Direct limits of a direct system of fuzzy complete Multialgebra

1 Introduction

Hyperstructure theory was born in 1934 when Marty defined hypergroups, began to analysis their properties and applied them to groups, relational algebraic functions [12].

Several aspects of homomorphisms, subalgebras and subdirect decompositions of multialgebras (also called hyperalgebras) has been studied by Picket in [18, 19] and by Hansoul in [9]. In [20] D. Schweigert studied the congruences of multialgebras and in [22] the exponentiation of universal hyperalgebras introduced.

In [5] R. Ameri and M.M. Zahedi introduced and studied notion of hyperalgebraic systems. Also some more basic properties of multialgebras such as, identities, term function and fundamental relation, and direct limit of multialgebras has been studied in [14–17]. R. Ameri and I.G. Rosenberg in [3] studied congruence and strong congruences of multialgebras.

As it is well known, in 1965 Zadeh [24] introduced the notion of a set μ on a nonempty set X as a function from X to the unite real interval I = [0, 1] as a fuzzy set. J.E. Goguen in [24] in the definition of fuzzy sets replaced I by a complete lattice L and introduced the notion of L-fuzzy sets. Also,

Rosenfeld [23] defined the concept of a fuzzy subgroup of a group and since then many researchers have worked in this area. In [4], R. Ameri and I.G. rosenfeld introduced and stidied L-multialgebras and fuzzy congruence relations on fuzzy multialgebras. Recently, R. Ameri and T. Nozari in [1] introduced and studied fuzzy multialgebras and fundamental relation of fuzzy hyperalgebras. In this regards we follow [1] and [2] to introduce the notion of fuzzy complete multialgebras. Also, we introduce the notion of direct limits of a direct system of complete fuzzy multialgebras and obtain some related results.

2 Preliminaries

In this section we gather necessary definitions and simple properties of multialgebras and fuzzy multialgebras. In the sequel A is a fixed nonvoid set, P^* (A) is the family of all nonvoid subsets of H, and for a positive integer n we denote by H ⁿ the set of n-tuples over A (for more details see [1]).

Definition 2.1. [1] Let τ = (n _γ ) _γ<o(τ) be a sequence over N = {0, 1, …}, where o (τ) is an ordinal, a fuzzy n _γ -ary multioperation f _γ on A is a map f _γ  : A ^{n _γ}  ⟶ F^* (A), which associated a nonzero fuzzy subset f _γ  (x₀, …, x_{n γ -1}) with ary n _γ -tuple (x₀, …, x_{n γ -1}) of elements of A. The couple (A, (f _γ ) _γ<o(τ)) is called a fuzzy n _γ -ary hypergroupoid. A fuzzy nullary multioperation on A is just an element of F^* (A), i.e. a nonzero fuzzy subset of A.

Definition 2.2. [1] Let A be a nonempty set and for any γ < o (τ), f _γ be a fuzzy n _γ -ary multioperation on A. Then U = (A, (f _γ ) _γ<o(τ)) is called fuzzy multialgebra, where γ < o (τ) is type of this fuzzy multialgebra.

Definition 2.3. [1] Let μ₀, …, μ_{n γ -1} be n _γ nonzero subset of a fuzzy multialgebra U = (A, (f _γ ) _γ<o(τ)) we define for all x ∈ A f γ ( μ 0 , … , μ n γ - 1 ) ( x ) = ∨ ( x 0 , … , x n γ - 1 ) ∈ A n γ ( μ 0 ( x 0 ) ∧ … ∧ μ n γ - 1 ( x n γ - 1 ) ∧ f γ ( x 0 , … , x n γ - 1 ) ( x ) ) .

Finally, if A₀, …, A_{n γ -1} are nonempty subsets of A, H = A₀ × … × A_{n γ -1} for all x ∈ A f γ ( A 0 , … , A n γ - 1 ) ( x ) = ∨ ( x 0 , … , x n γ - 1 ) ∈ H ( f γ ( x 0 , … , x n γ - 1 ) ( x ) ) .

If B is a nonempty subset of A, then we denote the characteristic function of B by χ _B . Note that, if f : A₁ ⟶ A₂ is a map and a ∈ A₁, then f (χ _a ) = χ_f(a).

Definition 2.4. [1] Let U = (A, (f _γ ) _γ<o(τ)) and B = (B, (f _γ ) _γ<o(τ)) be fuzzy multialgebras of the same types, and h :A ⟶ B be a map. We say that h is a homomorphism of fuzzy multialgebras if for γ < o (τ) and every x₀, …, x_{n γ -1} ∈ A one have h (f _γ  (x₀, …, x_{n γ -1})) ≤ f _γ  (h (x₀) , …, h (x_{n γ -1})).

