State monadic residuated lattices and their corresponding filters

1 Introduction

Non-classical logic is more suitable than classical logic to handle uncertain and fuzzy information. In the past several decade years, various fuzzy logical algebras have been proposed as the semantic units for systems of non-classical logics. For example, MV-algebras were introduced in [2] by Chang as algebraic models of the infinitely-valued logic of Łukasiewicz, while BL-algebras were introduced in [11] by Hájek as algebraic semantics of basic fuzzy logic, a general framework in which tautologies of continuous t-norm and their residua can be captured [3]. Inspired by Hájek’s work, Esteva and Godo proposed in [8] a new formal deductive system monoidal t-norm based logic, intended to cope with left-continuous t-norms and their residua [18]. However, all the above mentioned algebras are the particular case of residuated lattices, which were introduced by Dilworth in [28] and stemmed from attempts to generalize properties of the lattice of ideals of a ring, so residuated lattices are very basic and important algebraic structures. The filter theory plays an important role in studying the subdirect representation theorem of fuzzy logical algebras and in proving the completeness of their corresponding logical systems. In view of logic, various filters have natural interpretation as various sets of provable formulas. In recent years, some types of filters on residuated lattices have been studied in [32].

Monadic Boolean algebra (L, ∃), in the sense of Halmos [14], is a Boolean algebra equipped with a closure operator ∃, which abstracts algebraic properties of the standard existential quantifier “for some”. The name “monadic” comes from the connection with predicate logics for languages having one placed predicates and a single quantifier. After that, monadic MV-algebras, the algebraic counterpart of monadic Łukasiewicz logic, were introduced and studied in [6, 23]. Monadic BL-algebras, monadic residuated lattices, monadic residuated ℓ-monoids, monadic bounded hoops, monadic NM-algebras, monadic IMTL-algebras, monadic pseudo BCI-algebras and monadic pseudo equality algebras were introduced and investigated in [1, 30]. It is noted that both MV-algebras and basic algebras satisfy De Morgan and double negation laws, in the definition of the corresponding monadic algebras, it is possible to use only one of the existential and universal quantifiers as primitive, the other being definable as the dual of the one defined. However, definitions of monadic Heyting algebras, monadic BL-algebras, monadic residuated lattices and monadic bounded hoops require the introduction of both kinds of quantifiers simultaneously, because these quantifiers are not mutually interdefinable.

The axiomatisation of probability was done by Kolmogorov in 1933 and both probability and statistics developed into major fields. But new areas of science have appeared during the last century, such as information science, which do not satisfy the Kolmogorov axioms. Inference systems require a probability theory based on non-classical logics. For example, as we know, the probability of a classical event φ is usually presented as the truth value of a modal proposition P (φ) in fuzzy logic with modality P (interpreted as probably) in [11]. But for a fuzzy event like “the height of Mary is approximately 1.6 m”, the probability of it cannot be expressed by classical approach in fuzzy logic. Thus, there is a strong motivation to revise the classical probability theory and to introduce more general probability models based on non-classical logics. With the intent of measuring the average truth-value of propositions in Łukasiewicz logic, states on MV-algebras were introduced by Mundici [20] with the intent of measuring the average truth-value of propositions in Łukasiewicz logic, which are a generalization of probability measures on Boolean algebras. Inspired by this, serval authors studied states on residuated lattices and obtained some interesting results [4, 19]. Although these way can be expanded the scope of states, they both have as codomain the closed unit interval [0, 1]. However, fuzzy logical algebras with states are not Universal Algebra and hence they do not automatically induce an assertional logic. To present a unified approach to state and introduce in the many valued context a deduction apparatus able to reason by analogy in a logical and algebraic setting, a new approach to states on MV-algebras was introduced by Flaminio and Montagna [9], where they added an unary operation τ to the language of MV-algebras as an internal state satisfying some basic properties of state. The resulting algebras structures were so-called state MV-algebras. This approach generalizes the state, as a function on the algebra taking values in the interval [0, 1] with the addition property. Consequently, the notion of internal states has been extended to other fuzzy logical algebras such as BL-algebras [5], IMTL-algebras [30]. Motivated by this, He introduced state residuated lattices and study the lattice structure of their corresponding state filters [7, 31].

In this paper, we will extend internal states to monadic residuated lattices for providing an algebraic foundation for reasoning about probabilities of fuzzy events with respect to predicate variables in monadic substructural fuzzy predicate logics. The main focus of existing research about internal states is on MV-algebras [9], BL-algebras [5], IMTL–algebras [30] and residuated lattices [17]. All the above mentioned algebraic structures are the algebraic semantics of t-norm based on fuzzy propositional logics, which are particular cases of substructural fuzzy propositional logics. However, there is few research about internal states on monadic fuzzy logical algebras, which are the equivalent algebraic semantics of substructural fuzzy predicate logics so far. Therefore, it is interesting to study internal states on monadic residuated lattices for treating a variant of the concept of internal states within the framework of Universal Algebra and provide a sold algebraic foundation for reasoning about probabilities of fuzzy events with respect to predicate variables in monadic substructural fuzzy predicate logics. These are motivations for us to investigate internal states on monadic residuated lattices.

2 Preliminaries

In this section, we recall some definitions and results about residuated lattices and their corresponding monadic algebraic structures, which will be used in the sequel.

Definition 2.1. [28] A residuated lattice is an algebraic structure (L, ⊔ , ⊓ , ⊗ , → , 0, 1) with four binary operations such that

(1) (L, ⊔ , ⊓ , 0, 1) is a bounded lattice,

(2) (L, ⊗ , 1) is a commutative monoid,

(3) a ⊗ b ⩽ c if and only if a ⩽ b → c, for all a, b, c ∈ L.

In what follows, by L we denote the universe of a residuated lattice (L, ⊔ , ⊓ , ⊗ , → , 0, 1). In any residuated lattice L, we define ¬a = a → 0, a ⊕ b = ¬ (¬ a ⊗ ¬ b) , ¬¬ a = ¬ (¬ a), a⁰ = 1 and a ⁿ  = a^n-1 ⊗ a for n ≥ 1.

Let L be a residuated lattice and a ∈ L. Then order of a denoted ord (a) is the smallest integer n such that a ⁿ  = 0. If there is no such n, then ord (a) =∞. An element a ∈ L is called a nilpotent element of L if a ⁿ  = 0 for some n ∈ ℕ . Let B (L) be the set of all complemented elements of the residuated lattice L, recall that an element a ∈ L is called complemented if there is an element b ∈ L such that a ⊔ b = 1 and a ⊓ b = 0 [11].

Definition 2.2. [17]   Let L be a residuated lattice. Then L is called

(1) an involutive residuated lattice if it satisfies (INV)   ¬¬ a = a.

(2) an MTL-algebra if it satisfies (PRE)   (a → b) ⊔ (b → a) =1 .

(3) a BL-algebra if it is an MTL-algebra and satisfies (DIV)    a ⊓ b = a ⊗ (a → b).

(4) an IMTL-algebra if it is an MTL-algebra and satisfies (INV).

