Mathematics and Statistics 9(3): 302-308, 2021 DOI: 10.13189/ms.2021.090311 http://www.hrpub.org Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in Gamma-semigroups Wichayaporn Jantanan 1 , Anusorn Simuen 2 , Winita Yonthanthum 2 , Ronnason Chinram 2,* 1 Department of Mathematics, Faculty of Science, Buriram Rajabhat University, Buriram 31000, Thailand 2 Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand Received January 24, 2021; Revised April 14, 2021; Accepted April 27, 2021 Cite This Paper in the following Citation Styles (a): [1] Wichayaporn Jantanan, Anusorn Simuen, Winita Yonthanthum, Ronnason Chinram, ”Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in Gamma-semigroups,” Mathematics and Statistics, Vol.9, No.3, pp. 302-308, 2021. DOI: 10.13189/ms.2021.090311 (b): Wichayaporn Jantanan, Anusorn Simuen, Winita Yonthanthum, Ronnason Chinram, (2021). Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in Gamma-semigroups. Mathematics and Statistics, 9(3), 302-308. DOI: 10.13189/ms.2021.090311 Copyright ©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Γ-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Γ-semigroups. Many results in ideal theory of Γ-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Γ-semigroups. In Section 2, we define almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups by using the concept ideas of interior Γ-ideals and almost Γ-ideals of Γ-semigroups. Every almost interior Γ-ideal of a Γ-semigroup S is clearly a weakly almost interior Γ-ideal of S but the converse is not true in general. The notions of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups are generalizations of the notion of interior Γ-ideal of a Γ-semigroup S. We investigate basic properties of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Γ-semigroups. In Section 3, we introduce fuzzy almost interior Γ-ideals and fuzzy weakly almost interior Γ-ideals of Γ-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Γ-ideals [weakly almost interior Γ-ideals] and fuzzy almost interior Γ-ideals [fuzzy weakly almost interior Γ-ideals] of Γ-semigroups. Keywords Almost Interior Γ-ideals, Weakly Almost Inte- rior Γ-ideals, Fuzzy Almost Γ-interior Ideals, Fuzzy Weakly Almost Interior Γ-ideals. 1 Introduction and preliminaries In 1965, Zadeh [25] first introduced the notion of fuzzy sub- sets. Applications of fuzzy subsets have been developed in many fields. Rosenfeld applied the fuzzy subsets to define fuzzy subgroups of groups in [14]. Applications of fuzzy sub- sets in semigroups were first considered by Kuroki [11, 12]. Now, fuzzy subsets were studied in many algebraic structures (for example, in ternary semigroups ([21]), in non-associative ordered semigroups ([1]), etc). The definition of almost ideals of semigroups (or A-ideals) was first studied by Grosek and Satko [5, 6, 7] in 1980. Recently, Wattanatripop, Chinram and Changphas [23, 24] introduced the notion of quasi almost ide- als (or quasi-A-ideals) of semigroups and gave their basic prop- erties. Moreover, they applied fuzzy subsets to define fuzzy almost ideals, fuzzy almost bi-ideals and fuzzy quasi almost ideals of semigroups and showed relationship between almost ideals [almost bi-ideals, quasi almost ideals] and their fuzzifi- cations of semigroups. Furthermore, Kaopusek, Kaewnoi and Chinram [9] using the concepts of interior ideals and almost ideals of semigroups, defined the notions of almost interior ide- als and weakly almost interior ideals of semigroups. Moreover, they investigated their basic properties. Now, the notion of al- most ideals in semigroups were extend to some generalizations of semigroups, for example, almost ideals in ternary semigroup [21], almost hyperideals in semihypergroups [20], etc. The notion of Γ-semigroups were first introduced by Sen [16]. This algebraic structure generalized from the classi-