Let consider a direct system A of sets having the carrier (I, ≤) consisting of the family of fuzzy sets (A _i |i ∈ I) and the family of maps (φ _ij  : A _i  ⟶ A _j |i, j ∈ I, i ≤ j). So (I, ≤) is a directed preordered fuzzy set and the maps φ _ij  (i, j ∈ I, i ≤ j) are such that for any i, j, k ∈ I, with i ≤ j ≤ k, φ _jk oφ _ij  = φ _ik and φ _ii  = 1 _{A i} , for all i ∈ I. On the disjoint union A of the fuzzy sets A _i one defines the relation ≡ as follow: for any x, y ∈ A there exist i, j ∈ I, such that x ∈ A _i , y ∈ A _j , and x ≡ y if and only if φ _ik  (x) = φ _jk  (y) for some k ∈ I with i ≤ k, j ≤ k. This relation on A is an equivalence relation and the quotient fuzzy set A / ≡ = { x ˆ | x ∈ A }(denoted by A ∞ f) is the direct limit of the direct system of sets A.

Let us consider that each fuzzy set A _i is a support fuzzy set for a fuzzy multialgebra U _i and φ _ij are fuzzy multialgebra homomorphisms. The system A = ((U _i ) , (φ _ij  : A _i  ⟶ A _j |i, j ∈ I, i ≤ j)).

Obtained this way is a direct system of fuzzy multialgebras. Sometimes we will refer to A as the direct system (or the direct family) of fuzzy multialgebras (U _i |i ∈ I).

We will show that a general approach of the direct limit of a direct system in the case of fuzzy multialgebra, based on the results known for universal fuzzy algebras, is not only a step forward in the fuzzy hyperstructure theory, but also allows us to improve the results that already exist in the case of direct limits of direct systems of some particular fuzzy multialgebra.

Let us define the following fuzzy multioperations on A ∞ f: for any x 0 ˆ , … , x n - 1 ˆ ∈ A ∞ f,

we consider all of the elements m ∈ I such that for any j ∈ {0, …, n _γ  - 1}, there exists x j ′ ∈ x j ˆ ∩ A m then we define f γ ( x 0 ˆ , … , x γ - 1 ˆ ) ( x ˆ ) = ∨ x j ′ ∈ x j ˆ ∩ A m f γ ( x 0 ′ , … x n γ - 1 ′ ) ( x )

We obtain a fuzzy multialgebra U_∞ of type τ on A_∞. First, we characterize the fuzzy multioperations of U_∞ in a way which will prove to be very useful to our study.

Lemma 2.5. [1] Let x 0 ˆ , … , x n γ - 1 ˆ ∈ A ∞ f and for any j ∈ {0, …, n _γ  - 1} let us take i _j  ∈ I such that x _j  ∈ A _{i j} . The representative x′ of a class x ′ ^ such that f γ ( x 0 ˆ , … , x γ - 1 ˆ ) ( x ′ ˆ ) > 0 can be considered such that ∃m ∈ I, i₀ ≤ m, …, i _n  ≤ m with f γ ( x 0 ′ , … x n ′ ) ( x ′ ) > 0 where x j ′ = φ i j m ( x j ) ( j ∈ { 0 , … , n γ - 1 } ).

Remark 2.6. If for some γ < o (τ), f _γ is a fuzzy point in all the fuzzy multialgebras U _i , then f _γ is a fuzzy point in U_∞.

Remark 2.7. The maps φ_i∞ : A _i  ⟶ A_∞, φ i ∞ ( x ) = x ˆ are fuzzy multialgebras homomorphism.

Theorem. The fundamental relation of a multialgebra U is the transitive closure α^* of the relation α given on A as follows: for x, y ∈ A, xαy if and only if x, y ∈ p (a₀, …, a_n-1) for some n ∈ N, p ∈ P⁽ⁿ⁾ (P^* (U)) and a₀, …, a_n-1 ∈ A (see [16]). The relation α^* is the smallest equevalence relation on A such that the factor multialgebra U/α^* is a universal algebra. We denoted the class α^* (a) of a ∈ A modulo α^* by a ¯ and A/α^* by A ¯. We also denoted the algebra U/α^* by U ¯ and we called it the fundamental algebra of the multialgebra U.

Proposition. [16] The following conditions are equivalent for a multialgebra U = (A, (f _γ ) _γ<o(τ)) of type τ:

i) for all γ < o (τ), for all a₀, …, a_{n γ -1} ∈ A

a ∈ f γ ( a 0 , … , a n γ - 1 ) ⇒ a ¯ = f γ ( a 0 , … , a n γ - 1 ).

ii) for all m ∈ N, for all q, r ∈ P^(m) (τ) {X _i |i ∈ {0, …, m - 1}}, for all

a₀, …, a_m-1, b₀, …, b_m-1 ∈ A,

q (a₀, …, a_m-1) ∩ r (b₀, …, b_m-1) ¬ = ∅ ⇒ q (a₀, …, a_m-1) = r (b₀, …, b_m-1).

