Construction of some algebras of logic by using fuzzy ideals in mv-modules

Abstract

We study the lattice structure of fuzzy A-ideals in an mv-module M (fai (M), symbolically) and show that it is a complete Heyting lattice and so the set of its pseudocomplements forms a Boolean algebra. In the sequel, the properties of fuzzy congruences in an mv-module are investigated and using them some structural theorems are stated and proved. Finally, it is proved that fai (M) can be embedded into the lattice of fuzzy congruences.

Keywords

mv-module fuzzy A-ideal fuzzy congruence distributive lattice

1 Introduction

The concept of an mv-algebra was introduced by Chang [8], as an algebraic counterpart of ℵ₀-valued Łukasiewicz propositional calculus. After that many studies were done on mv-algebras by several researchers. For a complete information about the foundation of mv-algebras see [9]. In [20], Rachůnek studied the connection between mv-algebras and (bounded) DRℓ-semigroups and proved that they are equivalent. In [13], Di Nola et al. proved that perfect mv-algebras are categorically equivalent to abelian ℓ-groups. In [5], Belluce et al. studied mv-algebras from algebraic geometry point of view. Dvurečenskij et al. introduced product mv-algebras (briefly, pmv-algebra) [10]; i.e., an mv-algebra with a product defined on the whole mv-algebra, and proved that product mv-algebras and associative unital ℓ-rings are categorically equivalent. So, it is quite natural to consider modules over pmv-algebras, generalizing the divisible mv-algebras and the mv-algebras obtained from Riesz spaces, and to prove natural equivalence theorems. Hence, the notion of mv-module was introduced by Di Nola et al. [11], see also [4 , 26]. Among all respects, ideal theory plays a crucial role in studying these algebras; they correspond to the sets of provable formulas of related logic. Then, the notion of an ideal of an mv-algebra was introduced by Chang [8] and the notion of an A-ideal in an mv-module M, where A is a pmv-algebra was introduced by Di Nola et al. (see [11]). They proved that any A-ideal induces a congruence in M such that the congruence classes together with those operations induced from M form an mv-module. Up to now, mv-modules have studied by many researchers (see [3 , 22–24]).

To model uncertain problems in real life, many approaches have introduced. One of the most important tools is fuzzy set theory, introduced by Zadeh [27]. After that many researchers applied it to several branches of mathematics such as group theory, ring theory, topology, lattice theory, automata theory, and so on. In related to algebras of logic specially mv-algebras, many authors have studied these algebras and published interesting works, say [17, 18]. Forouzesh [15] introduced (prime) fuzzy A-ideals of mv-modules and obtained some characterizations. Bakhshi [1] also introduced the notion of L-fuzzy prime and L-fuzzy primary ideals of product mv-algebras and mv-modules and gave some related properties.

In this paper, we are going to investigate more properties of mv-modules from lattice theory and fuzzy set theory point of view, based on A-ideals. To do this, first we give an equivalent characterization of fuzzy A-ideals introduced in [15] and obtain some new results. In the sequel, we prove that fuzzy A-ideals form a complete Heyting lattice. Next, fuzzy congruences on an mv-module are introduced and several properties are obtained. Also, some structural results are obtained. In the third part, we give an embedding from fai (M) into the lattice of fuzzy congruences of M.

2 Preliminaries

In this section, we collect some useful information from the literature (see [8 , 16], for more details).

An algebra (M ; , ^★, 0) of signature (2,1,0) is called an mv-algebra if (M ; , 0) is a commutative monoid and for every a, b ∈ M, (a^★) ^★ = a, 0^★a = 0^★, and (a^★ b) ^★ b = (b^★ a) ^★ a. mv-algebra M with the binary relation ≼ defined as x ≼ y ⇔ x^★ y = 0^★ is a partially ordered set.

In an mv-algebra M, the operations ‘’, ‘⊓’ and ‘⊔’ are defined as follows: $\begin{matrix} x y & = & (x^{★} y^{★})^{★} \\ x ⊓ y & = & x (y x^{★}) \\ x ⊔ y & = & x (y x^{★}) \end{matrix}$ From the definitions, it is easy to verify that is commutative and associative and that 0 is a neutral element; i.e., x 0 = x, for all x ∈ M.

A nonempty subset I of mv-algebra (M ; , ^★, 0) is called an ideal if (i) x, y ∈ I implies that xy ∈ I and (ii) whenever a, b ∈ M are such that a ≼ b and b ∈ I, then a ∈ I. Assume that (M ; , ^★, 0) is an mv-algebra and + is a partial binary operation on M defined by a + b = a b provided that a ≼ b^★. Then the structure (M ; , • , ^★, 0) is called a pmv-algebra if (i) whenever a + b is defined in M, then a • c + b • c and c • a + c • b are defined and are equal to (a + b) • c and c • (a + b), respectively, (ii) (M ; • , 0, ≼) is a partially ordered monoid. mv-algebra (M ; , ^★, 0) is called an mv-module over pmv-algebra (A ; , • , ^★, 0) if there exists a function A × M→M with (α, a) ↦ αa such that (i) whenever x + y is defined in M, αx + αy is defined and α (x + y) = αx + αy, (ii) if α + β is defined in A, then αx + βx is defined in A and (α + β) x = αx + βx, (iii) (α • β) x = α (βx). M is called a unital mv-module if A has an identity 1_A and 1_Ax = x, for any x ∈ M.

Proposition 2.1. Let M be an mv-module over pmv-algebra A, x, y, ∈ M and α, β ∈ A. Then

x ≼ y implies αx ≼ αy.

α ≼ β implies αx ≼ βx.

α (x y) ≼ αx αy.

In mv-modules, homomorphisms, epimorphisms, monomorphisms, the kernel of a homomorphism, and congruences are introduced naturally. A nonempty subset I of M is called an A-ideal if it is an ideal of M and for all α ∈ A and a ∈ I, αa ∈ I. It must be noted that the kernel of any homomorphism is an A-ideal.

Let X be a nonempty set, [0, 1] the real unit interval, and let [0, 1] ^X denotes the set of all fuzzy subsets of X. We mention that any element of [0, 1] ^X×X is called a fuzzy relation in X; we denote the set of fuzzy relations in X by FR (X). For σ, υ ∈ [0, 1] ^X, σ and υ are said to have the same tip if $sup_{x \in X} σ (x) = sup_{x \in X} υ (x)$ . We say that X has the tip property if all fuzzy subsets of X have the same tip.

