩bτ′=a⊕bτ′.Proof.
(1)
We haveaτ′=aτ⟶0=x∈L:aτ∩xτ=0=Lemma1.19x∈L:a⊕τa∧x⊕τxτ=0=x∈L:a⊕τa∧x⊕τx=0.(2)
Letx∈aτ′∩bτ′. Then, by (1), we have a⊕τa∧x⊕τx=0 and b⊕τb∧x⊕τx=0. In addition, by Lemma 1 (P16), we have a⊕τa⊕b⊕τb∧x⊕τx≤a⊕τa∧x⊕τx⊕b⊕τb∧x⊕τx=0; hence, a⊕τa⊕b⊕τb∧x⊕τx=0. Then, by Lemma 1 (P16), 2a⊕τa⊕b⊕τb∧x⊕τx≤2a⊕τa⊕b⊕τb∧x⊕τx=0. Moreover, by (RL6) and (SO22), (2a⊕τa⊕b⊕τb=a⊕τa⊕b⊕τb⊕a⊕τa⊕b⊕τb≥a⊕b⊕τa⊕τb≥a⊕b⊕τa⊕b.
Hence,a⊕b⊕τa⊕b∧x⊕τx=0. Then, by (1), we have x∈a⊕bτ′. Therefore, aτ′∩bτ′⊆a⊕bτ′.
Conversely, letx∈a⊕bτ′. Then, by (1), a⊕b⊕τa⊕b∧x⊕τx=0, we have a≤a⊕b, so τa≤τa⊕b. It follows that a⊕τa≤a⊕b⊕τa⊕b. We obtain a⊕τa∧x⊕τx≤a⊕b⊕τa⊕b∧x⊕τx=0 and thus, a⊕τa∧x⊕τx=0. Analogously, b⊕τb∧x⊕τx=0. Therefore, x∈aτ′∩bτ′.Lemma 4.
LetL,τ be a state residuated lattice, x∈L and n∈ℕ,n≥2. Then, the following hold:(P17)nx″=nx;(P18)x″⊕τx″=x⊕τx.Proof.
(P17): By induction, we have2x=x⊕x=x′⊙x′′. It follows that 2x″=x′⊙x′″′=x′⊙x′′=2x. Let k∈ℕ such that k≥2. Assume that kx″=kx. Then, we have k+1x=kx⊕x=hypotesiskx″⊕x=kx″′⊙x′′=kx′⊙x′′. It follows that k+1x″=kx′⊙x′″′=kx′⊙x′′=kx⊕x=k+1x. Therefore, nx″=nx for all n∈ℕ,n≥2.
(P18): We havex″⊕τx″=SO21x″⊕τx″=x″′⊙τx″′′=RL11x′⊙τx′′=x⊕τx.Theorem 5.
LetL,τ be a De Morgan state residuated lattice. The following are equivalent:(i)
SℐL,∨,∧,′,0,L is a Boolean algebra;(ii)
Every state ideal ofL,τ is principal, and for every x∈L, there is n∈ℕ∗, such that x⊕τx∧nx⊕τx′⊕τnx⊕τx′=0.Proof.
i⇒ii. Assume that SℐL,∨,∧,′,0,L is a Boolean algebra. Then, for every I∈SℐL, I∨I′=L; thus, 1∈I∨I′. But according to Theorem 1 (4), I∨I′:=I∪I′τ=x∈L:x≤y⊕z,withy∈Iandz∈I′. Hence, there are y∈I and z∈I′ such that y⊕z=1. We will prove that I=yτ. Since y∈I, it follows that yτ⊆I. According to Corollary 1, I′=a∈L:x∧nx⊕τx=0,for allx∈Iandn∈ℕ∗. Thus, x∧nz⊕τz=0, for all x∈I and n∈ℕ∗. Then x⊙z≤x∧z=0 for every x∈I, that is, x⊙z=0 for every x∈I. Hence, x″⟶z′=RL11x⊙z′=1, that is, x″≤z′. Since y⊕z=1, we obtain that ny⊕τy⊕z=1 for every n∈ℕ∗ (because y⊕z≤ny⊕τy⊕z). Hence, by (P1), ny⊕τy′⟶z″=1, that is, ny⊕τy′≤z″. It follows that z′=RL11z″′≤RL4ny⊕τy″=P17ny⊕τy. Hence, z′≤ny⊕τy. Thus, we obtain that x≤RL9x″≤z′≤ny⊕τy, that is, x≤ny⊕τy, for every x∈I, that is, I⊆yτ. Therefore, I=yτ.