(5) is an MV-algebra if it is a BL-algebra and satisfies (INV)¹ 1

Proposition 2.3. [17]  Let L be a residuated lattice. Then the following properties are valid: for all a, b, c ∈ L,

(1) a ⊔ b ⩽ (a → b) → b (in particular a ⩽ ¬¬ a),

(2) a → b ⩽ a ⊗ c → b ⊗ c,

(3) (a → b) ⊗ (b → c) ⩽ (a → c),

(4) If a ⩽ b, then a ⊗ c ⩽ b ⊗ c,  c → a ⩽ c → b and b → c ⩽ a → c,

(5) a → (b → c) = b → (a → c) = (a ⊗ b) → c,

(6) a ↔ b ⩽ (b ↔ c) ↔ (a ↔ c),

(7) If ⊔_i∈Ia _i , ⊔ _i∈Ib _i and ⊓_i∈I (a _i  → b _i ) exist, then ⊓_i∈I (a _i  → b _i ) ⩽ ⊔ _i∈Ia _i  → ⊔ _i∈Ib _i ,

(8) If ⊓_i∈Ia _i , ⊓ _i∈Ib _i and ⊓_i∈I (a _i  → b _i ) exist, then ⊓_i∈I (a _i  → b _i ) ⩽ ⊓ _i∈Ia _i  → ⊓ _i∈Ib _i ,

(9) a ⊔ (b ⊗ c) ⩾ (a ⊔ b) ⊗ (a ⊔ c),

(10) a ^m  ⊔ b ⁿ  ⩾ (a ⊔ b)  ^mn ,  m,  n ⩾ 1,

(11) a ⊗ b = a ⊗ (a → (a ⊗ b)),

(12) a → b = a → (a ⊓ b),

(13) a ⩽ b if and only if a → b = 1.

Definition 2.4. [21] A monadic residuated lattice is an algebraic structure (L, ⊔ , ⊓ , ⊗ , → , ∀ , ∃ , 0, 1) such that (L, ⊔ , ⊓ , ⊗ , → , 0, 1) is a residuated lattice, and ∀, ∃ are two unary operations on L satisfying the following conditions: for all a, b ∈ L,

(M1) a → ∃ a = 1,

(M2) ∀a → a = 1,

(M3) ∀ (a → ∃ b) = ∃ a → ∃ b,

(M4) ∀ (∃ a → b) = ∃ a → ∀ b,

(M5) ∀ (a ⊔ ∃ b) = ∀ a ⊔ ∀ b,

(M6) ∃ ∀ a = ∀ a,

(M7) ∀∀ a = ∀ a,

(M8) ∃ (∃ a ⊗ ∃ b) = ∃ a ⊗ ∃ b,

(M9) ∃ (a ⊗ a) = ∃ a ⊗ ∃ a.

Monadic residuated lattices form a variety that we will denote by 𝕄𝕇𝕃, and for simplicity, if L is a residuated lattice and we enrich it with a monadic structure, we denote the resulting algebra by (L, ∀ , ∃). It is immediate to see that for each proper subvariety S of residuated lattices the algebras in 𝕄𝕇𝕃 whose RL-reducts are in 𝕊 form a proper subvariety 𝕄𝕊 of 𝕄𝕇𝕃. These algebras will be called monadic S-algebras.

Proposition 2.5. [21]  Let (L, ∀ , ∃) be a monadic residuated lattice. Then following hold: for all a, b ∈ L,

(1) ∀1 =1,

(2) ∀ ∃ a = ∃ a, ∃ ∀ a = ∀ a,

(3) ∃∃ a = ∃ a,

(4) ∀ (∃ a → ∃ b) = ∃ a → ∃ b,

(5) ∃ (∃ a → b) → (∃ a → ∃ b) =1,

(6) a ⩽ b ⇒ ∀ a ⩽ ∀ b, ∃ a ⩽ ∃ b,

(7) ∀a = a if and only if ∃a = a,

(8) ∀ (∃ a ⊔ ∃ b) = ∃ a ⊔ ∃ b, ∃ (∃ a ⊓ ∃ b) = ∃ a ⊓ ∃ b,

(9) ∃0 =0,

(10) ∃ (a ⊔ b) = ∃ a ⊔ ∃ b, ∀ (a ⊓ b) = ∀ a ⊓ ∀ b,

(11) a ⩽ ∃ b if and only if ∃a ⩽ ∃ b, ∀ a ⩽ b if and only if ∀a ⩽ ∀ b,

(12) ∀ (a → b) → (∀ a → ∀ b) =1,

(13) ∀ (a → b) → (∃ a → ∃ b) =1,

(14) (∀ a ⊗ ∃ b) → ∃ (a ⊗ b) =1,

(15) ∃ (∃ a ⊗ b) = ∃ a ⊗ ∃ b = ∃ (a ⊗ ∃ b),

(16) ∃ (a ⊗ ∀ b) = ∃ a ⊗ b, ∃ (∀ a ⊗ b) = ∀ a ⊗ ∃ b,

(17) ∃ (∃ a → b) → (∃ a → ∃ b) =1,

(18) ∃ (a → ∃ b) → (∀ a → ∃ b) =1,

(19) ∃ (∃ a → ∃ b) = ∃ a → ∃ b, ∃ (∀ a → ∀ b) = ∀ a → ∀ b,

(20) ∀ (∀ a → b) = ∀ a → ∀ b.

Definition 2.6. [26] A monadic MTL-algebra is an algebraic structure (L, ⊔ , ⊓ , ⊗ , → , ∀ , ∃ , 0, 1) such that (L, ⊔ , ⊓ , ⊗ , → , 0, 1) is an MTL-algebra, and ∀, ∃ are two unary operations on L satisfying the following conditions: for all a, b ∈ L,

(M2) ∀a → a = 1,

(M5) ∀ (a ⊔ ∃ b) = ∀ a ⊔ ∀ b,

(M9) ∃ (a ⊗ b) = ∃ a ⊗ ∃ a,

(M10) ∀ (∀ a → b) = ∀ a → ∀ b,

(M11) ∀ (a → ∀ b) = ∃ a → ∀ b.

Definition 2.7. [1] A monadic BL-algebra is an algebraic structure (L, ⊔ , ⊓ , ⊗ , → , ∀ , ∃ , 0, 1) such that (L, ⊔ , ⊓ , ⊗ , → , 0, 1) is a BL-algebra, and ∀, ∃ are two unary operations on L satisfying the following conditions (M2),(M5),(M9),(M10),(M11). The monadic algebras of involutive substructural fuzzy logics also studied by serval authors, for example, monadic involutive residuated lattice, monadic IMTL-algebra, monadic NM-algebra and monadic MV-algebra.

Definition 2.8. [26] A monadic involutive residuated lattice is an algebraic structure (L, ⊔ , ⊓ , ⊗ , → , ∀ , ∃ , 0, 1) such that (L, ⊔ , ⊓ , ⊗ , → , 0, 1) is an involutive residuated lattice, and ∀ is an unary operation on L satisfying the following conditions: for all a, b ∈ L,

(M2) ∀a → a = 1,

(M10) ∀ (∀ a → b) = ∀ a → ∀ b,

(M11) ∀ (a → ∀ b) = ∃ a → ∀ b,

(M12) ∀ (a ⊔ ∀ b) = ∀ a ⊔ ∀ b,

(M13) ∀ (¬ a → a) = ¬ ∀ a → ∀ a, Definition 2.9. [24] A monadic IMTL-algebra is an algebraic structure (L, ⊔ , ⊓ , ⊗ , → , ∀ , ∃ , 0, 1) such that (L, ⊔ , ⊓ , ⊗ , → , 0, 1) is an IMTL-algebra, and ∀ is an unary operation on L satisfying the following conditions: (M2),(M10),(M11),(M12),(M13). Definition 2.10. [23] A monadic MV-algebra is an algebraic structure (L, ⊔ , ⊓ , ⊗ , → , ∀ , ∃ , 0, 1) such that (L, ⊔ , ⊓ , ⊗ , → , 0, 1) is an MV-algebra, and ∀ is unary operation on L satisfying the following conditions (M2),(M10),(M13). Remark 2.11. The following figure shows that the relationship between monadic algebras of substructural fuzzy predicate logics, which can be directly from [26]

OPEN IN VIEWER

Fig. 1

Relations between monadic algebras of substructural fuzzy logics.

Then we recall some results on monadic filters of monadic residuated lattices in [21].