Let (L ; ⊓ , ⊔ , ≤) be a complete lattice. An element a ∈ L is said to be compact if whenever a ≤ ⨆ A in L, we have a ≤ ⨆ A₁, for some finite subset A₁ of A. L is said to be algebraic if it is complete and any element of L is a join of compact elements of L. L is called Brouwerian if ⊓ is distributive over the arbitrary joins. Homomorphisms and monomorphisms (or embeddings) are defined algebraically; i.e., any function that preserves joins and meets.

For nonempty set A, a mapping C : 2^A→2^A with three properties

X⊂C (X),

X⊂Y implies that C (X) sC (Y),

C (C (X)) = C (X)

is called a closure operator. If C (X) = X, X is said to be closed. Also, if C (X) = ⋃ {C (Y) : YsX isfinite}, for X⊂A, C is said to be algebraic.

In this paper, M will be denoted an mv-module over pmv-algebra A, unless otherwise stated.

3 Construction of some algebras of logic by using fuzzy A-ideals

Equivalent definitions of fuzzy A-ideals are given and some characterizations of them such as the properties of the lattice of them are obtained. Next, the notion of a fuzzy congruence is introduced and by using it we construct a new mv-module structure.

3.1 Fuzzy A-ideals

Let σ ∈ [0, 1] ^M and consider the following properties:

(FI1) σ (0) ≥ σ (x).

(FI2) σ (y) ≥ σ (x) ⊓ σ (y x^★).

(FI3) σ (αx) ≥ σ (x), for α ∈ A.

(FI4) σ (a b) ≥ σ (a) ⊓ σ (b).

(FI5) a ≼ b ⇒ σ (a) ≥ σ (b).

(FI6) z^★ y x = 0^★ ⇒ σ (z) ≥ σ (x) ⊓ σ (y).

(FI7) z ≼ x y ⇒ σ (z) ≥ σ (x) ⊓ σ (y).

According to [15], a fuzzy A-ideal of mv-module M is a fuzzy subset of M satisfying (FI1), (FI2), and (FI3). The set of all fuzzy A-ideals of M will be denoted by fai (M). Theorem 3.1 gives some equivalent conditions for a fuzzy A-ideal.

Theorem 3.1. Let σ ∈ [0, 1] ^M. Then, the following are equivalent:

σ ∈ fai (M).

σ satisfies (FI3), (FI4), and (FI5).

σ satisfies (FI3) and (FI6).

σ satisfies (FI3) and (FI7).

Proof. (1) Ra (2) Assume that a ≼ b, for a, b ∈ M. Then a^★ b = 0^★ and so a b^★ = 0. Now, by (FI2) and then (FI1) we have $σ (a) \geq σ (b) ⊓ σ (a b^{★}) = σ (b) ⊓ σ (0) = σ (b),$ proving (FI5). To prove (FI4), taking x = a and y = a b in (FI2) and then using (FI5) we get $\begin{matrix} σ (a b) & \geq & σ (a) ⊓ σ ((a b) a^{★}) \\ = & σ (a) ⊓ σ (a^{★} ⊓ b) \\ \geq & σ (a) ⊓ σ (b) . \end{matrix}$

(2) Ra (3) Let x, y, z ∈ M such that z^★ y x = 0^★. Then z ≼ y x and so by (FI5) and then (FI4) we get that σ (z) ≥ σ (y x) ≥ σ (y) ⊓ σ (x), proving (FI6).

(3) ⇔ (4) Straightforward.

(4) Ra (1) Let x, y ∈ M. From $y ≼ x ⊔ y = x (y x^{★}),$ by (FI7) we get that σ (y) ≥ σ (x) ⊓ σ (y x^★), proving (FI2). Taking z = 0 and y = x, in (FI7), we get σ (0) ≥ σ (x), proving (FI1). Hence, σ ∈ fai (M).□

Notation. The functions x ↦ 0 and x ↦ 1, which are obviously fuzzy A-ideals of M will be denoted by zero and unit, respectively.

Example 3.2. [15] Let M = A = 2^{1,2}. Then A is a pmv-algebra, where : =∪, • : =∩ and M is an mv-module over A together with (α, X) ↦ α ∩ X. We define the mapping σ by σ (∅) = t and for all x ∈ M \ {∅}, σ (x) = s, where 0 ≤ s < t < 1. It is routine to verify that σ is a fuzzy A-ideal of M.

Example 3.3. Let ℝ be the real line, M₂ (ℝ) the ring of 2-square matrices over ℝ, zero the zero matrix, and $v = (\begin{matrix} 1 / 2 & 1 / 2 \\ 1 / 2 & 1 / 2 \end{matrix}) .$ If we define the ordering relation on M₂ (ℝ) as $X = (x_{ij})_{i, j = 1, 2} \geq 0 LR x_{ij} \geq 0,$ then A = Γ (M₂ (ℝ) , v) is a pmv-algebra and M = Γ (ℝ², u), where u = (1, 1) is an A-module (see [11]). Let σ be a fuzzy subset of M defined by μ ((0, 0)) = t and μ ((x, y)) = s, for all (x, y) ∈ ℝ² \ {(0, 0)}, where 0 ≤ s < t ≤ 1. Obviously, μ is a fuzzy A-ideal of M. Moreover, every non-constant fuzzy ideal of M is of the given form. To see this, assume that a, b ∈ ℝ be such that at least one of them differ from 0. For convenience, assume that a ≠ 0. From (FI3) we have $σ ((\begin{matrix} 1 \\ 0 \end{matrix})) = σ ((\begin{matrix} 1 / a & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix} a \\ 0 \end{matrix})) \geq σ ((\begin{matrix} a \\ 0 \end{matrix}))$ Similarly, we can deduce that $σ ((\begin{matrix} a \\ 0 \end{matrix})) \geq σ ((\begin{matrix} 1 \\ 0 \end{matrix}))$ Moreover, if b also differ from 0, we can deduce that $σ ((\begin{matrix} a \\ b \end{matrix})) = σ ((\begin{matrix} 1 \\ 1 \end{matrix}))$ Therefore, for any (a, b) , (x, y) ∈ ℝ² \ {(0, 0)} we must have σ ((a, b)) = σ ((x, y)). Considering (FI1), we deduce that if σ is a non-constant fuzzy A-ideal of M, σ ((0, 0)) > σ ((a, b)), for all (a, b) ∈ ℝ² \ {(0, 0)} and so σ must be two-valued.