Now, letx∈L, since SℐL,∨,∧,′,0,L is a Boolean algebra, we have L=xτ∨xτ′=t∈L:t≤y⊕nx⊕τx,for somen∈ℕ∗,y∈xτ′. Hence, there exists y∈xτ′ and n∈ℕ∗, such that y⊕nx⊕τx=1. Since y∈xτ′, then by Proposition 5 (1), x⊕τx∧y⊕τy=0. From y⊕nx⊕τx=1, we deduce that y′⟶nx⊕τx″=1, which implies y′≤nx⊕τx″=P17nx⊕τx. Thus, by (RL4), nx⊕τx′≤y″, which implies τnx⊕τx′≤SO10τy″. Hence, nx⊕τx′⊕τnx⊕τx′≤P6y″⊕τy″=P18y⊕τy. It follows that x⊕τx∧nx⊕τx′⊕τnx⊕τx′≤x⊕τx∧y⊕τy=0. Therefore, x⊕τx∧nx⊕τx′⊕τnx⊕τx′=0.
ii⇒i. By Remark 5, SℐL,∨,∧,′,0 is a Heyting algebra. In order to prove that SℐL,∨,∧,′,0,L is a Boolean algebra, it is enough to prove that for every I∈SℐL, we have I′=0⇔I=L (according to ([30])). Let I∈SℐL with I′=0. By the hypothesis, every state ideal is principal. Hence, there is x∈L such that I=xτ. Thus, xτ′=0. Moreover, there is n∈ℕ∗ such that x⊕τx∧nx⊕τx′⊕τnx⊕τx′=0. By Proposition 5 (1), it follows that nx⊕τx′∈xτ′=0. Thus, nx⊕τx′=0, that is, nx⊕τx=P17nx⊕τx″=1. Since nx⊕τx∈xτ, we deduce that 1∈xτ=I. Therefore, I=xτ=L.


## 4. State Relative Annihilators in DMSRL

Consider a state filterF of a state residuated lattice L,τ, Pengfei et al. defined the notion of co-annihilator of a nonempty subset X of L with respect to F by XI⊥τ=a∈L:τa∨x∈F,for allx∈X, which is a state filter. Inspired by this work, in this section, we introduce the notion of state relative annihilator of a nonempty set X with respect to a state ideal I in a De Morgan state residuated lattice L,τ and study some properties.Definition 7.
LetL,τ be a De Morgan state residuated lattice, I a state ideal of L,τ, and X a nonempty subset of L.
Theτ-relative annihilator of X with respect to I is defined by XI⊥τ=a∈L:τa∧x∈I,for allx∈X.
We have0∈XI⊥τ. Hence, XI⊥τ≠∅. Particularly, the following facts hold:(1)
Ifτ=idL, I=0, then X⊥:=X0⊥idL=a∈L:a∧x=0,for allx∈X, and it is called the annihilator of X in L (see [2]);(2)
Ifτ=idL, then XI⊥:=XI⊥idL=a∈L:a∧x∈I,for allx∈X, and it is called the relative annihilator of X with respect to I (see [2]);(3)
IfX=x, then xI⊥τ:=xI⊥τ=a∈L:τa∧x∈I;(4)
IfI=0, then X⊥τ:=X0⊥τ=a∈L:τa∧x=0,for allx∈X.Example 5.