Given a monadic residuated lattice (L, ∀ , ∃), a filter F is called a monadic filter of (L, ∀ , ∃) if it closed under ∀. A proper monadic filter F of (L, ∀ , ∃) is called a maximal monadic filter if it is not strictly contained in any proper monadic filter of (L, ∀ , ∃). For any nonempty subset X of L, we denote by 〈X〉_∀ the monadic filter of (L, ∀ , ∃) generated by X, that is, 〈X〉_∀ is the smallest monadic filter of (L, ∀ , ∃) containing X. Indeed, 〈X〉_∀ = {x ∈ L|x ≥ ∀ x₁ ⊙ ∀ x₂ ⊙ ⋯ ⊙ ∀ x _n , x _i  ∈ X, n ≥ 1}. In particular, 〈a〉_∀ = {x ∈ L|x ≥ (∀ a)  ⁿ , n ≥ 1}.

Also, if F is a monadic filter of (L, ∀) and x ∉ F, then we put 〈F, x〉_∀ : = 〈F ⊔ {x} 〉_∀ = {y ∈ L|x ≥ f ⊙ (∀ x)  ⁿ , f ∈ F} = F ⊔ [∀ x).

The intersection of all maximal monadic filters of a monadic residuated lattice (L, ∀ , ∃) is called the radical of (L, ∀ , ∃) and it is denoted by Rad (L, ∀ , ∃). Also, let F be a proper state monadic filter of a state monadic residuated lattice (L, ∀ , ∃ , τ). Then the intersection of all maximal state monadic filters of (L, ∀ , ∃ , τ) obtaining F is called the radical of F and is denoted by Rad (F).

Proposition 2.12. [21] Let F be a monadic filter of a monadic residuated lattice (L, ∀ , ∃). Then Rad (F) = {a ∈ L |  ¬ ((∀ a)  ⁿ ) → ∀ a ∈ F, for all n ≥ 1}.

In this section, we introduce the notion of internal states on monadic residuated lattice and investigate some of their algebraic properties.

Definition 3.1. Let (L, ∀ , ∃) be a monadic residuated lattice. A mapping τ : (L, ∀ , ∃) → (L, ∀ , ∃) is said to be an internal state if it satisfies the following conditions: for all a, b ∈ L,

(SM1) τ (0) =0,

(SM2) τ (a → b) ⩽ τ (a) → τ (b),

(SM3) τ (a → b) = τ (a) → τ (a ⊓ b),

(SM4) τ (τ (a) ⊗ τ (b)) = τ (a) ⊗ τ (b),

(SM5) τ (τ (a) ⊔ τ (b)) = τ (a) ⊔ τ (b),

(SM6) τ (τ (a) ⊓ τ (b)) = τ (a) ⊓ τ (b),

(SM7) τ (∀ a) = ∀ τ (a),

(SM8) τ (∃ a) = ∃ τ (a).

The pair (L, ∀ , ∃ , τ) is said to be a state monadic residuated lattice. State monadic residuated lattices form a variety that we will denote by 𝕊𝕄ℝ𝕃 and for simplicity, if L is a residuated lattice and we enrich it with a state monadic structure, we denote the resulting algebra by (L, ∀ , ∃ , τ). It is immediate to see that for each proper subvariety S of residuated lattices the algebras in 𝕊𝕄ℝ𝕃 whose RL-reducts are in 𝕊 form a proper subvariety 𝕊𝕄 of 𝕊𝕄ℝ𝕃. These algebras will be called state monadic S-algebras.

Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Denote Ker₀ (τ) = {a ∈ L ∣ τ (a) =1},

Ker_∀ (τ) = {a ∈ L ∣ τ (∀ a) =1},

Ker_∃ (τ) = {a ∈ L ∣ τ (∃ a) =1}. The internal state τ is said to be 0-faithful, if Ker₀ (τ) = {1},

∀-faithful, if Ker_∀ (τ) = {1},

∃-faithful, if Ker_∃ (τ) = {1}.

Example 3.2. Any monadic residuated lattice (L, ∀ , ∃) can been as a state monadic residuated lattice, where τ = id _L . Example 3.3. Let L = {0, a, b, c, 1} with 0 ≤ c ≤ a ≤ 1, 0 ≤ c ≤ b ≤ 1 and the operations ⊗, → defined as follows

Then (L, ⊔ , ⊓ , ⊗ , → , 0, 1) is a residuated lattice.

Consider the mappings ∀ _j , ∃  _k , τ _i : (L, ∀ , ∃ , τ) → (L, ∀ , ∃ , τ), j = 1, 2, 3,  k = 1, 2, 3, i = 1, ⋯ 6, given in the tables below

Then we have

(1) (∀  _i , ∃  _i ) , i = 1, 2, 3 are all pairs forming monadic residuated lattices (L, ∀  _i , ∃  _i ). The structures (L, ∀  _i , ∃  _i ) , i = 1, 2, 3 are monadic residuated lattices.

(2) The mappings τ _i , i = 1, ⋯ , 6 are all internal states on the residuated lattices (L, τ). The structures (L, τ _i ) , i = 1, ⋯ , 6 are state residuated lattices.

(3) The following structures are all state monadic residuated lattices (L, ∀ ₃, ∃ ₃, τ₁) ,   (L, ∀ ₁, ∃ ₁, τ₂) ,   (L, ∀ ₂, ∃ ₂, τ₂) ,

(L, ∀ ₃, ∃ ₃, τ₂) ,   (L, ∀ ₃, ∃ ₃, τ₃) ,   (L, ∀ ₃, ∃ ₃, τ₄) ,

(L, ∀ ₃, ∃ ₃, τ₅) ,   (L, ∀ ₂, ∃ ₂, τ₆) ,   (L, ∀ ₃, ∃ ₃, τ₆) .

Definition 3.4. A subalgebra of a state monadic residuated lattice (L, ∀ , ∃ , τ) is a nonempty subset of L, closed under ⊔, ⊓ , ⊗ , → , τ, ∀ , ∃ and carrying the induced operations. Example 3.5. The state monadic residuated lattices

({0, c, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₃, ∃ ₃, τ₁),

({0, c, a, b, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₁, ∃ ₁, τ₂),

({0, c, a, b, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₂, ∃ ₂, τ₂),

({0, c, a, b, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₃, ∃ ₃, τ₂),

({0, c, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₃, ∃ ₃, τ₃),

({0, a, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₃, ∃ ₃, τ₄),

({0, b, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₃, ∃ ₃, τ₅),

({0, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₂, ∃ ₂, τ₆),

({0, 1} , ⊓ , ⊔ , ⊗ , → , ∀ ₃, ∃ ₃, τ₆), are subalgebras of state monadic residuated lattices in Example 3.3(3), respectively.

Proposition 3.6. If (L, ∀ , ∃ , τ) is a state monadic residuated lattice, then Ker_∀ (τ) = Ker₀ (τ) ⊆ Ker_∃ (τ).

Proof. We have ∀a ⩽ a ⩽ ∃ a, for all a ∈ L. Since a ∈ Ker_∀ (τ) if and only if ∀τ (a) =1 if and only if τ (a) =1, we get Ker_∀ (τ) = Ker₀ (τ). If a ∈ Ker₀ (τ), then τ (∃ a) = ∃ τ (a) = ∃1 = 1, hence a ∈ Ker_∃ (τ), and hence Ker₀ (τ) ⊆ Ker_∃ (τ). □ The converse of Proposition 3.6 is not true.

Example 3.7. Let (L, ∀ ₁, ∃ ₁, τ₂) be a state monadic residuated lattice in Example 3.3. Then Ker_∃ (τ) = {c, a, b, 1} ⊊ Ker_∀ (τ) = {1}. For an internal state τ we denote Ker (τ) = Ker₀ (τ) = Ker_∀ (τ) and it is called the kernel of τ, and an internal state τ is said to be faithful if Ker (τ) = {1}.