Remark 3.4. We observe that Faid(M) is closed with respect to the arbitrary intersections. Hence, for Σ, a family of fuzzy A-ideals of M, the least fuzzy A-ideal of M containing Σ exists; we call it the fuzzy A-ideal generated by Σ, denoted by Fg (∪ Σ). As usual, when Σ = {σ} is singleton we just write Fg (σ). Forouzesh [15] proved that the fuzzy A-ideal generated by σ ∈ [0, 1] ^M is characterized as

$\begin{matrix} 〈 σ 〉 (x) = ⨆ {σ (i_{1}) ⊓ \dots ⊓ \\ σ (i_{r}) ⊓ σ (a_{i + 1} i_{r + 1}) ⊓ \dots \\ σ (a_{k} i_{k}) : x ≼ i_{1} \dots i_{r} a_{r + 1} i_{r + 1} \\ \dots a_{k} i_{k}, \\ k \in ℕ, i_{1}, \dots, i_{k} \in M, a_{r + 1}, \dots, a_{k} \in A} \end{matrix}$ (3.1) Another characterization of Fg (σ) is as follows.

Proposition 3.5. Let σ ∈ [0, 1] ^M. Then

$\begin{matrix} Fg (σ) (x) \\ = ⨆ {σ (i_{1}) ⊓ \dots ⊓ σ (i_{r}) : x ≼ i_{1} \dots \\ i_{r}, r \in ℕ, i_{1}, \dots, i_{r} \in M} \end{matrix}$ (3.2) is the least fuzzy A-ideal of M which contains σ_α’s. If M is unital, (3.1) and (3.2) are equivalent.

Proof. We shall prove that Fg (σ) satisfies (FI3), (FI4) and (FI5). For x, y ∈ M and ɛ > 0 arbitrary, from x ≼ x, we always can find n ∈ ℕ and i₁, i₂, …, i_n ∈ M such that $x ≼ i_{1} i_{2} \dots i_{n}$ (3.3) and $Fg (σ) (x) < ɛ + (σ) (i_{1}) ⊓ (σ) (i_{2}) ⊓ \dots ⊓ (σ) (i_{n})$ (3.4) Similarly, there exists m ∈ ℕ and j₁, j₂, …, j_m ∈ M such that $y ≼ j_{1} j_{2} \dots j_{m}$ (3.5) and $Fg (σ) (y) < ɛ + (σ) (j_{1}) ⊓ (σ) (j_{2}) ⊓ \dots ⊓ (σ) (j_{m})$ (3.6) Now, (3.3) and (3.5) imply that $x y ≼ i_{1} \dots i_{n} j_{1} \dots j_{m}$ (3.7) Combining (3.4), (3.6) and (3.7) we get that $\begin{matrix} Fg (σ) (x y) = ⨆ {σ (z_{1}) ⊓ \dots ⊓ σ (z_{r}) : \\ x y ≼ z_{1} \dots z_{r}, r \in ℕ, z_{1}, \dots, z_{r} \in M} \\ \geq & σ (i_{1}) ⊓ \dots ⊓ σ (i_{n}) ⊓ σ (j_{1}) ⊓ \dots ⊓ σ (j_{m}) \\ > & (Fg (σ) (x) - ɛ) ⊓ (Fg (σ) (y) - ɛ) \\ = & Fg (σ) (x) ⊓ Fg (σ) (y) - ɛ, \end{matrix}$ whence Fg (σ) (x y) ≥ Fg (σ) (x) ⊓ Fg (σ) (y), proving (FI4).

To prove (FI5), let x ≼ y. Similar to the proof of the previous step we have $Fg (σ) (y) < ɛ + σ (i_{1}) ⊓ σ (i_{2}) ⊓ \dots ⊓ σ (i_{n})$ (3.8) for some n ∈ ℕ and i₁, i₂, …, i_n ∈ M such that y ≼ i₁ i₂ ⋯ i_n. From x ≼ y it follows that $x ≼ i_{1} i_{2} \dots i_{n}$ (3.9) and combining (3.8) and (3.9) we get $\begin{matrix} Fg (σ) (x) = ⨆ {σ (j_{1}) ⊓ σ (j_{2}) ⊓ \dots ⊓ σ (j_{r}) : \\ x ≼ j_{1} j_{2} \dots j_{r}, r \in ℕ, j_{1}, \dots, j_{r} \in M} \\ \geq & σ (i_{1}) ⊓ σ (i_{2}) ⊓ \dots ⊓ \cup σ (i_{n}) \\ > & Fg (σ) (y) - ɛ \end{matrix}$ proving Fg (σ) (x) ≥ Fg (σ) (y).

To prove (FI3), let x ∈ M, a ∈ A and ɛ > 0. Similar to the first part of the proof we can find a natural number n and i₁, i₂, …, i_n ∈ M such that x ≼ i₁ i₂ ⋯ i_n and $Fg (x) < ɛ + σ (i_{1}) ⊓ σ (i_{2}) ⊓ \dots ⊓ σ (i_{n})$ From Proposition 2 it follows that $ax ≼ {ai}_{1} a i_{2} \dots {ai}_{n},$ whence we conclude that $\begin{matrix} Fg (σ) (ax) & \geq & σ ({ai}_{1}) ⊓ σ ({ai}_{2}) ⊓ \dots ⊓ σ ({ai}_{n}) \\ \geq & σ (i_{1}) ⊓ σ (i_{2}) ⊓ \dots ⊓ σ (i_{n}) \\ > & Fg (σ) (x) - ɛ \end{matrix}$ Since ɛ is arbitrary it follows that Fg (σ) (ax) ≥ Fg (σ) (x). Thus, Fg (σ) is a fuzzy A-ideal of M. Now, let β be a fuzzy A-ideal of M containing σ. Then $\begin{matrix} Fg (σ) (x) = ⨆ {σ (i_{1}) ⊓ σ (i_{2}) ⊓ \dots ⊓ σ (i_{r}) : \\ x ≼ i_{1} i_{2} \dots i_{r}, r \in ℕ, i_{1}, . . ., i_{r} \in M} \\ \leq & ⨆ {β (i_{1}) ⊓ β (i_{2}) ⊓ \dots ⊓ β (i_{r}) : \\ x ≼ i_{1} i_{2} \dots i_{r}, r \in ℕ, i_{1}, \dots, i_{r} \in M} \\ \leq & ⨆ {β (i_{1} i_{2} \dots i_{r}) : \\ x ≼ i_{1} i_{2} \dots i_{r}, r \in ℕ, i_{1}, . . ., i_{r} \in M} \\ \leq & β (x), \end{matrix}$ proving that Fg (σ) is the least fuzzy A-ideal of M containing σ.