Consider the De Morgan residuated lattice of Example2. Assume that τ is a state operator on L. Then, τ0=0. Also, we have b⊙b=0, which implies by (SO12) that 0=τ0=τb⊙b=τb⊙τb. Hence by the table of ⊙, τb∈0,p,b. Suppose that τb=0. Since p≤b, by (SO10) τp≤τb=0, that is, τp=0. Moreover, τf=τc⟶d=SO3τc⟶τc∧d=τc⟶τb=τc⟶0=τc′=SO11τc′=τd. Thus, τf=τd. In addition, τe=τf⟶e=τf⟶τf∧e=τf⟶τd=τf⟶τf=RL11, that is, τe=1. Since e≤q≤1, by (SO10) 1=τe≤τq≤τ1, that is, τq=τ1=1. Also on the one hand, we have 1=τe=τe⟶c=SO14τe⟶τc=1⟶τc=RL1τc. Hence, τc=1. On the other hand, we have 1=τe=τc⟶a=SO14τc⟶τa=1⟶τa=RL1τa. Thus, τa=1. We have d⊙a=0, which implies that 0=τ0=τd⊙a=SO12τd⊙τa=τd⊙1=τd. It follows that τf=τd=0. Therefore, we obtain that the map τ defined on L by(4)τx=0,if x∈0,p,b,d,f;1,if x∈a,c,e,q,1.
Is a state operator onL. Thus, L,τ is a De Morgan state residuated lattice. One can easily check that 0, 0,p, 0,p,a, 0,p,c, 0,p,b,d,f, L are ideals of L and 0, I=0,p, J=0,p,b,d,f and L are state ideals of L,τ. In addition, cokerτ=0,p,b,d,f≠0. Thus, τ is a uncofaithful state operator.
We have0I⊥τ=pI⊥τ=L and aI⊥τ=cI⊥τ=dI⊥τ=eI⊥τ=fI⊥τ=qI⊥τ=1I⊥τ=0,p,b,d,f=J.
LetX=0,P,b, we have XI⊥τ=0,p,b,d,f=J.Example 6.
LetL=0,n,a,b,c,d,1 with his Hasse diagram and Cayley tables of ⊙ and ⟶ be the following (Figure 4):
Then ,L=L,∧,∨,⊙,⟶,0,1 is a nonregular De Morgan residuated lattice, which is Stonean. Now suppose that τ is a state operator on L[2]. Then, τ0=0 and by (SO3), τb=τc⟶b=τc⟶c∧b=τc⟶τa. If τc=1, we have τb=1⟶τa=τa, that is, τb=τa. Since c≤d≤1, by (SO10), we have 1=τc≤τd≤τ1, that is, τd=τ1=1. Also, 1=τd=τa⟶n=SO14τa⟶τn, that is, τa⟶τn=1. Hence, τa≤RL2τn. Since, n≤a, we have τn≤τa. Thus, τn=τa=τb=λ, for some λ∈L. By (SO7) and (SO8), we have, for ∗∈∨,∧, ττa∗τb=τa∗τb⇔τλ=τλ∗λ=λ∗λ=λ. That is, τλ=λ. Thus, λ∈0,n,a,b,1. In addition, if λ=a, we have ττa⊙τb=τa⊙a=τn=a≠n=a⊙a=τa⊙τb. Hence, by (SO5), λ≠a. By some calculations, we obtain that for λ=1, the map τ defined on L by(5)τx=0,if x∈0;1,if x∈n,a,b,c,d,1,is a state operator on L. Thus, L,τ is a De Morgan state residuated lattice. In addition, cokerτ=0. Hence, τ is cofaithful.
We have0⊥τ=L and n⊥τ=a⊥τ=b⊥τ=c⊥τ=d⊥τ=1⊥τ=0.
For anyX⊆L, X≠0, we have X⊥τ=0.Figure 4
A non regular De Morgan residuated lattice.Example 7.
Consider the De Morgan residuated lattice of Example1. Then, from Remark 3, L,idL is a De Morgan state residuated lattice. If I=0 and X=fidL=0,c,f, it is easy to check that X⊥:=X0⊥idL=0,a,b=bidL=b.Theorem 5.
LetI be a state ideal of L,τ. Given a nonempty subset X of L, XI⊥τ is a state ideal of L,τ.Proof.
Leta,b∈XI⊥τ. Then, τa∧x∈I and τb∧x∈I, for all x∈X. Hence, τa∧x⊕τb∧x∈I because I is an ideal. Since τa⊕b∧x≤SO22τa⊕τb∧x≤P16τa∧x⊕τb∧x∈I, we have τa⊕b∧x∈I. That is, a⊕b∈XI⊥τ.
