Research Article On Fuzzy Ordered Hyperideals in Ordered Semihyperrings O. KazancJ , 1 F.YJlmaz , 1 and B. Davvaz 2 1 Department of Mathematics, Karadeniz Technical University, 61080, Trabzon, Turkey 2 Department of Mathematics, Yazd University, Yazd, Iran Correspondence should be addressed to O. Kazancı; kazancio@yahoo.com Received 28 May 2018; Accepted 12 December 2018; Published 3 February 2019 Academic Editor: Antonin Dvor´ ak Copyright © 2019 O. Kazancı et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we introduce the concept of fuzzy ordered hyperideals of ordered semihyperrings, which is a generalization of the concept of fuzzy hyperideals of semihyperrings to ordered semihyperring theory, and we investigate its related properties. We show that every fuzzy ordered quasi-hyperideal is a fuzzy ordered bi-hyperideal, and, in a regular ordered semihyperring, fuzzy ordered quasi-hyperideal and fuzzy ordered bi-hyperideal coincide. 1. Introduction Te theory of algebraic hyperstructures is a well-established branch of classical algebraic theory which was initiated by Marty [1]. Since then many researchers have worked on algebraic hyperstructures and developed it [2, 3]. A short review of this theory appears in [4–8]. Te notion of semiring was introduced by Vandiver [9] in 1934, which is a generalization of rings. Semirings are very useful for solving problems in graph theory, automata theory, coding theory, analysis of computer programs, and so on. We refer to [10] for the information we need concerning semiring theory. In [11–13], quasi-ideals of semirings are studied and some properties and related results are given. In [8], Vougiouklis generalized the notion of hyperring and named it as semihyperring, where both the addition and multiplication are hyperoperations. Semihyperrings are a generalization of Krasner hyperrings. Davvaz, in [14], studies the notion of semihyperring in a general form. Ameri and Hedayati defne k-hyperideals in semihyperrings in [15]. In 2011, Heidari and Davvaz [16] studied a semihypergroup (,∘) with a binary relation ≤, where ≤ is a partial order so that the monotony condition is satisfed. Tis structure is called an ordered semihypergroup. Properties of hyperideals in ordered semihypergroups are studied in [17]. Also, the properties of fuzzy hyperideals in an ordered semihyper- group are investigated in [18, 19]. Yaqoop and Gulistan [20] study the concept of ordered LA-semihypergroup. In [21], Davvaz and Omidi introduce the basic notions and properties of ordered semihyperrings and prove some results in this respect. In 2018, Omidi and Davvaz [22] studied on special kinds of hyperideals in ordered semihyperrings. Some properties of hyperideals in ordered Krasner hyperrings can be found in [23]. Afer the introduction of fuzzy sets by Zadeh [24], recon- sideration of the concept of classical mathematics began. Because of the importance of group theory in mathematics, as well as its many areas of application, the notion of fuzzy subgroup is defned by Rosenfeld [25] and its structure is investigated. Tis subject has been studied further by many others [26, 27]. Fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now used in the world both on the theoretical point of view and for their many applications. Tere exists a rich bibliography: publications that appeared within 2015 can be found in “Fuzzy Algebraic Hyperstructures - An Introduction” by Davvaz and Cristea [28]. Recently, many researchers have considered fuzzifca- tion on many algebraic structures, for example, on semi- groups, rings, semirings, near-rings, ordered semigroups, semihypergroups, ordered semihypergroups, and ordered hyperrings [29–34]. Inspired by the study on ordered semihyperrings, we study the concept of fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals of an ordered semihyperring and we present some examples in this respect. Te rest of this paper is organized as follows. Hindawi Advances in Fuzzy Systems Volume 2019, Article ID 3693926, 7 pages https://doi.org/10.1155/2019/3693926