For the last part, it is enough we observe that if M is unital so every element x of M can be written as ax, for some a ∈ A. Hence the inequality $x ≼ i_{1} \dots i_{r} .$ holds if and only if the inequality $x ≼ i_{1} \dots i_{k} a_{k + 1} x_{k + 1} \dots a_{r} x_{r} .$ holds, where a_j ∈ A and x_j ∈ M, for j = k + 1, …, r.□

Theorem 3.6. The mapping C : 2^FI(M)ra2^FI(M) defined by C (A) = Fg (∪ A) is an algebraic closure operator.

Proof. Let X⊂FI (M). Then X⊂ ⋃ X⊂Fg (∪ X) = C (X). Now, let $\begin{matrix} X = {σ_{α} : α \in I, σ_{α} \in FI (M)} \\ Y = {σ_{β} : β \in J, σ_{β} \in FI (M)} \end{matrix}$ and X⊂Y. Also, let x ≼ i₁ i₂ ⋯ i_r, where r ∈ ℕ and i₁, . . . , i_r ∈ X. Obviously, $\begin{matrix} C (X) (x) = ⨆ {(\cup σ_{β}) (i_{1}) ⊓ \dots ⊓ (\cup σ_{β}) (i_{r}) : \\ x ≼ i_{1} i_{2} \dots i_{r}, r \in ℕ, i_{1}, . . ., i_{r} \in M} \\ \geq & ⨆ {(\cup σ_{α}) (i_{1}) ⊓ \dots ⊓ (\cup σ_{α}) (i_{r}) : \\ x ≼ i_{1} i_{2} \dots i_{r}, r \in ℕ, i_{1}, . . ., i_{r} \in M} \\ = & C (Y) (x) . \end{matrix}$ Next, for X⊂FI (M) and x ∈ M we have $\begin{matrix} C (C (X)) (x) = Fg (\cup C (X)) (x) \\ = Fg (\cup Fg (\cup X)) (x) \\ = & Fg (Fg (\cup X)) (x) = ⨆ {(Fg (\cup X) (i_{1}) ⊓ \dots \\ ⊓ (Fg (\cup X) (i_{r}) : x ≼ i_{1} \dots i_{r}, r \in ℕ, i_{1}, \\ \dots, i_{r} \in M} ≼ ⨆ {Fg (\cup X) (x)} = Fg (\cup X) (x) \\ = & C (X) (x) \end{matrix}$ Obviously, C (X) sC (C (X)). Thus C is a closure operator.

The second part follows from the fact that for any nonempty set A of fuzzy A-ideals we have ⋃A = ⋃ {∪ X : X∈A isfinite}. Indeed, for any x ∈ M, $(⋃ A) (x) = ⨆ {(\cup X) (x) : X∈A is finite}$ and so for any i₁, …, i₂ ∈ M such that x ≼ i₁ ⋯ i_r, $\begin{matrix} (\cup A) (i_{1}) ⊓ \dots ⊓ (\cup A) (i_{r}) = \\ ⨆ (\cup X) (i_{1}) ⊓ \dots ⊓ ⨆ (\cup X) (i_{r}) = \\ ⨆ {(\cup X) (i_{1}) ⊓ \dots ⊓ (\cup X) (i_{r})} . \end{matrix}$ Hence Fg (∪ A) (x) = ⋃ {Fg (∪ X) : X∈A isfinite}. □

From Theorem 3.6 and [21 Theorem I.5.5] it follows that

Corollary 3.7. FI (M) is an algebraic lattice in which the compact elements are precisely the fuzzy A-ideals Fg (A), where A is a finite set of fuzzy A-ideals of M.

Corollary 3.8. Let A be the set of all compact elements of FI (M). Then there is an algebraic closure operator C on A such that FI (M) is isomorphic to the lattice of closed subsets of A.

Proof. Assume that C : 2^Ara2^A is defined by C (X) = {σ ∈ A : σs ⨆ X}. Since FI (M) is algebraic, C is well-defined. The remain follows from [21 Theorem I.5.8].□

Theorem 3.9. The structure FI (M) = (FI (M) ; ⊓ , ⊔), where for all σ, ν ∈ FI (M),

$σ ⊓ ν = σ \cap ν σ ⊔ ν = [σ \cup ν]$ is a complete Brouwerian lattice.

Proof. We shall prove that σ ⊓ (⨆ ν_α) ≼ ⨆ (σ ⊓ ν_α). The converse inequality always holds. Let x ∈ M and ɛ > 0 be arbitrary. Then x ≼ a₁ a₂ ⋯ a_n and $\begin{matrix} (⨆ ν_{α}) (x) < ɛ + (\cup ν_{α}) (a_{1}) ⊓ \dots ⊓ (\cup ν_{α}) (a_{n}) \end{matrix}$ for some n ∈ ℕ and a₁, a₂, …, a_n ∈ M. By the definition of ∪ν_α we get (∪ ν_α) (a_i) < ν_{α_i} (a_i) + ɛ, for at least one α_i. Obviously, we can assume that

$\begin{matrix} (\cup ν_{α}) (a_{1}) & < & ν_{α_{1}} (a_{1}) + ɛ \\ (\cup ν_{α}) (a_{2}) & < & ν_{α_{2}} (a_{2}) + ɛ \\ ⋮ \\ (\cup ν_{α}) (a_{n}) & < & ν_{α_{n}} (a_{n}) + ɛ . \end{matrix}$