Example 3.8. In Example 3.3, it is clear that (L, ∀ ₁, ∃ ₁, τ₂), is 0-faithful and ∀₁-faithful, but it is not ∃₁-faithful. So Ker (τ₂) = Ker_∀1 (τ₂) = Ker₀ (τ₂) = {1} and τ₂ is faithful. Since Ker_∀3 (τ₄) = Ker₀ (τ₄) = {b, 1} is not faithful. Proposition 3.9. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Then for all a, b ∈ L, we have

(1) τ (1) =1,

(2) τ (¬ a) = ¬ τ (a),

(3) If a ⩽ b, then τ (a) ⩽ τ (b),

(4) τ (a ⊗ b) ⩾ τ (a) ⊗ τ (b). In particular, if a ⊗ b = 0, then τ (a ⊗ b) = τ (a) ⊗ τ (b),

(5) τ (a → b) ⩽ τ (a) → τ (b). In particular, if a, b are comparable, then τ (a → b) = τ (a) → τ (b),

(6) τ² (a) = τ (a), where τ² (a) = τ (τ (a)),

(7) (τ (L) , ∀ , ∃ , τ) is a subalgebra of state monadic residuated lattice (L, ∀ , ∃ , τ),

(8) τ (L) = {a ∈ L ∣ a = τ (a)},

(9) If ord (a)< ∞, then ord (τ (a)) ⩽ ord (a), and in monadic residuated lattices we have τ (a) ∉ Rad (L),

(10) τ (a → b) = τ (a) → τ (b) if and only if τ (b → a) = τ (b) → τ (a),

(11) If τ (L) = L, then τ is identity on L,

(12) If τ is faithful and a < b, then τ (a) < τ (b),

(13) If τ is faithful, then either τ (a) = a or τ (a) and a are not comparable,

(14) If L is linear and τ faithful, then τ (a) = a, for all a ∈ L,

(15) τ ((∀ a ⊗ τ (∀ a))  ⁿ ) ⩾ (τ (∀ a)) ²ⁿ, for n ⩾ 1,

(16) ∀a = 1 if and only if a = 1, and ∃a = 0 if and only if a = 0,

(17) a ∈ Ker_∀ (τ) if and only if ∀a ∈ Ker_∀ (τ),

(18) If a ∈ Ker (τ), then ∃a ∈ Ker_∀ (τ).

Proof. The proof of parts (1)=(14) is similar to [12, Proposition 7].

(15) By (4) and (6) we have τ ((∀ a ⊗ τ (∀ a))  ⁿ ) ⩾ (τ (∀ a ⊗ τ (∀ a)))  ⁿ  ⩾ (τ (∀ a) ⊗ τ τ (∀ a))  ⁿ  = (τ (∀ a)) ²ⁿ.

(16) Form ∀a = 1 we have 1 = ∀ a ⩽ a, that is a = 1. On the other hand ∀1 =1, hence ∀a = 1 if and only if  a = 1. Similarly, ∃a = 0 implies a ⩽ ∃ a = 0. Since ∃0 =0, ∃a = 0 if and only if a = 0.

(17) a ∈ Ker_∀ (τ) if and only if τ (∀ a) =1 if and only if ∀ (τ (∀ a)) =1 if and only if τ (∀∀ a) =1 if and only if ∀a ∈ Ker_∀ (τ) .

(18) Since ∀a ⩽ a ⩽ ∃ a, we have 1 = τ (∀ a) ⩽ τ (∃ a). It follows that τ (∃ a) =1, so τ (∀ ∃ a) =1. Hence ∃a ∈ Ker_∀ (τ). □ Proposition 3.10. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Then the following hold: for all a, b ∈ L,

(i) τ (a ⊓ b) = τ (a) ⊓ τ (b) if and only if τ (a → b) = τ (a) → τ (b) ,

(ii) τ (a ⊔ b) = τ (a) ⊔ τ (b) if and only if τ (a → b) = τ (a) → τ (b) ,

(iii) If τ (a → b) = τ (a) → τ (b), then τ (a ⊗ b) = τ (a) ⊗ τ (b) . Proof. It follows form Proposition 3.9. □ Proposition 3.11. If (L, ∀ , ∃ , τ) is a linearly ordered state monadic residuated lattice, then τ (a ⊗ b) = τ (a) ⊗ τ (b), for all a, b ∈ L.

Proof. Since L is a linearly ordered state monadic residuated lattice, by Proposition 3.9(5) we have τ (a → b) = τ (a) → τ (b), and hence by Proposition 3.10(3), τ (a ⊗ b) = τ (a) ⊗ τ (b). □ The converse of Proposition 3.11 is not true in general, let (L, ∀ ₃, ∃ ₃, τ₁) be a state monadic residuated lattice in Example 3.3. Then τ₁ (a ⊗ b) = τ₁ (a) ⊗ τ₁ (b)

for any a, b ∈ L. But L is not a linearly ordered residuated lattice.

Proposition 3.12. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. The following statements are equivalent: for all a ∈ L,

(1) τ (a) = τ (∀ b) , for some b ∈ L,

(2) τ (a) = τ (∃ b) , for some b ∈ L,

(3) τ (a) = τ (∀ a),

(4) τ (a) = τ (∃ a),

(5) τ (∀ a) = τ (∃ a).

Proof. By (SM7), (SM8) and Proposition 2.5(2), we have

(1) ⇒ (2) If τ (a) = τ (∀ (b)), then τ (a) = ∀ τ (b) = ∃ ∀ τ (b) = τ (∃ ∀ b).

(2) ⇒ (3) If τ (a) = τ (∃ b) , then τ (∀ a) = ∀ τ (a) = ∀ (τ ∃ b) = ∀ ∃ τ (b) = ∃ τ (b) = τ (∃ b) = τ (a) .

(3) ⇒ (4) If τ (a) = τ (∀ a) , then τ (∃ a) = ∃ τ (a) = ∃ (τ ∀ a) = ∃ ∀ τ (a) = ∀ τ (a) = τ (∀ a) = τ (a) .

(4) ⇒ (5) If τ (a) = τ (∃ a) , then τ (∀ a) = ∀ τ (a) = ∀ (τ ∃ a) = ∀ ∃ τ (a) = ∃ τ (a) = τ (∃ a) = τ (a) .

(5) ⇒ (1) If τ (∀ a) = τ (∃ a) , by ∀τ (a) ⩽ τ (a) ⩽ ∃ τ (a), then τ (a) = τ (∀ a). Hence τ (a) = τ (∀ b). □

Proposition 3.13. In every state monadic residuated lattices (L, ∀ , ∃ , τ), the following hold: for all a, b ∈ L,

(1) ∃ (τ (a) → τ (b)) ⩽ ∀ τ (a) → ∃ τ (b),

(2) ∀ (∀ τ (a) → τ (b)) = ∀ τ (a) → ∀ τ (b),

(3) τ (∀ (¬ a)) = ¬ ∃ τ (a) and τ (∃ (¬ a)) ⩽ ¬ ∀ τ (a).

Proof. (1) By Definition 3.1 (SM7), ∀τ (a) = ∀∀ τ (a) and ∀ (∃ τ (b)) = ∃ τ (b). Using Definition 3.1(SM3), we get ∀τ (a) → ∃ τ (b) = ∀ (∀ τ (a)) → ∀ (∃ τ (b)) = ∀ (∀ τ (a) → ∃ τ (b)).

Thus by Proposition 2.5(7),we have ∀τ (a) → ∃ τ (b) = ∃ (∃ τ (a) → ∃ τ (b)). Furthermore, we have τ (a) → τ (b) ⩽ ∀ τ (a) → ∃ τ (b), and hence ∃ (τ (a) → τ (b)) ⩽ ∃ (∀ τ (a) → ∃ τ (b)) = ∀ τ (a) → ∃ τ (b).

(2) The proof is a result of Proposition 3.9(10).

