Fuzzy Neighborhood Structures on Partially Ordered Groups

Ahsanullah (1988) showed the compatibility between group structures and I-fuzzy neighborhood systems. In this paper, we require not only that the I-fuzzy neighborhood systems be compatible with the group structures, but also compatible with the order relation, in one sense or another. 1. Introductions. In [8], Katsaras combine the concepts of [0, 1]-topology and order structure to bring out the so-called ordered fuzzy topological spaces. Several authors have continued on the work of Katsaras in the area of [0, 1]-topology and order [3, 4, 10]. In [2] Ahsanullah introduced the notion of I-fuzzy neighborhood groups. In this paper, we aim to introduce and study the concept of I-fuzzy neighborhood structures on ordered groups.

1. Introductions.In [8], Katsaras combine the concepts of [0, 1]-topology and order structure to bring out the so-called ordered fuzzy topological spaces.Several authors have continued on the work of Katsaras in the area of [0, 1]-topology and order [3,4,10].
In [2] Ahsanullah introduced the notion of I-fuzzy neighborhood groups.In this paper, we aim to introduce and study the concept of I-fuzzy neighborhood structures on ordered groups.

Preliminaries.
Let X be a nonempty set.A relation ≤ on X is said to be preorder if it is reflexive and transitive.An antisymmetric preorder is said to be a partially order.By a preordered (resp., an ordered) set, we mean a set X with a preorder (resp., a partially order) relation on it and we denote it by (X, ≤).Every set can be considered as a partially ordered set equipped with the discrete order (x ≤ y if and only if x = y).
A function f from a preordered set (X, ≤) to a preordered set (X , ≤ ) is called isotone or order-preserving (resp., antitone or order-inverting) if x ≤ y in X implies f (x) ≤ f (y) (resp., f (y) ≤ f (x)) in X .The function f is said to be order isomorphism if it is bijection and (∀x, y ∈ X) x ≤ y f (x) ≤ f (y).Suppose that (G, * ) is a semigroup and that G is endowed with an order ≤.We say that (G, * , ≤) is an ordered semigroup if the low of composition and the order are related by the property: for all x, y ∈ G (2.1) to be order-homomorphism if it is both isotone and semigroup homomorphism.By an ordered group we mean an ordered semigroup which is a group.In this paper, we use the multiplicative ordered group (G, •, ≤) which is sometimes written as (G, ≤).
Combining the notion of order-isomorphism and group isomorphism, we say that an ordered group (G 1 , ≤ 1 ) is OG-isomorphic to an ordered group (G 2 , ≤ 2 ) if there is a mapping f : G 1 → G 2 which is both order isomorphism and group isomorphism.
By an I-topological (resp., stratified I-topological) ordered space are we mean a triplet (X, ≤,τ), consisting of a partially ordered set (X, ≤) and an I-topology (resp., stratified I-topology) τ on X.
By |I-TopOS| (resp., |SI-TopOS|), we mean the category of all I-topological (resp., stratified I-topological) ordered spaces as object and all order-preserving continuous mappings between them as morphisms.
The order ≤, in an I-topological ordered space (X, ≤,τ), is said to be closed [8] if and only if the following condition holds: if x y, then there are neighborhoods µ, ρ of x, y, respectively, such that i(µ) ∧ d(ρ) = 0.
Let (X, ≤,τ) be an L-topological ordered space.If the order is closed, then X is Hausdorff [8].
An I-fuzzy quasi-uniformity [9] is a subset U of I X×X which is prefilter and has the following three properties: ( } is an I-fuzzy quasi-uniformity on X called the conjugate of U. We denote by U * the I-fuzzy uniformity which generated by U, that is, The I-fuzzy quasiuniformity U can generate an order, say ≤ u , by setting A triplet (X, ≤, U * ), consisting of an ordered set (X, ≤) and an I-fuzzy uniformity U * , is called an I-fuzzy uniform ordered space [10] if there exists an I-fuzzy quasiuniformity U on X such that Where f is called quasi-uniform equivalence if f is bijective and both f and f −1 are quasi-uniformly continuous.Definition 2.2 [10].A mapping f : (X, ≤, U * ) → (X 1 , ≤ 1 , U * ) is said to be uniformly order-mapping if there exist I-fuzzy quasi-uniformities u and u 1 on X and X 1 , respectively such that (i) Definition 2.3 [2].Let (G, •) be a group and let ℵ be an I-fuzzy neighborhood system on G.Then, the triplet (G, •,t(ℵ)) is called I-fuzzy neighborhood group if and only if the following conditions are fulfilled: (1) the mapping m : Proposition 2.4 [2].Let (G, •) be a group and let ℵ be an I-fuzzy neighborhood system on G.Then, (G, •,t(ℵ)) is an I-fuzzy neighborhood group if and only if the mapping is continuous
By |I − FNOGr |, we mean the category of all I-fuzzy neighborhood ordered groups as objects and all order-preserving homeomorphisms between them as morphisms.
In agreement with [1], a faithful functor T : A → Set is said to be topological (monotopological) if and only if, given any index class ((X j ,ξ j ) : j ∈ J) of A-objects indexed by a class J and any source (resp., mono-source) (f j : X → X j ) in Set, there exists a unique A-structure ξ on X which is initial with respect to (f j : X → (X j ,ξ j )) j∈J , that is, such that for any Also, we have that the constant function lift to morphism in A and the A-fibre T −1 (S) for any set S is small.
initial and let ≤ be the order defined by x ≤ y if and only if f α (x) ≤ α f α (y) for all α ∈ J. Then (G, ≤,t(ℵ)) ∈ |I − FNOGr |.Initiality of the mono-source can easily be checked; thus T is mono-topological.The other conditions for a monotopological category are clearly met.
Proof.The proof follows from Definition 2.3.
that is, I-fuzzy set µ • ρ is increasing.

Fuzzy quasi-uniformity on I-fuzzy neighborhood ordered groups. As given in
) is an I-fuzzy neighborhood group and µ ∈ ℵ(e), then µ L (resp., µ R ) is called the left (resp., right) I-fuzzy entourages associated with µ.We can easily note that the left (resp., right) (y, x).Also, µ R (x, y) ≠ µ R (y, x).
In the sequel, we use ℵ i (e) (resp., ℵ d (e)) to denote the system of all increasing (resp., decreasing) I-fuzzy neighborhoods of e. From the above discussion we have the following easily established result.
} is a basis for the conjugate left (resp., right) I-fuzzy quasi-uniformity We denote ) is an I-fuzzy uniformity on G called the left (resp., right) I-fuzzy uniformity generated by U L (resp., U R ).Also, the two-sided I-fuzzy uniformity U * = U * R ∨ U * L can be generated by the two-sided I-fuzzy quasi-uniformity (U R ∨ U L ).
It is known that the entourages of the above I-fuzzy quasi-uniformities can generate an order on G by setting The partial order ≤ * is said to be generated by the left I-fuzzy quasi-uniformity U L .

Call for Papers
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