1 st Annual International Interdisciplinary Conference, AIIC 2013, 24-26 April, Azores, Portugal - Proceedings- 17 ON THE TOPOLOGICAL STRUCTURES ON Γ-SEMIRINGS Kostaq Hila, Asoc. Prof. Department of Mathematics & Computer Science, Faculty of Natural Sciences, University of Gjirokastra, Albania Ilir Vardhami, PhD Kristaq Gjino, PhD Department of Mathematics, Faculty of Natural Sciences, University of Tirana, Tirana, Albania. Abstract: In this paper, we introduce some special classes of ideals in Γ-semirings called prime k-ideal, prime full k-ideal, prime ideals, maximal and strongly irreducible ideals. Considering and investigating properties of the collection A , T , M , B and S of all proper prime k-ideals, proper prime full k-ideals, maximal ideals, prime ideals and strongly irreducible ideals, respectively, of a Γ- semiring R, we construct the respective topologies on them by means of closure operator defined in terms of intersection and inclusion relation among these ideals of Γ-semiring R. The respective obtained topological spaces are called the structure spaces of the Γ-semiring R. We study a several principal topological axioms and properties in those structure spaces of Γ-semiring such as separation axioms, compactness and connectedness etc. Key Words: Γ-Semiring; Prime k-ideal (ideal); (strongly) irreducible ideal; Hull-Kernel topology; Structure space Introduction and preliminaries Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory etc. The theory of semiring was first developed by H. S. Vandiver [33] and he has obtained important results of the objects. Semiring constitute a fairly natural generalization of rings, with board applications in the mathematical foundation of computer science. Also, semiring theory has many applications to other branches. For example, automata theory, optimization theory, algebra of formal process, combinatorial optimization, Baysian networks and belief propagation (cf. [12, 13, 14]). It is well known that the concept of Γ-rings was first introduced and investigated by Nobusawa in 1964 [27], which is a generalization of the concept of rings. The class of Γ -rings contains not only all rings but also all Hestenes ternary rings. Later Barnes [2] weakened slightly the conditions in the definition of Γ-ring in the sense of Nobusawa. After these two papers were published, many mathematicians obtained interesting results on Γ-rings in the sense of Barnes and Nobusawa extending and generalizing many classical notions and results of the theory of rings. Γ-semirings were first studied by M. K. Rao [28] as a generalization of Γ-ring as well as of semiring. The concepts of Γ- semirings and its sub-Γ-semirings with a left(right) unity was studied by J. Luh [26] and M. K. Rao in [28]. The ideals, prime ideals, semiprime ideals, k -ideals and h -ideals of a Γ-semiring, regular Γ- semiring, respectively, were extensively studied by S. Kyuno [21, 22, 24] (cf. [23]) and M. K. Rao [28, 29]. In Γ-semirings, the properties of their ideals, prime ideals, semiprime ideals and their generalizations play an important role in their structure theory, however the properties of an ideal in semirings and Γ-semirings are somewhat differerent from the properties of the usual ring ideals. In order to amend these differs, the concepts of k-ideals and h-ideals in a semiring were introduced and considered by D. R. LaTorre [25] in 1965. For the properties of some h-ideals in Γ-semirings, the reader is referred to the recent papers of T. K. Dutta and S. K. Sardar, K. P. Shum in [7,8,9, 10, 31].