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 ideals of a DMSRL is a coherent frame. Then, we characterize the DMSRL for which the lattice 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.

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 , 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 [58]). 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 [1417]), 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. Section 2 is the part in which we consider the algebraic structure of the set of all state ideals in a DMSRL . It is shown that is a coherent frame. Also, we characterize the DMSRL for which the lattice is a Boolean algebra. In Section 3, we bring in the notion of state annihilator of a nonempty set with respect to a state ideal 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, 1821].

A nonempty set with four binary operations and two constants is called a bounded integral commutative residuated lattice or shortly residuated lattice if the following axioms are verified:(C1) is a bounded lattice;(C2) is a commutative monoid (with the unit element 1);(C3) For all , .

The following notations of residuated lattices will be used:

will stand for , a residuated lattice and for any and , , , and .

In Refs. [2,9,14,21], we have the following definitions and examples:(1)A residuated lattice satisfying the divisibility condition (div): is called a R -monoid.(2)A residuated lattice satisfying the prelinearity condition (pre): 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): 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 properties and is called a Boolean algebra;(8)A residuated lattice satisfying the Stone property is called a Stonean residuated lattice;(9)A residuated lattice satisfying the De Morgan property 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. Let endowed with the Hasse diagram and Caley tables (Figure 1):
Then, is a nonregular residuated lattice. One can easily check that is a De Morgan residuated lattice, which is Stonean.

Example 2. Let with his Hasse diagram and Cayley tables of and be the following (Figure 2):
Then, is a residuated lattice.
One can readily verify that is a De Morgan residuated, which is regular. But is not a BL-algebra, is not a MV-algebra, is not a Boolean algebra, is not a Stonean residuated lattice, is not a MTL-algebra, and is not a IMTL-algebra.
The following basic arithmetic of residuated lattices will be used for any (see [19,22]):(RL1): , , , ;(RL2): ;(RL3): ;(RL4): if , then , , and ;(RL5): ; ;(RL6): ; ; ;(RL7): and ;(RL8): , ;(RL9): ;(RL10): ;(RL11): , ;(RL12): , ; ;(RL13): ;(RL14): ;(RL15): .Let be a residuated lattice, and we set , for every .

Definition 1 (see [19]). An ideal of is a nonempty subset of , which satisfies the following conditions for every :(I1) If and , then ;(I2) If , then .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: , for every .
Note that in Ref. [2], it was shown that a nonempty subset of a residuated lattice is an ideal of if and only if the following conditions hold for any :(I3) If and , then ;(I4) If , then .The set of all ideals of a residuated lattice will be denoted by .
For a nonempty subset of a residuated lattice , the ideal generated by is [14].
Here are some properties of the operation (see [13,14]). Let be a residuated lattice. For any , we have(P1): ;(P2): , , ;(P3): , ;(P4): ;(P5): If , then ;(P6): If and , then .Let be a residuated lattice. For any and , we define , , and , for .
Then, the following relations hold for any and :(P7): . In particular, ;(P8): ;(P9): ;(P10): ;(P11): .

Lemma 1. In any De Morgan residuated lattice , the following relations hold for any AND [2]:(P12): ;(P13): ;(P14): ;(P15): ;(P16): .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 lattice is called Brouwerian if it satisfies the equality (whenever the arbitrary joins exist), for any , .

Definition 2. We call frame a complete lattice that satisfies the infinite distributive law , for all and [25].

Remark 1. (1)Every Brouwerian lattice is distributive;(2)A frame is a complete Brouwerian lattice.An element of a complete lattice is called compact if for all , implies that for some finite (see [9,26]). We will denote by the set of all compact elements of a complete lattice .

Proposition 1. Let be a frame and [27]. Then, if for all , implies that for some finite .

Definition 3. A frame is called coherent if the following conditions hold [27]:(i) is a sublattice of ; that is, for all , if , then ;(ii)For all , .Recall that if is a lattice with 0 its bottom element, and , then is said to be a pseudocomplement of if and for every , implies . is called pseudocomplemented if every element has a pseudocomplement. Every frame is pseudocomplemented (see [28]).
For every , we call a relative pseudocomplement of with respect to , the greatest element (if it exists) such that .
In what follows, we recall some results about state residuated lattices, which will be used in the sequel.