(3) By Definition 2.4(M3), we have ∀ (¬ a) = ¬ (∃ a) . Also, by Proposition 2.5(18), we have ∃ (¬ a) ⩽ ¬ (∀ a), and hence τ (∀ (¬ a)) = τ (¬ (∃ a)), τ (∃ (¬ a)) ⩽ τ (¬ (∀ a)).

By Proposition 2.5(2) and 3.9(2), we have τ (∀ (¬ a)) = τ (¬ (∃ a)) = ¬ (τ (∃ a)) = ¬ (∃ τ (a)),

τ (∃ (¬ a)) ⩽ τ (¬ (∀ a)) = ¬ τ (∀ a) = ¬ ∀ τ (a). □ We can also define state monadic MTL-algebras, state monadic BL-algebras, state monadic IMTL-algebras, state monadic MV-algebras, respectively, which are similar to Definition 3.1, and here hence we omit the exact definition of them.

Remark 3.14. Figure 2 shows that the relationship between state monadic algebras of substructural fuzzy logics completely maintains the relationship between corresponding monadic algebras.

OPEN IN VIEWER

Fig. 2

Relations between state monadic algebras of substructural fuzzy logics.

In this section, we introduce the notion of state monadic filter of a state monadic residuated lattice and characterize the state monadic filter generated by a nonempty subset, and also give some characterization of maximal and prime state monadic filters.

Definition 4.1. A nonempty subset F ⊆ L is called a state monadic filter of (L, ∀ , ∃ , τ) if it satisfies the following conditions:

(F1)  1 ∈ F,

(F2)  a ∈ F and a → b ∈ F imply b ∈ F,

(F3)  ∀a ∈ F, for all a ∈ F,

(F4)  τ (∀ a) ∈ F, for all a ∈ F.

A state monadic filter F of (L, ∀ , ∃ , τ) is called a proper state monadic filter if F ≠ L. A proper state monadic filter F of (L, ∀ , ∃ , τ) is called a maximal state monadic filter if it not strictly contained in any proper state monadic filter. We will denote the set of all state monadic filters of (L, ∀ , ∃ , τ) by F _τ  (L). We also denote the set of all maximal state monadic filters of (L, ∀ , ∃ , τ) by Max _τ  (L) and Rad _τ  (L) = ∩ _{F∈Max τ (L)}F.

Example 4.2. In Example 3.3, F = {b, 1} is a state monadic filter of a state monadic residuated lattice (L, ∀ ₃, ∃ ₃, τ₄). Proposition 4.3. Ker (τ) is a state monadic filter of a state monadic residuated lattice (L, ∀ , ∃ , τ).

Proof. Since Ker (τ) = Ker_∀ (τ) it is known that it is a monadic filter and also a state filter. Combining these together we get that a ∈ L implies ∀a ∈ L (monadic filter) and τ (∀ a) ∈ L (state filter) [or a ∈ F implies τ (a) ∈ L implies ∀ (τ (a)) = τ (∀ a) ∈ F]. Thus Ker (τ) is a state monadic filter. □ Proposition 4.4. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Then the following statements hold:

(i) If Ker (τ) ⊆ B (L), then B (L/Ker (τ)) = B (L)/Ker (τ).

(ii) If a is a nilpotent element of (L, ∀ , ∃), then a/Ker (τ) is a nilpotent element of L/Ker (τ).

Proof. (i) Let Ker (τ) ⊆ B (L). Then B ( L ) / Ker ( τ ) = { e / Ker ( τ ) | e ∈ B ( A ) } = { e / Ker ( τ ) | e ⊔ ¬ e = 1 } = { e / Ker ( τ ) | ( e ⊔ ¬ e ) / Ker ( τ ) = 1 / Ker ( τ ) } = B ( L / Ker ( τ ) ) . (ii) Let a ∈ L be a nilpotent element. Then there exists n ∈ ℕ such that a ⁿ  = 0. So 0/Ker (τ) = a ⁿ /Ker (τ) = (a/Ker (τ))  ⁿ . Hence a/Ker (τ) ∈ L/Ker (τ) is a nilpotent element. □

Remark 4.5. If (F _i ) _i∈I is a family of state monadic filters on a state monadic residuated lattice (L, ∀ , ∃ , τ), then ∩_i∈IF _i is also a state monadic filter. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. For any nonempty set X of L, we denote by [X)  _τ the smallest state monadic filter of (L, ∀ , ∃ , τ) containing X, called the state monadic filter of (L, ∀ , ∃ , τ) generated by X. If X = {a}, then we write [a)  _τ instead of [{a})  _τ . For F ∈ F _τ  (L), obviously [F)  _τ  = F. If F ∈ F _τ  (L) and a ∉ F, denote [F, a)  _τ  : = [F ⊔ {a})  _τ .

Theorem 4.6. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice and X be a nonempty subset of L. Then [X)  _τ  = {b ∈ L | b ≥ (∀ a₁ ⊗ τ (∀ a₁))  ^{n ₁}  ⊗ … ⊗ (∀ a _k  ⊗ τ (∀ a _k ))  ^{n _k} , for some a i ∈ X , n i ∈ ℕ , i ∈ { 1 , . . . , k } , k ≥ 1 } .

Proof. The proof is a consequence of Proposition 3.16 in [17] and Proposition 5.1 in [21]. □ Proposition 4.7. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Then the following hold: for any a, b ∈ L,

(1)    [a)  _τ  = {x ∈ L | x ≥ (∀ a ⊗ τ (∀ a))  ⁿ , n ≥ 1},

(2)   if a ≤ b, then [b)  _τ  ⊆ [a)  _τ ,

(3)    [τ (a))  _τ  ⊆ [a)  _τ ,

(4)    [a)  _τ  = [∀ a ⊗ τ (∀ a))  _τ ,

(5)    [[a)  _τ  ∪ [b)  _τ )  _τ  = [a)  _τ  ⊔ [b)  _τ  = [(∀ a ⊗ τ (∀ a)) ⊗ (∀ b ⊗ τ (∀ b)))  _τ ,

(6)    [a)  _τ  ∩ [b)  _τ  = [(∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)))  _τ .

Proof. The proof is similar to that of Proposition 4.1 in [17]. □ In the following example we show that the converse of part (3) of above proposition is not true.

Example 4.8. Consider Example 3.3:

(1) In (L, ∀ ₃, ∃ ₃, τ₃), we have [a)  _{τ 3}  = {a, 1} ⊈ [τ₃ (a))  _{τ 3}  = [1)  _{τ 3}  = {1}.

(2) In (L, ∀ ₂, ∃ ₂, τ₂) , [a)  _{τ 2}  = [b)  _{τ 2} , but neither a ≤ b, nor b ≤ a hold. Proposition 4.9. Let F, F₁, F₂ ∈ F _τ  (L) and a ∈ L such that a ∉ F. Then the following hold:

(1) [F, a)  _τ  = {x ∈ L | x ≥ f ⊗ (∀ a ⊗ τ (∀ a))  ⁿ , for some f ∈ F, n ≥ 1},

(2) [F₁ ∪ F₂)  _τ  = {x ∈ L | x ≥ f₁ ⊗ f₂, f₁ ∈ F₁, f₂ ∈ F₂}. Proof. The proof is a consequence of Proposition 3.17 in [17] and Proposition 5.1 in [21]. □ Theorem 4.10. Let F ∈ F _τ  (L). Then the following are equivalent:

(i) F is a maximal state monadic filter (L, ∀ , ∃ , τ),

(ii) for any a ∈ L ∖ F, there are f ∈ F and n ≥ 1 such that f ⊗ (τ (∀ a))  ⁿ  = 0,

(iii) for any a ∈ L ∖ F, there is n ≥ 1 such that ¬ (τ (∀ a))  ⁿ  ∈ F. Proof. The proof is a consequence of Proposition 3.18 in [17] and Proposition 5.1 in [21]. □ Proposition 4.11. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Then the following hold:

(i) If F is a (maximal) monadic filter of (τ (L) , ∀ , ∃), then τ^-1 (F) is a (maximal) state monadic filter of (L, ∀ , ∃ , τ),

(ii) If F is a (maximal) state monadic filter of (L, ∀ , ∃ , τ), then τ (F) is a (maximal) monadic filter of (τ (L) , ∀ , ∃).