Hence, $(\cup ν_{α}) (x) < n ɛ + ν_{α_{1}} (a_{1}) ⊓ \dots ⊓ ν_{α_{n}} (a_{n})$ and so $\begin{matrix} σ ⊓ (\cup ν_{α}) (x) & < & n ɛ + (σ (x) ⊓ ν_{α_{1}} (a_{1})) ⊓ \dots ⊓ \\ (σ (x) ⊓ ν_{α_{n}} (a_{n})) . \end{matrix}$ Now, let $\begin{matrix} b_{n}^{★} & = & a_{1} a_{2} \dots a_{n - 1} x^{★}, \\ b_{n - 1}^{★} & = & a_{1} a_{2} \dots a_{n - 2} b_{n} x^{★}, \\ ⋮ \\ b_{2}^{★} & = & a_{1} b_{n} b_{n - 1} \dots b_{3} x^{★}, \\ b_{1}^{★} & = & b_{n} b_{n - 1} \dots b_{2} x^{★} . \end{matrix}$ We observe that $b_{1} b_{2} \dots b_{n} x^{★} = b_{1} b_{1}^{★} = 0^{★}$ whence $x ≼ b_{1} b_{2} \dots b_{n} .$ (3.10) On the other hand, $b_{n}^{★} a_{n} = a_{1} a_{2} \dots a_{n} x^{★} = 0^{★}$ , whence b_n ≼ a_n. Also, $\begin{matrix} b_{1}^{★} a_{1} = & a_{1} b_{n} \dots b_{3} b_{2} x^{★} \\ = b_{2}^{★} b_{2} = 0^{★} \end{matrix}$ and so b₁ ≼ a₁. For i = 2, …, n - 1 we have $\begin{matrix} b_{i}^{★} a_{i} = & a_{1} \dots a_{i} b_{n} b_{n - 1} \dots \\ b_{i + 1} x^{★} = 0^{★} . \end{matrix}$ Thus b_i ≼ a_i, for each i ∈ {1, 2, …, n}, whence ν_{α_i} (a_i) ≤ ν_{α_i} (b_i). Obviously, $b_{i}^{★} x = 0^{★}$ , whence b_i ≼ x and so σ (x) ≤ σ (b_i), for all i ∈ {1, 2, …, n}. Thus σ (x) ⊓ ν_{α_i} (a_i) ≤ (σ ⊓ ν_{α_i}) (b_i) and so $\begin{matrix} σ ⊓ (\cup ν_{α}) (x) \\ < n ɛ + (σ ⊓ ν_{α_{1}}) (b_{1}) ⊓ \dots ⊓ (σ ⊓ ν_{α_{n}}) (b_{n}) . \end{matrix}$ Obviously, $σ ⊓ ν_{α_{i}} (b_{i}) \leq [\cup (σ ⊓ ν_{α})] (b_{i}), \forall i \in {1, 2, \dots, r} .$ Thus by (3.10), $\begin{matrix} σ ⊓ (\cup ν_{α}) (x) & < & n ɛ + ⊓_{i = 1}^{n} [\cup (σ ⊓ ν_{α})] (b_{i}) \\ \leq & n ɛ + [\cup (σ ⊓ ν_{α})] (x) \end{matrix}$ Since ɛ is arbitrary, we have $σ ⊓ (⨆ ν_{α}) (x) \leq ⨆ (σ ⊓ ν_{α}) (x) .$ □ By Theorem 3.9 and [6 Theorem 7.10] it follows that

Corollary 3.10. FI (M) = (FI (M) ; ⊓ , ⊔) is a Heyting lattice.

By Corollary 3.1, for any ν ∈ FI (M), the mapping α ↦ α ⊓ ν on FI (M) is residuated and so by [6 Theorem 1.3] sup {ν ∈ FI (M) : α ⊓ νsβ} = α→β exists, for any α, β ∈ FI (M). So, (FI (M) , ⊔ , ⊓ , →, zero, unit) is a Heyting algebra.

Furthermore, for ν ∈ FI (M), ν^★ = ν→0 is the pseudocomplement of ν. So, we have

Corollary 3.11. (FI (M) , ⊔ , ⊓) is a pseudocomplemented lattice.

Let S (FI (M)) = {ν^★ : ν ∈ FI (M)} and define α ⊔ β = (α^★ ⊓ β^★) ^★, for any α, β ∈ FI (M). Then by [6 Theorem 7.2] we have

Theorem 3.12. (S (FI (M)) , ⊓ , ⊔ , ^★, zero, unit) is a Boolean algebra.

As an applied example of Theorems 3.9 and 3.12 and Corollary 3.11 we get that FI (M) is a distributive lattice and so by [16, Theorem 101], FI (M) does not contain any pentagon or diamond as a sublattice. This means that there are not three fuzzy A-ideals (or three non-comparable fuzzy A-ideals) μ, ν and ρ with μ ⊂ ν and μ ⊓ ρ = ν ⊓ ρ and μ ⊔ ρ = ν ⊔ ρ. For convenience, we consider the crisp case. If I, J and K are ideals of M with I ⊂ J, then there exists an ideal K such that I ∩ K = J ∩ K. In other words, the elimination property for the intersection operation does not hold. Similarly, for the join operation we get a same result. Furthermore, for any ideal I the largest ideal K such that I ⊓ K = zero exists, which is called the pseudocomplement of I. Most important result is that these pseudocomplements form a Boolean lattice (or Boolean algebra).

3.2 Fuzzy congruences

An element ϱ ∈ FR (M) is called a fuzzy equivalence relation if for any x, y ∈ M, (i) ϱ (x, x) = ⨆ {ϱ (y, z) : y, z ∈ M}, (ii) ϱ (x, y) = ϱ (y, x), and ϱ (x, y) ≥ ⨆ _z∈M {ϱ (x, z) ⊓ ϱ (z, y)}. Let Fe (M) denotes the set of all fuzzy equivalence relations in M. The intersection, composition and the inclusion relation on [0, 1] ^M×M are defined as

ϱsσ ⇔ ϱ (x, y) ≤ σ (x, y),

(ϱ ∩ σ) (x, y) = ϱ (x, y) ⊓ σ (x, y), and

ϱccσ (x, y) = ⨆ _z∈X ϱ (x, z) ⊓ σ (z, y).

for all x, y ∈ M.

Definition 3.13. Let ϑ ∈ Fe (M). ϑ is called a fuzzy congruence if for any x, y, u, v ∈ M,

ϑ (x^★, y^★) ≥ ϑ (x, y),

ϑ (x y, u v) ≥ ϑ (x, u) ⊓ ϑ (y, v),

ϑ (αx, αy) ≥ ϑ (x, y),

for all α ∈ A.