Leta∈XI⊥τ and b∈L such that b≤a. Then, τb≤τa, from (SO10). It follows that τb∧x≤τa∧x, for all x∈X. Since τa∧x∈I and I is and ideal, τb∧x∈I, for all x∈X. Therefore, b∈XI⊥τ.
Leta∈XI⊥τ. Since ττa=τ2a=SO16τa, we have ττa∧x=τa∧x∈I, for all x∈X. Hence, τa∈XI⊥τ. Therefore, XI⊥τ is a state ideal of L,τ.Corollary 4.
LetI be a state ideal of L,τ. Given a nonempty subset X of L, we have(1)
X⊥τ is a state ideal of L,τ;(2)
XI⊥ and X⊥ are state ideals of L;(3)
For allx∈L, xI⊥τ is a state ideal of L,τ and I⊆xI⊥τ.Proposition 6.
LetI,J be state ideals of L,τ. Given nonempty subsets X,Y of L, we have(1)
I⊆J⇒XI⊥τ⊆XJ⊥τ;(2)
X⊆Y⇒YI⊥τ⊆XI⊥τ;(3)
∪k∈KXkI⊥τ=∩k∈KXkI⊥τ;(4)
X∩k∈KIk⊥τ=∩k∈KXIk⊥τ;(5)
XI⊥τ=XI⊥τ. Particularly, ∅I⊥τ=0I⊥τ=L;(6)
cokerτ⊆X⊥τ,L⊥τ=cokerτ;(7)
XI⊥τ=∩x∈XxI⊥τ;(8)
X⊥τ=∩x∈Xx⊥τ;(9)
XI⊥τ=L if X⊆I. Specifically, 0I⊥τ=II⊥τ=L;(10)
∩k∈KXkI⊥τ⊆∩k∈KXkI⊥τ;(11)
I⊆XI⊥τ;(12)
X⊥τ⊆XI⊥τ.Proof.
(1)
LetI⊆J and a∈XI⊥τ. Then, τa∧x∈I,for allx∈X. Thus, τa∧x∈J,for allx∈X, that is, a∈XJ⊥τ. Therefore, XI⊥τ⊆XJ⊥τ.(2)
LetX⊆Y and a∈YI⊥τ. Then, τa∧x∈I,for allx∈Y. Hence, τa∧x∈I for all x∈X, which implies a∈XI⊥τ. Therefore, YI⊥τ⊆XI⊥τ.(3)
SinceXk⊆∪k∈KXk, it follows from (2) that ∪k∈KXkI⊥τ⊆XkI⊥τ for all k∈K. We deduce that ∪k∈KXkI⊥τ⊆∩k∈KXkI⊥τ.Conversely, leta∈∩k∈KXkI⊥τ; we have a∈XkI⊥τ for all k∈K. Hence, τa∧xk∈I for all xk∈Xk and k∈K, which implies a∈∪k∈KXkI⊥τ. That is, ∩k∈KXkI⊥τ⊆∪k∈KXkI⊥τ. Therefore, ∪k∈KXkI⊥τ=∩k∈KXkI⊥τ.(4)
We havea∈X∩k∈KIk⊥τ⇔τa∧x∈∩k∈KIk,for allx∈X⇔τa∧x∈Ik,for allx∈Xandk∈K⇔a∈XIk⊥τ,for allk∈K⇔a∈∩k∈KXIk⊥τ. Thus, X∩k∈KIk⊥τ=∩k∈KXIk⊥τ.(5)
SinceX⊆X, by (2), we obtain XI⊥τ⊆XI⊥τ. Conversely, let a∈XI⊥τ and z∈X. Then, τa∧x∈I, for all x∈X. Since z∈X, there are n∈ℕ∗ and x1,x2,….,xn∈X such that z≤x1⊕x2⊕,….,⊕xn. It follows that τa∧z≤τa∧x1⊕x2⊕,….,⊕xn≤τa∧x1⊕τa∧x2⊕,….,⊕τa∧xn. But τa∧xi∈I, for all 1≤i≤n. Thus, τa∧z∈I, which implies that a∈XI⊥τ. Therefore, XI⊥τ=XI⊥τ.(6)
Leta∈cokerτ. Then, τa=0. Since τa∧x≤τa=0, for all x∈X, we have τa∧x=0, for all x∈X. That is, a∈X⊥τ. Thus, cokerτ⊆X⊥τ. Taking X=L, we have cokerτ⊆L⊥τ. Now, let a∈L⊥τ, and then, τa∧x=0, for all x∈L. In particular, taking x=τa, we have τa=0, which implies that a∈cokerτ, that is, L⊥τ⊆cokerτ. Therefore, cokerτ⊆X⊥τ and L⊥τ=cokerτ.(7)