Proof. The proof is similar to that of Proposition 3.19 in [17]. □ Definition 4.12. A proper state monadic filter P of (L, ∀ , ∃ , τ) is called a prime state monadic filter if for a, b ∈ L such that (∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)) ∈ P, then a ∈ P or b ∈ P. We will denote by Spec _τ  (L) the set of all prime state monadic filters of (L, ∀ , ∃ , τ).

Example 4.13. Consider Example 3.3, in (L, ∀ ₃, ∃ ₃, τ₃), clearly F = {a, 1} is a prime state monadic filter of (L, ∀ , ∃ , τ). Proposition 4.14. Let P be a proper state monadic filter of (L, ∀ , ∃ , τ). Then the following are equivalent:

(i) If P₁, P₂ ∈ F _τ  (L) and P = P₁ ∩ P₂, then P = P₁ or P = P₂,

(ii) P∈ Spec _τ  (L). Proof. (i) ⇒ (ii) Let (∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)) ∈ P, for all a, b ∈ L. Let P₁ = [P, a)  _τ and P₂ = [P, b)  _τ . Clearly, P ⊆ P₁ ∩ P₂. Let x ∈ P₁ ∩ P₂ and u = ∀ a ⊗ τ (∀ a), v = ∀ b ⊗ τ (∀ b). By Proposition 4.9, there are p, q ∈ P and m, n ≥ 1, such that x ≥ p ⊗ u ^m and x ≥ q ⊗ v ⁿ . Then we have x ≥ ( p ⊗ u m ) ⊔ ( q ⊗ v n ) ≥ ( p ⊔ q ) ⊗ ( u m ⊔ q ) ⊗ ( p ⊔ v n ) ⊗ ( u m ⊔ v n ) ≥ ( p ⊔ q ) ⊗ ( u m ⊔ q ) ⊗ ( p ⊔ v n ) ⊗ ( u ⊔ v ) mn . Since (p ⊔ q) , (u ^m  ⊔ q) , (p ⊔ v ⁿ ) , (u ⊔ v)  ^mn  ∈ P,

we get x ∈ P. Hence P = P₁ ∩ P₂, so that P = P₁ or P = P₂, that is a ∈ P or b ∈ P. (ii) ⇒ (i) Let P₁, P₂ ∈ F _τ  (L) such that P = P₁ ∩ P₂. If P ≠ P₁ and P ≠ P₂, then there are a ∈ P₁ ∖ P, b ∈ P₂ ∖ P, so that ∀a ⊗ τ (∀ a) ∈ P₁, ∀ b ⊗ τ (∀ b) ∈ P₂.

Thus (∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)) ∈ P₁ ∩ P₂ = P. It follows that a ∈ P or b ∈ P, which is a contradiction. Thus P = P₁ or P = P₂. □ Proposition 4.15. Let F be a proper state monadic filter of (L, ∀ , ∃ , τ). Then there exists a maximal state monadic filter F₀ of (L, ∀ , ∃ , τ) such that F ⊆ F₀. Proof. Clearly, the set L _F  = {F′ | F′is a proper state monadic filter and F ⊆ F′} is nonvoid and inductively ordered by inclusion. So by Zorn’s lemma, L _F has a maximal element F₀. We prove that F₀ is a maximal state monadic filter of (L, ∀ , ∃ , τ). If F₁ is a proper state monadic filter of (L, ∀ , ∃ , τ) such that F₀ ⊆ F₁, then F₁ ∈ L _F and the maximality of F₀ implies that F₁ = F₀. □ Proposition 4.16. Let a ∈ L, a < 1. Then there is a prime state monadic filter P of (L, ∀ , ∃ , τ), that a ∉ P. Proof. Let F _a  = {F | F is state monadic filter and a ∉ F}. Then {1} ∈ F _a , so F _a ≠ ∅. We know that F _a is inductively ordered and by Zorm’s lemma, there is a maximal element P of F _a . We will prove that P is a prime state monadic filter. Let x, y ∈ L such that (∀ x ⊗ τ (∀ x)) ⊔ (∀ y ⊗ τ (∀ y)) ∈ P and x, y ∉ P. Consider the sets [P, x)  _τ and [P, y)  _τ . It is clear that P is strictly contained in [P, x)  _τ and [P, y)  _τ . The maximality of P implies a ∈ [P, x)  _τ  ∩ [P, y)  _τ . Then there are a₁, b₁ ∈ P and m, n ≥ 1 such that a ≥ a₁ ⊗ (∀ x ⊗ τ (∀ x))  ^m and a ≥ b₁ ⊗ (∀ y ⊗ τ (∀ y))  ⁿ . Similar to Proposition 4.14, we have a ∈ P, which is a contradiction. Hence P is a prime state monadic filter.

□ Proposition 4.17. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice, F ∈ F _τ  (L)) and a ∈ L ∖ F. Then the following hold:

(i) There is P∈ Spec _τ  (L) such that F ⊆ P and a ∉ P.

(ii) F = ∩ _{P∈Spec τ (L),F⊆P}P,

(iii) F = ∩ _{P∈Spec τ (L)}P = {1}.

Proof. (i) Similar as in Proposition 4.16 using L_F,a = {F′|F′ is a proper monadic filter in (L, ∀ , ∃) containing F, a  ∉ F}.

(ii) Clearly, F ⊆ ∩ _{P∈Spec τ (L),F⊆P}P. Conversely, if a ∈ ∩ _{P∈Spec τ (L),F⊆P}P and a ∉ F, then from (i) that there is P∈ Spec _τ  (L) such that F ⊆ P and a ∉ P, which is a contradiction.

(iii) It follows from (i), for F = {1}. □ Proposition 4.18. Let P be a proper state monadic filter of (L, ∀ , ∃ , τ). Then the following are equivalent

(i) P∈ Spec _τ  (L),

(ii) for every a, b ∈ L for which ∀a ⊗ τ (∀ a) , ∀ b ⊗ τ (∀ b) ∈ L ∖ P,there is c ∈ L ∖ P such that ∀a ⊗ τ (∀ a) ≤ c and ∀b ⊗ τ (∀ b) ≤ c,

(iii) if [a)  _τ  ∩ [b)  _τ  ⊆ P, then a ∈ P or b ∈ P. Proof. (i)   ⇔   (ii) Let a, b ∈ L such that ∀a ⊗ τ (∀ a) , ∀ b ⊗ τ (∀ b) ∈ L ∖ P and c = (∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)). If c ∈ P, then a ∈ P or b ∈ P, so ∀a ⊗ τ (∀ a) ∈ P or ∀b ⊗ τ (∀ b) ∈ P, which is a contradiction. Thus c ∈ L ∖ P and ∀a ⊗ τ (∀ a) ≤ c, ∀ b ⊗ τ (∀ b) ≤ c.

Conversely, let a, b ∈ L such that (∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)) ∈ P, a ∉ P and b ∉ P. Then ∀a ⊗ τ (∀ a) , ∀ b ⊗ (τ (∀ b) ∈ L ∖ P. So there is c ∈ L ∖ P such that ∀a ⊗ τ (∀ a) ≤ c and ∀b ⊗ τ (∀ b) ≤ c. Hence (∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)) ≤ c, that is c ∈ P, which is a contradiction.

(i)   ⇔   (iii) By Proposition 4.7 (6), we have [a)  _τ  ∩ [b)  _τ  = [(∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)))  _τ . Thus (∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)) ∈ P and a ∈ P or b ∈ P.