The multialgebras which verify one of the equivalent condition (i) and (ii) from the previous proposition are generalizations for the complete semihypergroups. This suggests the following:

Definition 2.8. A multialgebras which satisfies one of the equivalent conditions of the previous proposition is called a complete multialgebra.

Definition 2.9. [1] Let p, q ∈ P⁽ⁿ⁾ (F^* (H)). We can consider that the n-ary identity q = p is satisfied in the fuzzy multialgebra U of type τ if p ( χ a 1 , … , χ a n ) = q ( χ a 1 , … , χ a n ) for all a₀, …, a_n-1 ∈ A we can consider that the n-ary weak identity q∩ p ¬ = ∅ is satisfied in the fuzzy multialgebra U of type τ if p ( χ a 1 , … , χ a n ) ∧ q ( χ a 1 , … , χ a n ) > 0

3 Fuzzy complete multialgebras

Definition 3.1. Let A = (A, (f _γ ) _γ<o(τ)) and B = (B, (f _γ ) _γ<o(τ)) be to fuzzy multialgebras of the same type, and h : A ⟶ B be a mapping then:

i) h is a homomorphism of fuzzy multialgebras if for γ < o (τ) and every x₀, …, x_{n γ -1} ∈ A we have h (f _γ  (x₀, …, x_{n γ -1})) ≤ f _γ  (h (x₀) , …, h (x_{n γ -1})).

We say that h is a strong homomorphismof fuzzy multialgebras if h (f _γ  (x₀, …, x_{n γ -1})) = f _γ  (h (x₀) , …, h (x_{n γ -1})) and h is a weakhomomorphism of fuzzy multialgebras if h (f _γ  (x₀, … , x_{n γ -1})) ∩ f _γ  (h (x₀) , …, h (x_{n γ -1})) ¬ = ∅

ii) h is a (resp. monomorphism) epimorphism if h is (resp. one-to-one) onto.

iii) h is a isomorphism of fuzzy multialgebras if h is a bijection fuzzy homomorphism. If there exists a fuzzy isomorphism from A to B, we write A ≅ B clearly the relation ≅ is an equivalent relation on class of fuzzy multialgebras.

Proposition. Let h : A ⟶ B be a fuzzy homomorphism then:

i) If h is a fuzzy isomorphism, then h^-1 is a fuzzy homomorphism.

ii) If h is a fuzzy isomorphism, then h^-1 is a strong fuzzy homomorphism.

Proof.

i) Let h : A ⟶ B be a fuzzy homomorphismand h^-1 (b _i ) = a _i so h (a _i ) = b _i . h^-1 (f (b₁, …, b _n ))   =  h^-1 (f (h (a₁) , …, h (a _n )))   ≥  h^-1 (h (f (a₁, …, a _n )))   =  f (a₁, …, a _n )   =  f (h^-1 (b₁) , …, h^-1 (b₁)).

ii) Let h : A ⟶ B be a fuzzy homomorphism and h (a _i ) = b _i then h (f (a₁, …, a _n )) ≤ f (h (a₁) , …, h (a _n )) f (a₁, …, a _n ) ≤ h^-1 (f (h (a₁) , …, h (a _n ))) f (h^-1 (b₁) , …, h^-1 (b _n )) ≤ h^-1 (f (b₁, …, b _n )). since h^-1 is homomorphism we have h^-1 (f (b₁, …, b _n )) ≤ f (h^-1 (b₁) , …, h^-1 (b _n )). So h^-1 is a strong fuzzy homomorphism.

Remark 3.2. Based on definition there are three type of fuzzy homomorphism of fuzzy multialgebras homomorphism, weak homomorphism and strong homomorphism. There for agains the category of (fuzzy) algebras there three various categories of multialgebras, that morphisms are different for multialgebras A and B we denote by Hom (A, B) , Hom _w  (A, B) and Hom _s  (A, B) respectively for all homomorphism, weak homomorphism and strong homomorphism from A into B, respective. We denote the relevant category to these three kinds of morphisms by F MA , F MA w and F MA s respectively. where clearly F MA s ≤ F MA ≤ F MA w.

A ≤ B, means that A is a subcategory of B. Obviously, these subcategories are not full.

Proposition. If < A , f γ ︷ > be a fuzzy multialgebra, then <A, f _γ > is a multialgebra such that f γ ( a 1 , … , a n ) = { b | f γ ︷ ( a 1 , … , a n ) ( b ) > 0 }.