According to (3), we have XI⊥τ=∪x∈XxI⊥τ=∩x∈XxI⊥τ.(8)
TakingI=0 in (1), we obtain X⊥τ=∩x∈Xx⊥τ.(9)
AssumeXI⊥τ=L, and let x∈X. Then, x=τ1∧x∈I, that is, x∈I.Conversely, supposeX⊆I, and a∈L. Then, for any x∈X, we have x∈I. Since τa∧x≤x∈I, we obtain τa∧x∈I. Hence, a∈XI⊥τ, and therefore, XI⊥τ=L.(10)
Since∩k∈KXk⊆Xk for each k∈K, it follows from (2) that XkI⊥τ⊆∩k∈KXkI⊥τ for all k∈K. Thus, ∩k∈KXkI⊥τ⊆∩k∈KXkI⊥τ.(11)
Leta∈I. Since I is a state ideal, we have τa∈I. In addition, τa∧x≤τa∈I, for all x∈X. Then, τa∧x∈I, for all x∈X. That is a∈XI⊥τ. Therefore, I⊆XI⊥τ.(12)
LetI be an ideal of L. We have a∈X⊥τ implies τa∧x=0∈I, for any x∈X. Hence, τa∧x∈I, for any x∈X. Thus, a∈XI⊥τ. Therefore, X⊥τ⊆XI⊥τ.Example 8.
Consider the De Morgan state residuated latticeL,τ of Example 6. By taking X=c and Y=d, for I=0, we have XI⊥τ=YI⊥τ=0. It follows that XI⊥τ∩YI⊥τ=0. But X∩YI⊥τ=∅I⊥τ=L⊈0=XI⊥τ∩YI⊥τ. So, the inclusion may be strict for (10).
SetZ=0,n,a,b, then Z=0∪n∪a∪b. It follows by (3) that ZI⊥τ=0I⊥τ∩nI⊥τ∩aI⊥τ∩bI⊥τ=L∩0∩0∩0=0.Remark 6.
For the reader’s convenience, we shall notice that forτ=idL in the above Proposition 6, we get all the properties of relative annihilators established in De Morgan residuated lattices in ([2], Proposition 4.51). Also, for τ=idL and I=0, we get all the important results about annihilators in De Morgan residuated lattices investigated in Ref. [2] (pages 462–472).
The following properties of relative annihilators always hold forτ=idL but may not in general be the case for some state operators of De Morgan residuated lattices.Proposition 7.
LetL be a De Morgan residuated lattice and I,J,K be ideals of L and X,Y two nonempty subsets of L. Then,(1)
JI⊥∩J⊆I;(2)
J∩K⊆I⇔K⊆JI⊥;(3)
X∩XI⊥⊆I;(4)
I⊆X if X∩XI⊥=I. Particularly, XI⊥∩XI⊥I⊥=I;(5)
YI⊥∪XI⊥⊆YXI⊥⊥,XYI⊥⊥ and YXI⊥⊥∪XYI⊥⊥⊆X∧YI⊥, where X∧Y=x∧y:x∈X,y∈Y;(6)
LI⊥=I and 1I⊥=I;(7)
X⊆XI⊥I⊥. Particularly, II⊥I⊥=I and LI⊥I⊥=L;(8)
XI⊥=XI⊥I⊥I⊥;(9)
XXI⊥⊥⊆XXYI⊥;(10)
IfX⊆I, then XXI⊥⊥=XXXI⊥⊥⊥=XXYI⊥=L, whenever X⊆Y.Proof.