Example 3.14. Consider the mv-module M defined in Example 3.1 and define the fuzzy relation ϱ by ϱ (x, x) = t for all x ∈ M and ϱ (x, y) = s, for all x ≠ y, where 0 ≤ s < t ≤ 1. It is easy to check that ϱ is a fuzzy congruence in M.

We say that M is fuzzy congruence-permutable if ϱccσ = σccϱ, for all ϱ, σ ∈ Fc (M). It is easy to verify that for ϱ, σ ∈ FR (M) with the same tip t, ϱ ∩ σ and ϱccσ have the same tip t.

Let con (M), FR (M) (Fe (M), Fc (M)) denote the set of all congruences and fuzzy (equivalence, congruence) relations in M, respectively.

Obviously, by (i) and the definition of the operation ‘ ’, the condition (ii) of Definition 3.13 can be replaced by $ϑ (x y, u v) \geq ϑ (x, u) ⊓ ϑ (y, v) .$ (11)

By a routine verification we find that for ϑ ∈ FR (M), ϑ ∈ Fc (M) if and only if ϑ_t ∈ con (M), for all t ∈ [0, 1], with ϑ_t≠ ∅.

Let ϑ ∈ FR (M). For x ∈ M, ϑ_x ∈ [0, 1] ^M is defined as ϑ_x (y) = ϑ (y, x), for all y ∈ M. Now, we have the following lemma.

Lemma 3.15. Let ϑ ∈ Fc (M). Then

ϑ_x = ϑ_y if and only if $ϑ (x, y) = ⨆_{(u, v) \in M^{2}} ϑ (u, v) .$ Particularly, ϑ (0, 0) = ⨆ _(u,v)∈M² ϑ (u, v).

If ϑ_t≠ ∅, for t ∈ [0, 1], then 0/ϑ_t = (ϑ₀) _t.

Proof. (i) Assume that ϑ_x = ϑ_y, for x, y ∈ M. Then $\begin{matrix} ϑ (x, y) & = & ϑ_{y} (x) = ϑ_{x} (x) = ϑ (x, x) \\ = & ⨆_{(u, v) \in M^{2}} ϑ (u, v) . \end{matrix}$ Conversely, for w ∈ M, we have $\begin{matrix} ϑ_{x} (a) & = & ϑ (a, x) = ϑ (x, a) \geq ϑ (x, y) ⊓ ϑ (y, a) \\ = & (⨆_{(u, v) \in M^{2}} ϑ (u, v)) ⊓ ϑ (y, a) \\ = & ϑ (y, a) = ϑ_{y} (a) . \end{matrix}$ Similarly, it is proved that for all u ∈ M, ϑ_y (u) ≥ ϑ_x (u). Hence, ϑ_x = ϑ_y.

(ii) Assume that ϑ_t≠ ∅, for t ∈ [0, 1] and x ∈ 0/ϑ_t. Then (x, 0) ∈ ϑ_t; i.e., ϑ₀ (x) = ϑ (x, 0) ≥ t, whence x ∈ (ϑ₀) _t. Then 0/ϑ_ts (ϑ₀) _t. Conversely, if x ∈ (ϑ₀) _t, then ϑ (x, 0) = ϑ₀ (x) ≥ t, whence x ∈ 0/ϑ_t. Hence, (ϑ₀) _ts0/ϑ_t, completes the proof.□

Theorem 3.16. Let ϑ ∈ Fc (M). Then M/ϑ = {ϑ_x : x ∈ M} together with the following operations forms an mv-module.

$\begin{matrix} ϑ_{x} ϑ_{y} = ϑ_{x y}, (ϑ_{x})^{★} = ϑ_{x^{★}} \\ and \forall α \in A, α ϑ_{x} = ϑ_{α x} . \end{matrix}$

Proof. It is enough to prove that the mentioned operations are well-defined. Assume that ϑ_x = ϑ_z and ϑ_y = ϑ_w, for ϑ_x, ϑ_y, ϑ_z, ϑ_w ∈ M/ϑ. By Lemma 3.15 we have ϑ (x, z) = ⨆ _(u,v)∈^M2 ϑ (u, v) = ϑ (y, w). This implies that (x, z) , (y, w) ∈ ϑ_t, where t = ⨆ _(u,v)∈^M2 ϑ (u, v), whence (x y, z w) ∈ ϑ_t. Hence, $\begin{matrix} ⨆ {ϑ (u, v) : (u, v) \in M^{2}} \geq ϑ (x y, z w) \geq t \\ = & ⨆ {ϑ (u, v) : (u, v) \in M^{2}} . \end{matrix}$ Then ϑ (x y, z w) = ⨆ _(u,v)∈^M2 ϑ (u, v) and so $ϑ_{x} ϑ_{y} = ϑ_{x y} = ϑ_{z w} = ϑ_{z} ϑ_{w} .$ Now, if ϑ_x = ϑ_y, then $\begin{matrix} ϑ (x^{★}, y^{★}) & \geq & ϑ (x, y) \\ = & ⨆ {ϑ (u, v) : (u, v) \in M^{2}}, \end{matrix}$ which implies that ϑ (x^★, y^★) = ⨆ _(u,v)∈^M2 ϑ (u, v). Thus, (ϑ_x) ^★ = ϑ_{x
^★} = ϑ_{y
^★} = (ϑ_y) ^★. The other operation is obviously well-defined.□ We call the mv-module M/ϑ, in Theorem 3.16, the fuzzy quotient mv-module of M.

Proposition 3.17. If ϑ ∈ Fc (M), then ϑ₀ ∈ fai (M).