Example 3.3. [21] If (G, o) is a fuzzy hypergroup, then (G, *) is a hypergroup.

Example 3.4. [11] If (R, ⊕, ⨀) is a fuzzy hyperring, then (R, + , .) is a hyperring.

Denote by FMA and MA the classes of all fuzzy multialgebras and multialgebras respectively. Define the following two mappings: ψ : MA ⟶ FMA ψ ( < A , f γ > ) = < A , f γ ︷ > where for all a₁, …, a _n  ∈ A we have f γ ︷ ( a 1 , … , a n ) = χ f γ ( a 1 , … , a n ).

φ : FMA ⟶ MA φ ( < A , f γ ︷ > ) = < A , f γ >

where for all a₁, …, a _n  ∈ A we have f γ ( a 1 , … , a n ) = { b | f γ ︷ ( a 1 , … , a n ) ( b ) > 0 }.

Remark 3.5. We have φψ = 1 _MA . Clearly, ψφ ¬ =1 _FMA . But, if we consider a suitable equivalence relation ∼ on FMA, then we can obtain a bijection between FMA∼ and MA, let us define the following equevalence relation FMA.

A 1 = < A 1 , f 1 γ ︷ > ∼ A 2 = < A 2 , f 2 γ ︷ > iff φ ( < A 1 , f 1 γ ︷ > ) = φ ( < A 2 , f 2 γ ︷ > )

Hence

< A 1 , f 1 γ ︷ > ∼ < A 2 , f 2 γ ︷ > iff f 1 γ ︷ ( a 1 , … , a n ) ( b ) > 0 ⇔ f 2 γ ︷ ( a 2 , … , a n ) ( b ) > 0.

Then we obtain the following bijection:

φ ¯ : FMA ∼ ⟶ MA

φ ¯ < A , f γ ︷ > ¯ = φ < A , f γ ︷ >

Where < A , f γ ︷ > ¯ is the equivalence class of < A , f γ ︷ >, with respect to the equivalence relation “∼”. We also denote the fuzzy algebra A ∼ = < A , f γ ︷ > / ∼ by A ¯ = < A , f γ ︷ > ¯.

The fundamental relation of a fuzzy multialgebra U is the transitive closure α^* of the relation α given on A as follows: for x, y ∈ A, xαy if and only if p (a₀, …, a_n-1) (x) >0, p (a₀, …, a_n-1) (y) >0 for some n ∈ N, p ∈ P⁽ⁿ⁾ (F^* (U)) and a₀, …, a_n-1 ∈ A. The relation α^* is the smallest equivalence relation on A such that the factor fuzzy multialgebra U/α^* is a universal fuzzy algebra. We denoted the class a ∈ A modulo α^* by a ¯ and A/α^* by A ¯. We also denoted the fuzzy algebra U/α^* by U ¯ and we called it the fundamental algebra of the fuzzy multialgebra U.

Proposition. The following conditions are equivalent for a fuzzy multialgebra U = ( A , ( f γ ︷ ) γ < o ( τ ) ) of type τ:

i) for all γ < o (τ), for all a₀, …, a_{n γ -1} ∈ A

f γ ︷ ( a 0 , … , a n γ - 1 ) ( a ) > 0 ⇒ a ¯ = f γ ( a 0 , … , a n γ - 1 ). ii) for all m ∈ N, for all q, r ∈ P^(m) (τ) {X _i verti ∈ {0, …, m - 1}}, for all a₀, …, a_m-1, b₀, …, b_m-1 ∈ A, ∀ a ∈ A : q ︷ ( a 0 , … , a m - 1 ) ( a ) > 0 , r ︷ ( b 0 , … , b m - 1 ) ( a ) > 0 ⇒ q ( a 0 , … , a m - 1 ) = r ( b 0 , … , b m - 1 ).

Proof. (i) ⇒ (ii). If (i) holds for U then ∀minN, ∀ P ∈ P^(m) (τ) ∖ {X _i |i ∈ {0, …, m - 1}} , ∀ a₀, …, a_m-1 ∈ A, we have q ︷ ( a 0 , … , a m - 1 ) ( a ) > 0 , r ︷ ( b 0 , … , b m - 1 ) ( a ) > 0 ⇒ a ¯ = q ( a 0 , … , a m - 1 ) = r ( b 0 , … , b m - 1 ).

which justify (ii) .

(ii) ⇒ (i). Let us consider γ < o ( τ ) , a 0 , … , a n γ - 1 ∈ A , f γ ︷ ( a 0 , … , a n γ - 1 ) ( a ) > 0 and b ∈ Awith b ∈ a ¯ ( i . e . aα^*b). It follows that n ∈ N, x₀, …, x _n  ∈ A exist so that a = x₀αx₁α … αx_n-1αx _n  = b hence for any i ∈ {0, …, n - 1}, there exist m i ∈ N , P i ∈ P ( m i ) ( τ ) , a 0 i , … , a m i - 1 i ∈ A with P i ( a 0 i , … , a m i - 1 i ) ( x i ) > 0 , P i ( a 0 i , … , a m i - 1 i ) ( x i + 1 ) > 0.