For the proof of (1) and (2), see [2] (Proposition 4.51 (9) and (10)).(3)
Leta∈X∩XI⊥. Then, for any x∈X, we have a∧x∈I. Particularly, for x=a, we obtain a=a∧a∈I, implying a∈I. Thus, X∩XI⊥⊆I.(4)
Firstly,X∩XI⊥⊆I, from (3). Conversely, assume I⊆X. We know that I⊆XI⊥, from Proposition 6 (11). Therefore, I⊆X∩XI⊥. Hence, X∩XI⊥=I.In another hand, assumeX∩XI⊥=I. Then, I=X∩XI⊥⊆X(5)
We haveI⊆YI⊥. Applying Proposition 6 (1), we obtain XI⊥⊆XYI⊥⊥. Since from (1), YI⊥⊆XYI⊥⊥, it follows that YI⊥∪XI⊥⊆XYI⊥⊥. Similarly, we obtain YI⊥∪XI⊥⊆YXI⊥⊥.In addition, leta∈YXI⊥⊥, and z∈X∧Y. Then, there are x∈X, y∈Y such that z=x∧y. Since, a∧y∈XI⊥, we have a∧z=a∧x∧y=a∧x∧y=a∧y∧x∈I, which means that a∧z∈I. Hence, a∈X∧YI⊥, that is, YXI⊥⊥⊆X∧YI⊥. Analogously, we show that XYI⊥⊥⊆X∧YI⊥. Therefore, YXI⊥⊥∪XYI⊥⊥⊆X∧YI⊥.(6)
From Proposition6 (11), we have I⊆LI⊥. Conversely, for any x∈LI⊥, we have x=1∧x∈I. Hence, x∈I. Similarly, we obtain 1I⊥=I.(7)
Letx∈X. For any a∈XI⊥, we have a∧x∈I, which implies x∈XI⊥I⊥.(8)
By (7), we have XI⊥⊆XI⊥I⊥I⊥. On the other hand, applying Proposition 6 (2) to the above (7), we obtain XI⊥I⊥I⊥⊆XI⊥.(9)
We always haveI⊆YI⊥, from Proposition 6 (11). Applying Proposition 6 (1) twice, we obtain XXI⊥⊥⊆XXYI⊥.(10)
Now assumeX⊆I. First of all, we show that XXI⊥⊥=L.
We know thatX⊆I⊆XI⊥, that is, X⊆XI⊥, which implies XXI⊥⊥=L, from Proposition 6 (9).
In addition, sinceI⊆XI⊥, applying Proposition 6 (1) twice, we have XXI⊥⊥⊆XXXI⊥⊥⊥. Therefore, L=XXI⊥⊥⊆XXXI⊥⊥⊥, which implies XXXI⊥⊥⊥=L. Thus, XXI⊥⊥⊆XXXI⊥⊥⊥=L.
Moreover, supposeX⊆Y. Then, from Proposition 6 (2), YI⊥⊆XI⊥. This implies XXYI⊥⊆Prop.3.71XXXI⊥⊥⊥=XXI⊥⊥.
SinceXXI⊥⊥⊆9XXYI⊥, it follows that XXYI⊥=XXI⊥⊥.Example 8.
Consider the De Morgan state residuated latticeL,τ of Example 5. I=0,p and J=0,p,b,d,f are two state ideals of L,τ. We will show that item (3) and (4) do not hold with τ.(3)
SetX=b, and then, XI⊥τ=J. But X∩XI⊥τ=b⊈I.(4)
LetX=0,P,b, we obtain XI⊥τ=J. I⊆X, but X∩XI⊥τ=X∩J=X≠I.Theorem 6.
LetI be an ideal of L and X a nonempty set of L. Then, we have X⟶I=x∈L:X∩x⊆I=XI⊥ in the frame IL,⊆.Proof.
Letx∈X⟶I. Then, X∩x⊆I. It follows from Proposition 6 (5) and Proposition 7 (1) that x∈X⊆XI⊥τ=XI⊥. Therefore, we obtain X⟶I⊆XI⊥.