Proof. Assume that ϑ ∈ Fc (M). Then, for every a, b ∈ M, $\begin{matrix} ϑ_{0} (a b) & = & ϑ (a b, 0) \geq ϑ (a, 0) ⊓ ϑ (b, 0) \\ = & ϑ_{0} (a) ⊓ ϑ_{0} (b) . \end{matrix}$ Now, let a ≼ b, for a, b ∈ M. Then b^★ a = 0 and since 0^★ a = a we get $\begin{matrix} ϑ_{0} (a) & = & ϑ (a, 0) = ϑ (0^{★} a, b^{★} a) \\ \geq & ϑ (0^{★}, b^{★}) ⊓ ϑ (a, a) \geq ϑ_{0} (b) . \end{matrix}$ Finally, for a ∈ M and α ∈ A we have $ϑ_{0} (α a) = ϑ (α a, 0) = ϑ (α a, α 0) \geq ϑ (a, 0) = ϑ_{0} (a) .$ Therefore, ϑ₀ ∈ fai (M).□

Example 3.18. Consider the mv-module M defined in Example 3.1. We define the σ ∈ FR (M) by σ (x, x) = t₃, for all x ∈ M, σ (∅ , {2}) = σ ({2} , ∅) = t₁, and σ (x, y) = t₂, for the other elements of M, where 0 ≤ t₁ < t₂ < t₃ ≤ 1. It is obvious that σ_∅ = σ is a fuzzy A-ideal of M, while σ ∉ Fc (M) because σ ({1} ^★, {1, 2} ^★) = σ ({2} , ∅) = t₁notget₂ = σ ({1} , {1, 2}).

This shows that the converse of Proposition 3.2 is not true in general.

For fuzzy relation ϑ in M, let $M_{ϑ} = {x \in M : ϑ (x) = ϑ (0)} .$

Theorem 3.19. Let ϑ ∈ Fc (M) and f : M→N be a homomorphism of mv-modules with _{M_ϑ0}sKerf. Then the mapping $\hat{f} : M / ϑ \toN$ with ϑ_x ↦ f (x) is the unique homomorphism with $Im \hat{f} = Imf$ and $Ker \hat{f} = Kerf / ϑ$ . Furthermore, $\hat{f}$ is an isomorphism if and only if f is an epimorphism and Kerf = _Mϑ₀.

Proof. Assume that $\hat{f}$ is defined as $\hat{f} (ϑ_{x}) = f (x)$ . Assume that ϑ_x = ϑ_y, for ϑ_x, ϑ_y ∈ M/ϑ. Then ϑ (x, y) = t, where t = ⨆ _(u,v)∈^M2 ϑ (u, v). Hence $\begin{matrix} ϑ_{0} (x y^{★}) & = & ϑ (x y^{★}, 0) = ϑ (x y^{★}, y y^{★}) \\ \geq & ϑ (x, y) \land ϑ (y^{★}, y^{★}) = t \\ = & ϑ (0, 0) = ϑ_{0} (0), \end{matrix}$ by Lemma 3.15(i), whence x y^★ ∈ _Mϑ₀⊂Kerf and so f (x) f (y) ^★ = f (x y^★) =0; i.e., f (x) ≼ f (y). Similarly, it is proved that f (y) ≼ f (x). Thus, $\hat{f}$ is well-defined. It is clear that $Im \hat{f} = Imf$ . Now, $ϑ_{x} \in Ker \hat{f}$ if and only if $f (x) = \hat{f} (ϑ_{x}) = 0$ ; i.e., if and only if x ∈ Kerf. This means that $Ker \hat{f} = Kerf / ϑ$ .

We observe that if $\hat{f}$ is an epimorphism, f is also an epimorphism, and conversely. Also, $\hat{f}$ is one-to-one if and only if $Kerf / ϑ = Ker \hat{f} = {ϑ_{0}}$ and this is true if and only if Kerf = _Mϑ₀.□

Theorem 3.20. Let ϑ ∈ Fc (M) and f : M→N an epimorphism of mv-modules such that Kerf = _Mϑ₀. Then M/ϑ ≃ N.

Proof. From Theorem 3.19, taking N = Imf, the proof is obvious.□

Theorem 3.21. Let ϑ ∈ Fc (M). Then M/ϑ ≃ M/_{M_ϑ0}.

Proof. We first observe that _{M_ϑ0} is an A-ideal of M and under the congruence ∼_{_{M_ϑ0}}, M/_{M_ϑ0} is an mv-module. It is clear that φ : M→M/_{M_ϑ0} with x ↦ x/_{M_ϑ0} is an epimorphism of mv-modules with the kernel _{M_ϑ0}. So, by Theorem 3.20, the proof is complete.□

Lemma 3.22. Let ϑ, S ∈ Fc (M). Then ϑ/S ∈ Fc (M/S), where ϑ/S (_Sx, _Sy) = ϑ (x, y).

Proof. Straightforward.□

Theorem 3.23. Let ϑ, S ∈ Fc (M) with the same tip such that S⊂ϑ. Then (M/S)/(ϑ/S) ≃ M/ϑ.

Proof. Let φ : M/S→M/ϑ be defined by _Sx ↦ ϑ_x. We first show that φ is well-defined. Let _Sx = _Sy. Then $\begin{matrix} ϑ (x, y) \geq ⋃ (x, y) & = & ⨆ {⋃ (u, v) : (u, v) \in M^{2}} \\ = & ⨆ {ϑ (u, v) : (u, v) \in M^{2}}, \end{matrix}$ whence ϑ (x, y) = ⨆ {ϑ (u, v) : (u, v) ∈ ^M2}. Now, by Lemma 3.15 we have ϑ_x = ϑ_y, proving φ is well-defined. Clearly, φ is an epimorphism. Now, $\begin{matrix} Ker φ & = & {⋃_{x} \in M / ⋃ : ϑ_{x} = φ (⋃_{x}) = ϑ_{0}} \\ = & {⋃_{x} \in M / ⋃ : ϑ (x, 0) = ϑ (0, 0)} \\ = & {⋃_{x} \in M / ⋃ : ϑ / ⋃ (⋃_{x}, ⋃_{0}) = ϑ / ⋃ (⋃_{0}, ⋃_{0})} \\ = & {⋃_{x} \in M / ⋃ : (ϑ / ⋃)_{⋃_{0}} (⋃_{x}) = (ϑ / ⋃)_{⋃_{0}} (⋃_{0})} \\ = & M_{(ϑ / ⋃)_{⋃_{0}}} . \end{matrix}$ Thus, by Theorem 3.20, we get (M/S)/(ϑ/S) ≃ M/ϑ.

□

In Theorem 3.16, we proved that if ϑ ∈ Fc (M), M/ϑ together with the operations induced from those of M forms an mv-module without giving any information about the existence of ϑ. Here, we show that any fuzzy A-ideal induces such a fuzzy congruence.

For x, y ∈ M define m (x, y) = (x y^★) y ( x^★). It is routine to verify that

(m1) m (x, 0) = x.