We can consider that every two consequent elements from x₀, …, x _n are distinguished, thus no P _i is equal to an X j i , ( j < m i ) , ∀ i ∈ { 0 , … , n - 1 }. Hence ∀ i ∈ { 0 , … , n - 1 } , P i ( a 0 i , … , a m i - 1 i ) ( x i ) > 0 , P i + 1 ( a 0 i + 1 , … , a m i - 1 i + 1 ) ( x i ) > 0 and so ∀ i ∈ { 0 , … , n - 1 } , P i ( a 0 i , … , a m i - 1 i ) = P i + 1 ( a 0 i + 1 , … , a m i - 1 i + 1 ) which leads us to P 0 ( a 0 0 , … , a m i - 1 0 ) ( b ) > 0. But P 0 ( a 0 0 , … , a m i - 1 0 ) ( a ) > 0 , f γ ︷ ( a 0 , … , a n γ - 1 ) ( a ) > 0 thus P 0 ( a 0 0 , … , a m i - 1 0 ) = f γ ︷ ( a 0 , … , a n γ - 1 ) hence f γ ︷ ( a 0 , … , a n γ - 1 ) ( b ) > 0 and the proof is finished.

Definition 3.6. A multialgebras which satisfies one of the equivalent conditions from the previous proposition is called a fuzzy complete multialgebra.

Definition 3.7. The category of fuzzy complete multialgebras of the same type τ, where the objects are fuzzy complete multialgebras, the homomorphisms are fuzzy multialgebras homomorphisms and the product is the usual composition of mappings. Clearly, fuzzy complete multialgebras are a full subcategory of category of fuzzy multialgebras.

Theorem 3.8. The category of fuzzy complete multialgebras is a full subcategory of category of fuzzy multialgebras which is closed under direct limits of a direct system.

Proof. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) be a direct system of fuzzy complete multialgebras. Suppose that n ∈ N, q, r ∈ P^(m) (τ) {X _j |j ∈ {0, …, n - 1}   and a 0 ˆ , … , a n - 1 ˆ , b 0 ˆ , … , b n - 1 ˆ ∈ A ∞ f. We can consider that the representatives a₀, …, a_n-1, b₀, …, b_n-1 of the given classes are from the set A _k  (k ∈ I). If q ︷ ( a 0 ˆ , … , a n - 1 ˆ ) ( a ) ˆ > 0 ,

r ︷ ( b 0 ˆ , … , b n - 1 ˆ ) ( a ) ˆ > 0 From q ︷ ( a 0 ˆ , … , a n - 1 ˆ ) ( a ) ˆ > 0 it results that there exist m ′ ∈ I , m ′ ≥ k, and a ′ ≡ a such that q ︷ ( φ km ′ ( a 0 ) , … , φ km ′ ( a n - 1 ) ) ( a ′ ) > 0 .

Analogously, from r ︷ ( b 0 ˆ , … , b n - 1 ˆ ) ( a ) ˆ > 0 it results that there exist m ″ ∈ I , m ″ ≥ k, and a ″ ≡ a such that r ︷ ( φ km ″ ( b 0 ) , … , φ km ″ ( b n - 1 ) ) ( a ″ ) > 0 .

Let x ˆ be an arbitrary fuzzy point from q ︷ ( a 0 ˆ , … , a n - 1 ˆ ). Then there exists l ∈ I with k ≤ l such that q ︷ ( φ kl ( a 0 ) , … , φ kl ( a n - 1 ) ) ( x ) > 0

From a ′ ≡ a ″ ≡ a we deduce the existence of an element m ″ ∈ I with, m ″ ≤ m ‴, such that φ m ′ m ‴ ( a ′ ) = φ m ″ m ‴ ( a ″ ). Since (I, ≤) is directed, there exists m ∈ I with m ″ ≤ m and l ≤ m. We have

q ︷ ( φ km ( a 0 ) , … , φ km ( a n - 1 ) ) ( φ lm ( x ) ) = q ︷ ( φ lm ( φ kl ( a 0 ) ) , … , φ lm ( φ kl ( a n - 1 ) ) ) ( φ lm ( x ) ) ≥ φ lm ( q ︷ ( φ kl ( a 0 ) , … , φ kl ( a n - 1 ) ) ) ( φ lm ( x ) ) > 0.