Definition 4. (see [9]). A map is said to be a state operator on if the following conditions hold for any [25]:(SO1): ;(SO2): implies ;(SO3): ;(SO4): ;(SO5): ;(SO6): ;(SO7): ;(SO8): .The pair is called a state residuated lattice.
We shall notice that for any residuated lattice , the identity map is a state operator on , which is an endomorphism, but in general, a state operator is not an endomorphism.

Definition 5. Let be a state residuated lattice [9, 13]. An ideal of is said to be a state ideal of if (i.e., for all , ). Similarly, a filter of is called a state filter if .
will stand for the set of all state ideals of . It is obvious that .
For computational issues, we will use the following properties (see [9, 13]).
Let be a state residuated lattice. Then, for any , for all , we have(SO9): ;(SO10): implies ;(SO11): ;(SO12): and if , then ;(SO13): and if , then ;(SO14): . Particularly, if are comparable,  then ;(SO15): If is faithful, then implies ;(SO16): ;(SO17): , where ;(SO18): is a subalgebra of ;(SO19): is a state filter of ;(SO20): is a state ideal of ;(SO21): ;(SO22): ;(SO23): If , then .(SO24): .For any nonempty subset of , we denote by the state ideal of generated by ,=; that is, is the smallest state ideal of containing , and for an element , is called the principal state ideal of . If and , we denote by .
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]). Let be a nonempty subset of , and . Then,(1);(2);(3);(4).

Lemma 2 (see [13]). Let be a state residuated lattice. For all , we have(5);(6);(7);(8);(9).

Proposition 2. Let be a state residuated lattice [13]. Then,
is a bounded complete lattice with the bottom element and the top element .
Now, we introduce the concept of De Morgan state residuated lattice.

Definition 6. A state residuated lattice is called De Morgan if 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. Let be a De Morgan residuated lattice.(1)If a map is an idempotent De Morgan-endomorphism (i.e., is an endomorphism of such that ), then is a state operator on and it is said to be a state-morphism operator. Therefore, the couple is called a De Morgan state-morphism residuated lattice. So, by taking a state-morphism operator (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)Let be a state operator on . From Definition 1.9, we have(SO1) ;(SO3) , for any .Thus, from Ref. [18], Proposition 8.1 (iii) and Definition 8.1 (1), is a generalized Bosbach state of type 1. Moreover, from (SO10), we have implies , for any . Therefore, any state operator on a De Morgan residuated lattice can be seen as an ordering-preserving generalized Bosbach state of type 1.
A state operator is called cofaithful if and uncofaithful otherwise.

Example 3. Set with . Then, is De Morgan residuated lattice that is a BL-algebra but not an MV-algebra with the operations (Figure 3):
Let define the unary operator on byOne can easily check that is a state operator on . Hence, the couple is a DMSRL. In addition, and verified the following properties: and , for any . Therefore, is a cofaithful De Morgan state-morphism operator. Furthermore, the state ideals of are and .

Example 4. Let and be two nontrivial De Morgan residuated lattices and a homomorphism. We define the map Then, one can check that is an idempotent endomorphism on . So, is a state-morphism operator and the couple is a De Morgan state-morphism residuated lattice. In addition, . Hence, is a uncofaithful state operator.

Remark 3. (1)For every De Morgan residuated lattice , is a De Morgan state residuated lattice. That is, a De Morgan residuated lattice can be seen as a De Morgan state residuated lattice. One can see that all ideals of is a state ideal of .(2)Let be a De Morgan residuated lattice, a sub-De Morgan residuated lattice of and be a state operator on such that . Then, the restriction of on is a state operator on . We can see that is a state ideal of .This means that if a state operator preserves a substructure, then its restriction to this substructure is a state operator.(3)Let be a nonempty family of De Morgan residuated lattices and a family of state operators such that is a state operator on for each . We define the map .Then, is a state operator on . We can check that .
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. If is a De Morgan state residuated lattice, then for all , we have .