(m2) m (x, y) ≼ m (x, z) m (z, y).

(m3) m (x u, y v) ≤ m (x, y) m (u, v).

(m4) m (ax, ay) ≼ am (x, y).

Definition 3.24. For fuzzy subset σ of M we define ϑ_σ ∈ FR (M) by $ϑ_{σ} (x, y) = σ (m (x, y)), \forall x, y \in M .$

Theorem 3.25. ϑ_σ ∈ Fc (M).

Proof. We first prove that ϑ_σ ∈ Fe (M). Let x ∈ M. From σ (m (x, x)) = σ (0) ≥ σ (u), for all u ∈ M, it follows that σ (m (x, x)) ≥ σ (m (y, z)), for all y, z ∈ M. Hence, $\begin{matrix} ϑ_{σ} (x, x) & = & σ (m (x, x)) \\ = & ⨆ {σ (m (y, z)) : (y, z) \in M^{2}} \\ = & ⨆ {ϑ_{σ} (y, z) : (y, z) \in M^{2}}, \end{matrix}$ i.e., ϑ_σ is fuzzy reflexive. Obviously, ϑ_σ is fuzzy symmetric. Now, let x, y ∈ M. By (m2) it follows that $\begin{matrix} ϑ_{σ} (x, y) & = & σ (m (x, y)) \geq σ (m (x, z) m (z, y)) \\ \geq & σ (m (x, z)) ⊓ σ (m (z, y)) \\ = & ϑ_{σ} (x, z) ⊓ ϑ_{σ} (z, y) \end{matrix}$ and so ϑ_σ (x, y) ≥ ⨆ _z∈M ϑ_σ (x, z) ⊓ ϑ_σ (z, y), proving that ϑ_σ is fuzzy transitive. Therefore, ϑ_σ ∈ Fe (M). That ϑ_σ is a fuzzy congruence is a direct consequence of (m3) and (m4).□

Theorem 3.26. Let σ and ν be fuzzy A-ideals of M. Then

$ϑ_{σ \cap ν} = ϑ_{σ} \cap ϑ_{ν} ϑ_{σ ⊔ ν} = ϑ_{σ} ⊔ ϑ_{ν}$

Proof. We prove that ϑ_σ⊔ν is the least fuzzy congruence on M which contains ϑ_σ and ϑ_ν. From Theorem 3.25, we know that ϑ_σ, ϑ_ν and ϑ_σ⊔ν are fuzzy congruence on M. Now, for x, y ∈ M, $\begin{matrix} ϑ_{σ} (x, y) \\ = σ (m (x, y)) \leq σ ⊔ ν (m (x, y)) = ϑ_{σ ⊔ ν} (x, y) . \end{matrix}$ Similarly, ϑ_ν (x, y) ≤ ϑ_σ⊔ν (x, y). Hence ϑ_σ ⊔ ϑ_νsϑ_σ⊔ν. Now, let SFc (M) with ϑ_σ⊂S and ϑ_ν⊂S. Then σ (m (x, y)) = ϑ_σ (x, y) ≤ S (x, y) and ν (m (x, y)) = ϑ_ν (x, y) ≤ S (x, y). Also, $\begin{matrix} ϑ_{σ ⊔ ν} (x, y) & = & σ ⊔ ν (m (x, y)) \\ = & ⨆ {σ \cup ν (i_{1}) ⊓ \dots ⊓ σ \cup ν (i_{n}) : \\ m (x, y) ≼ i_{1} i_{2} \dots i_{n}} \\ = & ⨆ {(σ (i_{1}) ⊓ \dots ⊓ σ (i_{n})) \\ \cup (ν (i_{1}) ⊓ \dots ⊓ ν (i_{n})) : \\ m (x, y) ≼ i_{1} i_{2} \dots i_{n}} \\ \leq & ⨆ {σ (i_{1} i_{2} \dots i_{n}) \\ \cup ν (i_{1} i_{2} \dots i_{n}) : \\ m (x, y) ≼ i_{1} i_{2} \dots i_{n}} \\ \leq & ⊔ {σ (m (x, y)) \cup ν (m (x, y)) : \\ m (x, y) ≼ i_{1} i_{2} \dots i_{n}} \\ \leq & ⋃ (x, y) \end{matrix}$ proving ϑ_σ⊔ν = ϑ_σ ⊔ ϑ_ν. Obviously, ϑ_σ ∩ ϑ_ν = ϑ_σ∩ν.□

Remark 3.27. From [2 Theorem 3.9], we know that if M has tip property and is fuzzy congruence-permutable, then ϑ_σ ⊔ ϑ_ν = ϑ_σ ∘ ϑ_ν, where ∘ is the composition operation of fuzzy relations.

Theorem 3.28. Assume that M has tip property and is fuzzy congruence-permutable. Then the lattice FI (M) can be embedded into the lattice FC (M) = (FC ; ∩ , cc).

Proof. We define the mapping φ : FI→FC by σ ↦ ϑ_σ. By Theorem 3.26, φ preserves ⊓ and ⊔. Now, define ψ : FC→FI by ϑ ↦ ϑ₀. By Proposition 3.2, ψ is well-defined and also $\begin{matrix} ψ φ (σ) (x) & = & ψ (ϑ_{σ}) (x) = (ϑ_{σ})_{0} (x) = ϑ_{σ} (x, 0) \\ = & σ (m (x, 0)) = σ (x), \end{matrix}$ whence φ is an embedding.□

By Theorems 3.9 and 3.28 it follows that

Corollary 3.29. If M has tip property and is fuzzy congruence-permutable, then FI (M) is a Heyting sublattice of FC (M).

Open Problem 3.30. Under what conditions FI and FC are isomorphic?

4 Conclusions

One of the most important tools in studying those algebras that arose from logic is (fuzzy) A-ideal theory. And what properties these (fuzzy) A-ideals may have which are useful to study structural aspects of these algebras. To answer these questions, first, fuzzy A-ideals of mv-modules were studied and showed that they form a Heyting lattice and as an algebraic structure, it contains their pseudocomplemented of elements as a Boolean algebra. Second, using fuzzy congruences in mv-modules, we characterized them and after investigating homomorphism theorems we proved that fuzzy A-ideals can be considered as a sublattice of the lattice of fuzzy congrurences.

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