Also

q ︷ ( φ km ( a 0 ) , … , φ km ( a n - 1 ) ) ( φ m ′ m ( a ′ ) ) = q ︷ ( φ m ′ m ( φ km ′ ( a 0 ) ) , … , φ m ′ m ( φ km ′ ( a n - 1 ) ) ) ( φ m ′ m a ′ ) ≥ φ m ′ m ( q ︷ ( φ km ′ ( a 0 ) , … , φ km ′ ( a n - 1 ) ) ) ( φ m ′ m a ′ ) > 0.

and analogously,

r ︷ ( φ km ( b 0 ) , … , φ km ( b n - 1 ) ) ( φ m ″ m a ″ ) > 0

But

ϕ m ′ m ( a ′ ) = ϕ m ‴ m ( ϕ m ′ m ‴ ( a ′ ) ) = ϕ m ‴ m ( ϕ m ″ m ‴ ( a ″ ) ) = ϕ m ″ m ( a ″ ),

and, since the fuzzy multialgebraU m f is complete it follows that q ︷ ( φ km ( a 0 ) , … , φ km ( a n - 1 ) ) ( φ lm ( x ) ) = r ︷ ( φ km ( b 0 ) , … , φ km ( b n - 1 ) ) ( φ lm ( x ) ).

Consequently, r ︷ ( b 0 ˆ , … , b n - 1 ˆ ) ( x ˆ ) > 0. Thus we have proved that

q ( a 0 ˆ , … , a n - 1 ˆ ) ⊆ r ( b 0 ˆ , … , b n - 1 ˆ )

Similarly, one can show that q ( a 0 ˆ , … , a n - 1 ˆ ) ⊇ r ( b 0 ˆ , … , b n - 1 ˆ ), so, we have

q ( a 0 ˆ , … , a n - 1 ˆ ) = r ( b 0 ˆ , … , b n - 1 ˆ )

Thus the U ∞ f is complete.

Corollary 3.9. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) be a direct system of fuzzy complete multialgebras. If any i, j ∈ I have an upper bound k ∈ I such that U _k is a fuzzy complete multialgebra, then U_∞ is a fuzzy complete multialgebra.

4 Direct limits of a direct system of fuzzy multialgebras

Lemma 4.1. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) be a direct system of fuzzy multialgebras and U_∞ = underrightarrowlimA. If all fuzzy homomorphisms φ _ij are fuzzy strong homomorphisms then the fuzzy multioperations from U_∞ can be defined as follows: for any γ < o (τ) and for any x 0 ˆ , … , x n γ - 1 ˆ ∈ A ∞ with x₀ ∈ A _{i 0} , …, x_{n γ -1} ∈ A_{i n γ -1} we consider an element m ∈ I, i₀, …, i_{n γ -1} ≤ m and we define f γ ︷ ( x 0 ˆ , … , x n γ - 1 ˆ ) ( x ˆ ) > 0 ⇒ f γ ︷ ( φ i 0 m ( x 0 ) , … , φ i m γ - 1 ( x m γ - 1 ) ) ( x ) > 0

Proof. From Lemma 2.5 it is enough to prove that the definition the set dose not depended on m ∈ I. Indeed, taking any other m ′ ∈ I, with i _j ≤m′, for all j ∈ {0, …, n _γ  - 1} and x j ′ = φ i j m ( x j ) , x j ″ = φ i j m ′ ( x j ), we have { x ˆ ′ | f γ ︷ ( x 0 ′ , … , x n γ - 1 ′ ) ( x ′ ) > 0 } = { x ˆ ″ | f γ ︷ ( x 0 ″ , … , x n γ - 1 ″ ) ( x ″ ) > 0 }.

For, if q ∈ I such that m ≤ q , m ′ ≤ q then φ mq ( f γ ︷ ( x 0 ′ , … , x n γ - 1 ′ ) ) = f γ ︷ ( φ mq ( x 0 ′ ) , … , φ mq ( x n γ - 1 ′ ) ) = f γ ︷ ( φ i 0 q ( x 0 ) , … , φ i n γ - 1 q ( x n γ - 1 ) ) = f γ ︷ ( φ m ′ q ( x 0 ″ ) , … , φ m ′ q ( x n γ - 1 ″ ) ) = φ m ′ q ( f γ ︷ ( x 0 ″ , … , x n γ - 1 ″ ) ) thus for each f γ ︷ ( x 0 ′ , … , x n γ - 1 ′ ) ( x ′ ) > 0 there exists f γ ︷ ( x 0 ″ , … , x n γ - 1 ″ ) ( x ″ ) > 0 such that x ′ ≡ x ″ and conversely, for each f γ ︷ ( x 0 ″ , … , x n γ - 1 ″ ) ( x ″ ) > 0 there exists f γ ︷ ( x 0 ′ , … , x n γ - 1 ′ ) ( x ′ ) > 0 such that .

