Journal of Mathematical Sciences and Applications, 2018, Vol. 6, No. 1, 1-5 Available online at http://pubs.sciepub.com/jmsa/6/1/1 ©Science and Education Publishing DOI:10.12691/jmsa-6-1-1 A Study on Anti Bipolar Q – Fuzzy Normal Semi Groups Mourad Oqla Massa'deh * Department of Applied Science, Ajloun College, Al – Balqa Applied University, Jordan *Corresponding author: mourad.oqla@bau.edu.jo Abstract In this manuscript, we present notions of anti bipolar Q - fuzzy normal semi groups and we present characterization of anti bipolar Q – fuzzy normal semi groups. Further, we have characterized a semi group that a semi lattice of groups in terms of anti bipolar Q – fuzzy ideals, generalized anti bipolar Q – fuzzy bi – ideals and anti bipolar Q – fuzzy quasi – ideals. Keywords: semi lattice of groups, semi lattice of archimedean semi groups, bipolar Q – fuzzy sets, anti bipolar Q – fuzzy ideals, anti bipolar Q – fuzzy co – normal, anti bipolar Q – fuzzy bi – normal Cite This Article: Mourad Oqla Massa'deh, “A Study on Anti Bipolar Q – Fuzzy Normal Semi Groups.” Journal of Mathematical Sciences and Applications, vol. 6, no. 1 (2018): 1-5. doi: 10.12691/jmsa-6-1-1. 1. Introduction The fundamental concept of bipolar valued fuzzy sets popularized by Lee [3]. Bipolar valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [-1, 1]. In bipolar valued fuzzy set, the membership degree 0 means that elements are irrelevant to the corresponding property, the membership degree (0, 1] indicates that element some what satisfy the property and the membership degree [-1, 0) indicate that elements some what satisfy the implicit counter property, the study of bipolar fuzzy algebras has achieved great success, Many wonderful results have been obtained by some mathematical researchers, such as Yasodara and Sathappan [12], Shanmugapriya and Arjunan [8,9], Uma Maheswari et al [11] and Massa'deh and Fora [5]. On the other hand, in the pest several decades, studies on the subject of anti fuzzy ideals in semi groups introduced by Shabir and Nawaz [10]. Hee et al [1] extended the concept of bipolar fuzzy set to semi groups theory, while Muthuraj et al [7], Khizar et al [2] Massa'deh [4] and Massa'deh and Al naser [6] extend the concept of bipolar fuzzy set to subring , hemi – ring and Γ- near rings. In this paper, we deal with anti bipolar Q – fuzzy sets to a semi groups and investigate related properties. Further, we discussed anti bipolar Q – fuzzy normal semi groups. We studied some characteristics of anti bipolar Q – fuzzy normal semi groups and discussed the some of its properties. 2. Preliminaries Definition 2.1 [5] Let X and Q be a non-empty arbitrary sets. A bipolar-valued Q- fuzzy set or bipolar fuzzy set μ in X×Q is an object having the form μ = {(x, q), μ + (x, q), μ − (x, q) | x ∈ X & q ∈Q}, where μ + : X×Q →[0, 1] and μ − : X×Q →[−1, 0] are mappings. The positive membership degree μ+ (x) denotes the satisfaction degree of an element x to the property corresponding to a bipolar-valued fuzzy set μ = {(x,q), μ + (x, q), μ − (x, q) | x ∈ X & q ∈Q}, and the negative membership degree μ − (x, q) denotes the satisfaction degree of an element x to some implicit counter property corresponding to a bipolar-valued fuzzy set μ = {(x,q), μ + (x, q), μ − (x, q) | x ∈ X & q ∈Q }. If μ + (x, q) ≠ (0, q) and μ − (x, q) = (0, q), then it is the situation that x is regarded as having only positive satisfaction for μ = {(x,q), μ + (x, q), μ − (x, q) | x ∈ X & q ∈Q }. If μ + (x, q) = (0, q) and μ − (x, q) ≠ (0, q), then it is the situation that x does not satisfy the property of μ = {(x,q), μ + (x, q), μ − (x, q) | x ∈ X & q ∈Q }, but somewhat satisfies the counter property of μ = {(x,q), μ + (x, q), μ − (x, q) | x ∈ X & q ∈Q}. It is possible for an element x to be such that μ + (x, q) ≠ (0, q) and μ − (x, q) ≠ (0, q), when the membership function of property overlaps with its counter property over some portion of X. For the sake of simplicity, we shall use the symbol μ = (μ + , μ − ) for the bipolar-valued fuzzy set μ = {(x,q), μ + (x, q), μ − (x, q) | x ∈ X & q ∈Q}. Definition 2.2 Let A be a subset of a semi group G and Q be an arbitrary set, then the complement characteristic function of A, that is χ A c is defined by (1, );( , ) (, ) (0, );( , ) c A q xq A Q xq q xq A Q χ ∈ ×  =  ∉ ×  Definition 2.3 A semi group G is called semi lattice of groups if it is the theoretical union of a family of mutually disjoint subgroups G i ; i ∈ I such that for all i, j ∈ I the product G i G j are both contained in the same subgroup G k (k ∈ I). Definition 2.4 A semi group G is called Archimedean if, for all x, y ∈ G there exists a positive integer n such that x n ∈ GyG. Definition 2.5 A semi group G is called weakly commutative if, for all x, y ∈ G there exists a positive integer n such that (xy) n ∈ yGy. Definition 2.6 A bipolar Q – fuzzy subset µ of a semi group G is said to be anti bipolar Q- fuzzy sub – semi