Conversely, if (∀ a ⊗ τ (∀ b)) ⊔ (∀ b ⊗ τ (∀ b)) ∈ P, a, b ∈ L, then [(∀ a ⊗ τ (∀ a)) ⊔ (∀ b ⊗ τ (∀ b)))  _τ  ⊆ P. Hence [a)  _τ  ∩ [b)  _τ  ⊆ P and a ∈ P or b ∈ P, so that P∈ Spec _τ  (L). □

5 Some classes of state monadic residuated lattices

In this section, as an application of state monadic filters, we characterize the simple, semisimple and local state monadic residuated lattices.

Definition 5.1. For a state monadic residuated lattice (L, ∀ , ∃ , τ) and any subalgebra (as state monadic subalgebra) B of (L, ∀ , ∃), if a ∈ B implies τ (a) ∈ B, then B is closed upon τ. Proposition 5.2. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice and let B be a subalgebra of (L, ∀ , ∃) closed upon τ and F ∈ F _τ  (L). Then B ∩ F ∈ [B)  _τ . Proof. It is clear that B ∩ F is nonempty. Since F is an order filter, a ∈ B ∩ F, b ∈ B, b ≥ a implies b ∈ F. But b ∈ B, thus b ∈ F ∩ B. This shows that F ∩ B is an order filter of B. Let a, b ∈ F ∩ B. Then ∀a, ∀ b ∈ F ∩ B (B is a subalgebra and F is a monadic filter), thus ∀a ⊗ ∀ b ∈ B ∩ F (B is a subalgebra, F is a RL-filter). This show that F ∩ B is a monadic filter of B. Finally, since B is closed upon τ, ∀ (as subalgebra) and F is a state monadic filter, then a ∈ F ∩ B implies τ (∀ a) ∈ F ∩ B, thus F ∩ B is a state monadic filter of B. □ It is clear that |F (B) | ≤ |F (L) |,  |Spec(B) | ≤ |Spec(L) | and |Max (B) | ≤ |Max (L) |. So we have | [F)  _τ  (B) | ≤ | [F)  _τ  (L) |,  |Spec _τ  (B) | ≤ |Spec _τ  (L) | and |Max _τ  (B) | ≤ |Max _τ  (L) |

Proposition 5.3. |Max _τ  (L) | = |Max (τ (L)) |. Proof. Clearly, if B is a subalgebra of (L, ∀ , ∃) closed upon τ, then |Max _τ  (B) | ≤ |Max (τ (L)) |. If B = τ (L), then we have |Max _τ  (τ (L)) | ≤ |Max _τ  (L) |. Since in τ (L) the concepts of monadic filter and state monadic filter coincide, Max _τ  (τ (L)) = Max (τ (L)). Thus |Max (τ (L)) | ≤ |Max _τ  (L) |.

Conversely, let F ∈ Max _τ  (L). We must prove that F ∩ τ (L) ∈ Max (τ (L)). If F ∩ τ (L) = τ (L), then 0 ∈ F, which is a contradiction. Thus F ∩ τ (L) is a proper state monadic filter, and so there is a ∈ τ (L) ∖ (F ∩ τ (L)) = τ (L) ∖ F. Since F ∈ Max _τ  (L), by Theorem 4.10, there are f ∈ F and n ≥ 1 such that f ⊗ (τ (∀ a))  ⁿ  = 0. Thus τ (f ⊗ (τ (∀ a))  ⁿ ) =0 and so τ (f) ⊗ (τ (∀ a))  ⁿ  = 0. Since τ (a) = a, τ (f) ⊗ (∀ a)  ⁿ  = 0. We know that τ (f) ∈ F ∩ τ (L) so F ∩ τ (L) ∈ Max (τ (L)). We define Q : Max _τ  (L) ⟶ Max (τ (L)) by Q (F) = F ∩ τ (L). Let F₁, F₂ ∈ Max _τ  (L) such that Q (F₁) = Q (F₂). Suppose that there is a ∈ F₁ ∖ F₂. Since a ∈ F₁, τ (a) ∈ F₁. So τ (a) ∈ F₁ ∩ τ (L) = F₂ ∩ τ (L). Thus τ (a) ∈ F₂. Since a ∉ F₂, from the maximality of F₂, there are n ≥ 1 and f ∈ F₂ such that f ⊗ (∀ τ (a))  ⁿ  = f ⊗ (τ (∀ a))  ⁿ  = 0. But f ⊗ (τ (∀ a))  ⁿ  ∈ F₂ thus 0 ∈ F₂ and F₂ = L, which is a contradiction. Therefore F₁ ⊆ F₂, similarly F₂ ⊆ F₁ so F₁ = F₂. Hence Q is injective and |Max _τ  (L) | ≤ |Max (τ (L)) |. □ Definition 5.4. A monadic residuated lattice (L, ∀ , ∃) is called simple if it has only two monadic filters {1} and (L, ∀ , ∃). A state monadic residuated lattice (L, ∀ , ∃ , τ) is called simple if τ (L) is a simple monadic residuated lattice. A state monadic residuated lattice (L, ∀ , ∃ , τ) is called simple relative to F _τ  (L) if it has only state monadic filters {1} and (L, ∀ , ∃). Example 5.5. Consider Example 3.3, clearly (L, ∀ ₁, ∃ ₁, τ₂) is simple. Proposition 5.6. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice and let (L, ∀ , ∃) be simple (as a monadic residuated lattice), then τ (L) is simple, i.e., (L, ∀ , ∃ , τ) is simple (as a state monadic residuated lattice). Proof. Let F be a state monadic filter of τ (L), F ≠ {1}. By Proposition 4.11(i), the set τ^-1 (F) is a monadic filter of (L, ∀ , ∃). Hence τ^-1 (F) = {1} or τ^-1 (F) = L. If a ∈ F, then τ (a) = a, that is a ∈ τ^-1 (F). Hence F ⊑ τ^-1 (F) = L. It follows that 0 ∈ τ^-1 (F) and τ (0) ∈ F, thus 0 ∈ F, so that F = τ (L).

□ Proposition 5.7. Let (L, ∀ , ∃ , τ) be simple relative to F _τ  (L). Then (L, ∀ , ∃ , τ) is simple. Proof. Let F be a state monadic filter of τ (L), F ≠ {1}. By Proposition 4.11(i), the set τ^-1 (F) is a state monadic filter of (L, ∀ , ∃ , τ). Since F ⊆ τ^-1 (F), τ^-1 (F) ≠ {1}. So τ^-1 (F) = L and 0 ∈ τ^-1 (F). Hence 0 = τ (0) ∈ F and F = τ (L). □ Corollary 5.8. Let (L, ∀ , ∃ , τ) be a simple state monadic residuated lattice relative to F _τ  (L). Then τ is faithful. Proof. We know that Ker (τ) ∈ F _τ  (L) and Ker (τ) ≠ L. Thus Ker (τ) = {1}, and so τ is faithful. □

Theorem 5.9. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Then the following are equivalent

(i) (L, ∀ , ∃ , τ) is simple relative to F _τ  (L),

(ii) (L, ∀ , ∃ , τ) is simple and τ is faithful. Proof. (i) ⇒ (ii) It is clear by Propositions 5.7 and 5.8.