Remark 4.2. In this case is easier to observe that if for some γ < o (τ) we have for any two elements from I an upper bound m ∈ I such that f _γ is an operation in U _m then f _γ is an operation in U_∞.

Corollary 4.3. Let p ∈ P⁽ⁿ⁾ (τ) and a₀, …, a_n-1 ∈ A. If i₀, …, i_n-1 ∈ I are such that a _j  ∈ A _{i j} for all j ∈ {0, …, n - 1} and m ∈ I with i₀, …, i_n-1 ≤ m then p ︷ ( a 0 ˆ , … , a n - 1 ˆ ) ( a ˆ ) > 0 ⇒ p ︷ ( φ i 0 m ( a 0 ) , … , φ i n - 1 m ( a n - 1 ) ) ( a ) > 0 .

Lemma 4.4. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) be a direct system of fuzzy multialgebras and U_∞ = underrightarrowlimA. If all the fuzzy homomorphisms φ _ij are fuzzy strong homomorphisms then the homomorphisms φ i ∞ : A i ⟶ A ∞ , φ i ∞ ( x ) = x ˆ are fuzzy strong homomorphisms.

Proof. Indeed, for any γ < o (τ) and x₀, …, x_{n γ -1} ∈ A _i , we have

0 < φ i ∞ ( f γ ︷ ( x 0 , … , x n γ - 1 ) ) ( φ i ∞ ( x ) ) = ∨ t ∈ φ i ∞ - 1 ( φ i ∞ ( x ) ) f γ ︷ ( x 0 , … , x n γ - 1 ) ( t ) = ∨ t ∈ φ im - 1 ( φ m ∞ - 1 ( x ˆ ) ) f γ ︷ ( x 0 , … , x n γ - 1 ) ( t ) = φ im ( f γ ( x 0 , … , x n γ - 1 ) ) ( φ m ∞ - 1 ( x ˆ ) ) = f γ ︷ ( φ im ( x 0 ) , … , φ im ( x n γ - 1 ) ) ( φ m ∞ - 1 ( x ˆ ) ) ) = ∨ y ∈ φ m ∞ - 1 ( x ˆ ) f γ ︷ ( φ im ( x 0 ) , … , φ im ( x n γ - 1 ) ) ( y ) = f γ ︷ ( x 0 ˆ , … , x n γ - 1 ˆ ) ( x ˆ ) = f γ ︷ ( φ i ∞ ( x 0 ) , … , φ i ∞ ( x n γ - 1 ) ) ( φ i ∞ ( x ) ) .

Moreover, from the properties of the direct limit of a direct system of sets it follows that for any i, j ∈ I, with i ≤ j, φ_j∞oφ _ij  = φ_i∞.

Theorem 4.5. The subcategory fuzzy multialgebra with strong homomorphism of category fuzzy multialgebra is closed under the formation of the direct limit of a direct system.

Proof. Let us consider the following diagrams: A ∞ A ′ A i [ u ] φ i ∞ [ r ] φ ij [ ul ] φ j ∞ A j A i [ ur ] α j [ r ] φ ij A j [ u ] α j A ∞ [ r ] μ A ′ A i [ u ] φ i ∞ [ ur ] α i

The first diagram is commutative and whenever a fuzzy multialgebra U ′ of type τ, together with a family (α _i  : A _i ⟶A′|i ∈ I) of homomorphism make the second diagram commutative, there exists unique homomorphisms μ : A_∞⟶A′ such that the third diagram is commutative. The unique homomorphism μ wich make i ∈ I such that x ∈ A _i , and μ ( x ˆ ) = μ ( φ i ∞ ( x ) ) = α i ( x ).

If all the homomorphisms φ _ij and α _i are strong homomorphisms then, as we have Lemma 4.4 all the homomorphisms φ_i∞ are strong homomorphisms, and μ is an strong homomorphism, too. Let us take γ < o (τ) and x 0 ˆ , … , x n γ - 1 ˆ ∈ A ∞. We can consider that all the representatives x₀ ∈ A _{i 0} , …, x_{n γ -1} ∈ A _{i n γ -1} . For every x ˆ ∈ μ - 1 ( y ) we have μ ( x ˆ ) = y and since there exists m ′ ∈ I such that x ∈ A m ′ and μ ( x ˆ ) = α m ′ ( x ) then y = α m ′ ( x ) and x ∈ α m ′ − 1 ( y ). Therefore 0 < μ ( f γ ︷ ( x 0 ˆ , … , x n γ - 1 ˆ ) ) ( y ) = ∨ x ˆ ∈ μ - 1 ( y ) f γ ︷ ( x 0 ˆ , … , x n γ - 1 ˆ ) ( x ˆ ) = ∨ x ∈ α m ′ - 1 ( y ) f γ ︷ ( x 0 ˆ , … , x n γ - 1 ˆ ) ( x ˆ ) = ∨ x ˆ ∈ μ - 1 ( y ) ∨ x ∈ x ˆ f γ ︷ ( φ mm ′ ( x 0 ) , … , φ mm ′ ( x n γ - 1 ) ) ( x ) = ∨ x ∈ α m ′ - 1 ( y ) ∩ x ˆ f γ ︷ ( φ mm ′ ( x 0 ) , … , φ mm ′ ( x n γ - 1 ) ) ( x ) = α m ′ ( f γ ︷ ( φ mm ′ ( x 0 ) , … , φ mm ′ ( x n γ - 1 ) ) ( y ) = f γ ︷ ( α m ′ ( φ mm ′ ( x 0 ) ) , … , α m ′ ( φ mm ′ ( x n γ - 1 ) ) ) ( y )