Proof. Let . Then, from Lemma 2 (8), we have . Now, let . Then, and . Hence, from Theorem 1 (2), there exists such that, and . Therefore,That is, . Hence, . Thus, .

3. The Lattice of All State Ideals of a DMSRL

In this section, we focus on the study of the algebraic structure of the set of all state ideals of a De Morgan state residuated lattice .

From now on, unless otherwise specified, will always denote a De Morgan state residuated lattice ; that is, is a De Morgan residuated lattice and is a state operator on .

Proposition 3. is a Brouwerian lattice.

Proof. Let be an index set, , and be a family of state ideals of . We will show that . That is,
. Clearly, .
Let . Then, and . It follows that there exist , , , such that . Then, . Since , we have , for every . We deduce that . Hence, , that is, . Therefore, is a Brouwerian lattice.

Theorem 2. The lattice is a frame.

Proof. From Proposition 2, is a complete lattice. From Proposition 3, is a Brouwerian lattice. Combining them, we have by Remark 1 (2) that is a frame.
In the following result, we establish a concrete description of the right adjoint of the map:Now, for any , we put .

Proposition 4. In the frame , for any , we have(1)(2), that is, and is the right adjoint of ;(3).

Proof. (1)We will show that is a state ideal of . We have . Indeed, and . Hence, .Now, let , then and . We obtain . From Theorem 2, it follows that , which implies (by Lemma 3 (9)) that . Thus, .Assume and . Then, and . It follows that . Hence, . Finally, let ; then, . Since , we obtain that . Therefore, is a state ideal of . That is, .(2)Now, we show that , for any . Assume and , we obtain (since )). It follows that . That is, . Conversely, let and . Then, we have . That is, . Since , we deduce that , which implies . Therefore, and is the right adjoint of .(3)Set . First, let . Then, . For , and , we have (since )). It follows that , which implies that . Hence, .Conversely, let and . Then, there exists such that . Hence, . That is, . Hence, . Therefore, .
For every , we put . Then, from Proposition 1 (3), we have the following corollary.

Corollary 1. For all , we have .

Theorem 3. Let . Then, .

Proof. ). Assume . Set . Since , there are such that (Proposition 1). By Lemma 2 (9), we have . Thus, , that is, .
). Let . Then, there exists such that . Assume and . Then, . It follows that there exist , , for all such that . That is, . Hence, . Hence, , that is, . Therefore, .

Remark 4. Theorem 3 means that a state ideal is a compact element of the frame if and only if it is principal.

Theorem 4. The lattice is a coherent frame.

Proof. (i)From Theorem 2, we have that is a frame.(ii)From Theorem 3, we obtain that . By Lemma 3, we have , and by Lemma 2 (9), we have for all . Therefore, is a sublattice of .(iii)For any , we have .(i), (ii), and (iii) combined with Definition 3 imply is a coherent frame.
We have immediately the following results.

Corollary 2. The lattice is pseudocomplemented. Clearly, for all , we have that is the pseudocomplement of .
According to Definition 3 and Corollary 2, we have the following result in a De Morgan residuated lattice .

Corollary 3. (1) is a coherent frame;(2) is a pseudocomplemented lattice.We recall that a Heyting algebra ([30]) is a lattice with 0 such that for every , there exists an element (call the pseudocomplement of with respect to y) such that for every (i.e., ).

Remark 5. From Proposition 4, is a Heyting algebra, where for , .

Proposition 5. For any , we have(1);(2).

Proof. (1)We have .(2)Let . Then, by (1), we have and . In addition, by Lemma 1 (P16), we have ; hence, . Then, by Lemma 1 (P16), . Moreover, by (RL6) and (SO22), (.Hence, . Then, by (1), we have . Therefore, .
Conversely, let . Then, by (1), , we have , so . It follows that . We obtain