(ii) ⇒ (i) Let F ∈ F _τ  (L), F ≠ {1}. Then τ (F) = F ∩ τ (L) is a monadic filter of τ (L). Since τ (L) is simple, τ (F) = {1} or τ (F) = τ (L). But τ is faithful and F ≠ {1}. It follows that τ (F) ≠ {1}, hence τ (F) = τ (L). Thus 0 ∈ τ (F) and 0 ∈ F. Therefore F = L. □ Definition 5.10. A monadic residuated lattice (L, ∀ , ∃) is called semisimple if Rad (L) = {1}. A state monadic residuated lattice (L, ∀ , ∃ , τ) is called semisimple if τ (L) is semisimple, that is Rad (τ (L)) = {1}. A state monadic residuated lattice (L, ∀ , ∃ , τ) is called semisimple relative to F _τ  (L) if Rad _τ  (L) = {1} (the intersection of all maximal state monadic filters). Proposition 5.11. Rad (τ (L)) = τ (Rad _τ  (L)) . Proof. Let a ∈ τ (Rad _τ  (L)). Then a = τ (b) , b ∈ Rad _τ  (L). Let M be a maximal state monadic filter of τ (L). By Proposition 4.11(i), τ^-1 (M) is a maximal state monadic filter of (L, ∀ , ∃ , τ). Then b ∈ τ^-1 (M), τ (b) ∈ M. So a ∈ M and a ∈ Rad (τ (L)). Conversely, let a ∈ Rad (τ (L)) and M be a maximal state monadic filter of (L, ∀ , ∃ , τ). By Proposition 4.11(ii), τ (M) is a maximal state monadic filter of τ (L). So a ∈ τ (M) = M ∩ τ (L) and a ∈ M. Thus a ∈ Rad _τ  (L). Since a = τ (a), a ∈ τ (Rad _τ  (L)). □ Proposition 5.12. (L, ∀ , ∃ , τ) is semisimple and τ is faithful iff (L, ∀ , ∃ , τ) is semisimple relative to F _τ  (L). Proof. By Proposition 5.10, τ (Rad _τ  (L)) = Rad (τ (L)) = τ ({1}) = {1}. So Rad _τ  (L) ⊆ Ker (τ) = {1} and Rad _τ  (L) = {1}.

Conversely, Rad (τ (L)) = τ (Rad _τ  (L)) = τ ({1}) = {1},so (L, ∀ , ∃ , τ) is semisimple. Let a ∈ L such that τ (a) =1. If a ≠ 1, then a ∉ Rad _τ  (L). There is a maximal state monadic filter M so that a ∉ M. By Theorem 4.10, there are f ∈ M and m, n ≥ 1 such that 0 = f ⊗ (τ (∀ a))  ⁿ  = f ⊗ (∀ (τ (a)))  ⁿ  = f. Therefore f = 0 and 0 ∈ M, a contradiction. So a = 1 and τ is faithful. □ Definition 5.13. A monadic residuated lattice (L, ∀ , ∃) is said to be local iff has exactly one maximal monadic filter. Definition 5.14. A state monadic residuated lattice (L, ∀ , ∃ , τ) is called local if τ (L) is local. A state monadic residuated lattice (L, ∀ , ∃ , τ) is called local relative to F _τ  (L) if it has only one maximal state monadic filter.

Example 5.15. In Example 3.3, (L, ∀ ₁, ∃ ₁, τ₂) is local. {1} is the only maximal state monadic filter of τ₂ (L) = L. Theorem 5.16. (L, ∀ , ∃ , τ) is local relative to F _τ  (L) iff (L, ∀ , ∃ , τ) is local. Proof. Let F be the only maximal state monadic filter of (L, ∀ , ∃ , τ). We must show that τ (F) is the only maximal state monadic filter of τ (L). If τ (F) = τ (L), then 0 ∈ τ (F) and 0 ∈ F, a contradiction. So τ (F) is an proper monadic filter of τ (L). Let G be a monadic filter of τ (L) , G ≠ τ (L) and a ∈ G. By Proposition 4.11(i), we have τ^-1 (G) is a state monadic filter of (L, ∀ , ∃ , τ). Let τ^-1 (G) = L. Then 0 ∈ τ^-1 (G) and 0 ∈ G, a contradiction. So τ^-1 (G) is an proper monadic filter of (L, ∀ , ∃ , τ). Thus τ^-1 (G) ⊆ F. Since ∀a = τ^-1 (G) ∈ G, ∀a ∈ τ^-1 (G) so ∀a ∈ F. We know that a = τ (a), so ∀a ∈ τ (F). Thus G ⊑ τ (F) and (L, ∀ , ∃ , τ) is local. Conversely, let G be the only maximal state monadic filter of τ (L). By Proposition 4.11(i), τ^-1 (G) is a maximal state monadic filter of (L, ∀ , ∃ , τ). We prove that τ^-1 (G) is the only maximal state monadic filter. Let F ∈ F _τ  (L) , F ≠ L. Then F ∩ τ (L) = τ (F) is a proper monadic filter of τ (L). Thus F ∩ τ (L) ⊆ G. If a ∈ F, then τ (∀ a) ∈ F ∩ τ (L) ⊆ G. So ∀a ∈ τ^-1 (G), which implies a ∈ τ^-1 (G), thus F ⊆ τ^-1 (G). □ Proposition 5.17. Let F be a proper state monadic filter of (L, ∀ , ∃ , τ). Then F ⊆ M _τ  (L) : = {a ∈ L | ord (τ (∀ a)) = ∞}. Proof. Let a ∈ F, that is τ (∀ a) ∈ F. If ord (τ (∀ a)) is finite, then there is m ∈ ℕ such that 0 = (τ (∀ a))  ^m  ∈ F, which is a contradiction. Therefore F ⊆ M _τ  (L).

□

Theorem 5.18. Let (L, ∀ , ∃ , τ) be a state monadic residuated lattice. Then (L, ∀ , ∃ , τ) is local iff M _τ  (L) is a state monadic filter of (L, ∀ , ∃ , τ). In this case M _τ  (L) is the only maximal state monadic filter of (L, ∀ , ∃ , τ). Proof. Let (L, ∀ , ∃ , τ) be local. Then (L, ∀ , ∃ , τ) has only a maximal state monadic filter F. By Proposition 5.16, F ⊆ M _τ  (L). If a ∈ M _τ  (L) and a ∉ F, then [a)  _τ  ⊆ F. If [a)  _τ is a proper state monadic filter, then by Proposition 4.15 there is a maximal state monadic filter G containing [a)  _τ . Since F is the only maximal state monadic filter in (L, ∀ , ∃) we have that [a)  _τ  ⊆ G = F, which is a contradiction. Thus [a)  _τ is not proper, i.e., [a)  _τ  = L so 0 ∈ [a)  _τ . By Proposition 4.7(1) there is n ≥ 1 such that 0 ≥ (∀ a ⊗ τ (∀ a))  ⁿ , so 0 = τ (0) ≥ τ (∀ a ⊗ τ (∀ a))  ⁿ  ≥ ⋯ ≥ (τ (∀ a)) ²ⁿ. Thus (τ (∀ a)) ²ⁿ = 0, that is a ∉ M _τ  (L), which is a contradiction. So M _τ  (L) ⊆ F and M _τ  (L) = F. Therefore M _τ  (L) is a state monadic filter of (L, ∀ , ∃ , τ). Conversely, let M _τ  (L) is a state monadic filter of (L, ∀ , ∃ , τ) and F be a proper state monadic filter of (L, ∀ , ∃ , τ). By Proposition 5.16, F ⊆ M _τ  (L), so M _τ  (L) is the only maximal state monadic filter of (L, ∀ , ∃ , τ). Hence (L, ∀ , ∃ , τ) is local relative to F _τ  (L) and so (L, ∀ , ∃ , τ) is local. □

6 Conclusions and future research topics

Motivated by previous research on monadic and state residuated lattices, we further investigated algebraic properties of state monadic residuated lattices. In this paper, we introduced the notion of state monadic residuated lattice and study some of their algebraic properties and discuss the relationship between state monadic residuated lattice and other state monadic fuzzy logical algebras. Then we studied state monadic filter of state monadic residuated lattices and give some characterizations of them. Since the above topics are of current interest, we suggest further directions of research:

Introducing and studying state polyadic residuated lattices, namely generalizations of state monadic residuated lattices given by polyadic structures.