But the commutativity of the second diagram holds for any i, j ∈ I, i ≤ j, thus, from m ′ ∈ I , m ≤ m ′ it follows that: f γ ︷ ( α m ′ ( φ mm ′ ( x 0 ) ) , … , α m ′ ( φ mm ′ ( x n γ - 1 ) ) ) ( y ) = f γ ︷ ( α m ( x 0 ) ) , … , α m ( x n γ - 1 ) ) ( y )

So, μ ( f γ ︷ ( x 0 ˆ , … , x n γ - 1 ˆ ) ) ( y ) = f γ ︷ ( α m ( x 0 ) , … , α m ( x n γ - 1 ) ) ( y ) = f γ ︷ ( μ ( x 0 ˆ ) , … , μ ( x n γ - 1 ˆ ) ) ( y )

Which ends the proof.

Lemma 4.6. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) a direct system of fuzzy multialgebras and let q, r ∈ P⁽ⁿ⁾ (τ). If the weak identity q∩ r ¬ = ∅ is satisfied in each fuzzy multialgebra U_∞ then q∩ r ¬ = ∅ is satisfied in U _i  (i ∈ I).

Proof. Let us consider a 0 ˆ , … , a n - 1 ˆ ∈ A ∞, let us suppose that a₀ ∈ A _{i 0} , …, a_n-1 ∈ A _{i n-1} and let m ∈ I be such that i₀, …, i_n-1 ≤ m. Since the q∩ r ¬ = ∅ is satisfied in U_∞, it following that ( q ︷ ( a 0 ˆ , … , a n - 1 ˆ ) ∧ r ︷ ( a 0 ˆ , … , a n - 1 ˆ ) ) ( a ˆ ) > 0 and it follows immediatly that ( q ︷ ( a 0 ′ , … , a n - 1 ′ ) ∧ r ︷ ( a 0 ′ , … , a n - 1 ′ ) ) ( a ) > 0 where a j ′ = φ i j m ( a j ) which ends the proof.

Corollary 4.7. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) be a direct system of fuzzy multialgebras and let q, r ∈ P⁽ⁿ⁾ (τ). If any i, j ∈ I have an upper bound k ∈ I such that q∩ r ¬ = ∅ is satisfied in U_∞, then q∩ r ¬ = ∅ is satisfied in U _k .

Lemma 4.8. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) be a direct system of fuzzy multialgebras and let q, r ∈ P⁽ⁿ⁾ (τ). If the identity q = r is satisfied in each fuzzy multialgebra U _i  (i ∈ I), then q = r is satisfied in U_∞.

Proof. Let us consider a 0 ˆ , … , a n - 1 ˆ ∈ A ∞. Now suppose that a₀ ∈ A _{i 0} , …, a_n-1 ∈ A _{i n-1} . Let us take an arbitrary element a ˆ ∈ q ( a 0 ˆ , … , a n - 1 ˆ ) and a ˆ ∉ r ( a 0 ˆ , … , a n - 1 ˆ ). So a ∈ q ( a 0 ′ , … , a n - 1 ′ ) and a ∉ r ( a 0 ′ , … , a n - 1 ′ ) where a j ′ = φ i j m ( a j ), thus q ¬ = r is satisfied in U _i  (i ∈ I), a contradiction.

Corollary 4.9. Let A = ((U _i |i ∈ I) , (φ _ij |i, j ∈ I, i ≤ j)) be a direct system of fuzzy multialgebras and let q, r ∈ P⁽ⁿ⁾ (τ). If any i, j ∈ I have an upper bound k ∈ I such that q = r is satisfied in U _k , then q = r is satisfied in U_∞.

Acknowledgments

The first author is partially supported by Center of Excellence of Algebraic Hyperstructures and its Applications of Tarbiat Modares University (CEAHA), Tehran, Iran.