Conversely, letx∈XI⊥. Since XI⊥=XI⊥, we have x∈XI⊥, which implies that x⊆XI⊥. By Proposition 7 (2), we have X∩x⊆I. It follows that x∈X⟶I. Hence, XI⊥⊆X⟶I. Therefore, we obtain X⟶I=XI⊥ in the frame IL,⊆.
In Ref. [31], the authors investigated the concept of f-relative annihilators in residuated lattices, where f is an endomorphism. Knowing that the notion of endomorphism already existed in universal algebras and due to the fact that any idempotent endomorphism is a state operator, it seems reasonable that all the results of this section can be extended to a general universal algebras and in particular to residuated lattice, where the state operator τ is replaced by an idempotent endomorphism.


## 5. Prime State Ideals in DMSRL

This section proceeds with the study of the notion of prime state ideals in De Morgan state residuated lattices. We establish the prime state ideal theorem and make investigation for some related properties.Proposition 8.
LetP be a proper state ideal of L,τ. Then, the following statements are equivalent:(1)
IfP1,P2∈SℐL and P=P1∩P2, then P=P1 or P=P2;(2)
IfP1,P2∈SℐL and P1∩P2⊆P, then P1⊆P or P2=P;(3)
Ifa,b∈L so that a⊕τa∧b⊕τb∈P, then a∈P or b∈P.Proof.
1⇒2 Let P1,P2∈SℐL and P1∩P2⊆P. Then, P1∩P2∨P=P. From Proposition 3, the lattice SℐL,⊆ is Brouwerian, so it is distributive. It follows that P1∨P∩P2∨P=P. Now by (1), P1∨P=P or P2∨P=P. Thus, P1⊆P or P2⊆P.
1⇒3. Let a,b∈L. Assume a⊕τa∧b⊕τb∈P. Set (P1=P,aτ and (P2=P,bτ. Obviously, P⊆P1∩P2. Let x∈P1∩P2, then by Theorem 1 (3), there are l,k∈P and m,n≥1 such that x≤k⊕ma⊕τa and x≤l⊕nb⊕τb. Hence, we have(6)x≤k⊕ma⊕τa∧l⊕nb⊕τb≤P16k⊕ma⊕τa∧l⊕k⊕ma⊕τa∧nb⊕τb≤P16k∧l⊕ma⊕τa∧l⊕k∧nb⊕τb⊕ma⊕τa∧nb⊕τb≤P15k∧l⊕l∧ma⊕τa⊕k∧nb⊕τb⊕mna⊕τa∧b⊕τb.
Butk∧l,l∧ma⊕τa,k∧nb⊕τb,mna⊕τa∧b⊕τb∈P. Thus, x∈P. Hence, P=P1∩P2. Therefore, by (1), P=P1 or P=P2, that is, a∈P or b∈P.
3⇒1 Let P1,P2∈SℐL such that P=P1∩P2. Suppose that P≠P1 and P≠P2 and let a∈P1/P and b∈P2/P. Then, a⊕τa∧b⊕τb∈P1∩P2=P, that is, a contradiction. Thus, P=P1 or P=P2.Definition 8.
A proper state idealP of L,τ is said to be prime if it satisfies one of the equivalent conditions of Proposition 8.
We denote the set of all prime state ideals ofL,τ by SpectτL.
The following example shows a prime state ideal ofL,τ, which is not a prime ideal of L.Example 10.
Consider the De Morgan state residuated latticeL,τ of Example 5. 0, 0,p, 0,p,a, 0,p,c, 0,p,b,d,f, L are ideals of L, and the state ideals of L,τ are 0, I=0,p, J=0,p,b,d,f and L. The ideal I is not a prime ideal of L, because 0,p=0,p,a∩0,p,c but 0,p≠0,p,a and 0,p≠0,p,c. Still as a state ideal of L,τ, 0,p is a prime state ideal (according to Proposition 8, (1)).
Now, we give one of our main theorems.Theorem 7.
(Prime state ideal theorem) LetF be a filter in the lattice L,⊆, and I be a state ideal of L,τ such that I∩F=∅. Then, there is a prime state ideal P of L,τ such that I⊆P and P∩F=∅.Proof.
SetμI=J:J∈SℐL,I⊆JandJ∩F=∅. We have I∈SℐL,I⊆I and I∩F=∅. Then, I∈μI, that is, μI≠∅. It is easy to check that the set μI is inductively ordered by inclusion. Hence by Zorn’s lemma, it has a maximal element P. Let’s show that that P∈SpectτL. Since P∈μI, it follows that P is a proper state ideal and P∩F=∅. Let x,y∈L, such that a⊕τa∧b⊕τb∈P. Assume a∉P and b∉P, and consider the sets P,aτ and P,bτ. Then, P is strictly contained in P,aτ and P,bτ and the maximality of P implies that P,aτ∉μI and P,bτ∉μI. Thus, P,aτ∩F≠∅ and P,bτ∩F≠∅. Let x∈P,aτ∩F and y∈P,bτ∩F≠∅. According to Theorem 1 (3), there are k,l∈P and m,n≥1 such that x≤k⊕ma⊕τa and y≤l⊕nb⊕τb. Then,(7)x∧y≤k⊕ma⊕τa∧l⊕nb⊕τb≤P16k⊕ma⊕τa∧l⊕k⊕ma⊕τa∧nb⊕τb≤P16k∧l⊕ma⊕τa∧l⊕k∧nb⊕τb⊕ma⊕τa∧nb⊕τb≤P15k∧l⊕l∧ma⊕τa⊕k∧nb⊕τb⊕mna⊕τa∧b⊕τb.
Butk∧l,l∧ma⊕τa,k∧nb⊕τb,mna⊕τa∧b⊕τb∈P; hence, x∧y∈P. Moreover, since F is a filter of the lattice L,⊆, it follows that x∧y∈F, and therefore, P∩F≠∅, which is a contradiction. Therefore, P∈SpectτL.Proposition 9.
LetI be a proper state ideal of L,τ. Then, there is a maximal state ideal M of L,τ such that I⊆M.Proof.
Let us consider the setθI=J:Jisaproper state ideal such thatI⊆J. I is a proper state ideal of L,τ and I⊆I. So, I∈θI, that is, θI≠∅. One can easily see that θI is inductively ordered by inclusion. Then, by Zorn’s lemma, θI has a maximal element M. Let show that θI is a maximal state ideal of L,τ. In fact, if N is a proper state ideal of L,τ such that M⊆N, then N∈θI and the maximality of M implies that N=M.Proposition 10.
Leta∈L, a>0. Then, there is a prime state ideal P of L,τ such that a∉P.Proof.
Since0 is a state ideal and 0∩a=∅ (where a is the filter generated by a in the lattice L,⊆). Hence by Theorem 7, there exists a prime state ideal P such that P∩a=∅. Thus, a∉P.


## 6. Conclusion

In this article, we have introduced the notion of De Morgan state residuated lattice (DMSRL) and made investigations on certain related properties and examples. We have proved that the lattice of all state idealsSℐL,⊆ of a DMSRL is a coherent frame, and for any state ideal I, the pseudocomplement of I is I′=I⟶0=x∈L:I∩xτ=0=x∈L:i∧nx⊕τx=0,for alli∈Iandn∈ℕ∗. Furthermore, we have illustrated that the set of compacts element of the sublattice SℐL is CSℐL=xτ:x∈L. Also, we characterized the DMSRL for which the lattice SℐL is a Boolean algebra. In addition, we brought in the concept of state relative annihilator of a given nonempty subset with respect to a state ideal in DMSRL and investigated some of its properties. We proved that state relative annihilators are a particular kind of state ideals. After all, we have studied the notion of prime state ideals in DMSRL and established the prime state ideal theorem.Knowing that in universal algebras, any idempotent endomorphism is a state operator [29], it seems reasonable that all the results in this article can be extended to the more general class of residuated lattice and even in general case of universal algebras, where the state operator τ is replaced by an idempotent endomorphism. So, our further work will consist to look this direction. In the same view as the work in Ref. [32], another way on this topic will focus on the study of the lattices of L-fuzzy state ideals in De Morgan state residuated lattices.



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