A REMARK ON FACTORIZABLE SUBSTRUCTURES OF A PRODUCT SEMIGROUP Nistala V.E.S. Murthy 1 and Chundru Maheswari 1* 1 Department of Mathematics, J.V.D. College of Science and Technology Andhra University, Visakhapatnam-530003, A.P. State, INDIA e-mail: drnvesmurthy@rediffmail.com, cmaheswari2014@gmail.com URL: http://andhrauniversity.academia.edu/NistalaVESMurthy URL: https://www.researchgate.net/profile/Nistala_VES_Murthy * Corresponding Author: cmaheswari2014@gmail.com Abstract— In this paper, we study the lattice theoretic properties of all factorizable substructures of a product semigroup, which will be crucial in the representation of soft substructures of a soft semigroup by certain crisp cousins. Keywords— (Product) Semigroup, factorizable subsemigroup (left ideal, right ideal, ideal, quasi-ideal, bi-ideal) AMS Classification— 20M10, 20M12, 06B2. 1. INTRODUCTION Introduced by de Seguier[4] in 1904 and by L.E. Dickson[5] in 1905, the semigroup, a set with an associative binary operation, is one of the earliest algebraic objects of study. Some of the first results which we now take for granted were actually laid by A.K. Suschkewitsch[13] as early as in 1920. However, by the middle of 1960s with the work of such people as D. Rees, J.A. Green, E.S. Lyapin, A.H. Clifford, G. Preston etcetera, the theory of semigroup evolved as a completely independent branch of study within Mathematics. From a historical point of view, it may be interesting to note that in 1956, the notion quasi-ideal of a semigroup was introduced by Steinfeld[12] as a generalization of the notions (left, right) ideal of a semigroup and interestingly, the notion of bi-ideal of a semigroup which further generalizes the notion of quasi-ideal of a semigroup was introduced by Good-Hughes[6] much earlier in 1952. In 1999, Molodtsov[10] introduced the notion of a soft set as a mathematical tool for modelling uncertainties. Since its introduction, several mathematicians imposed various algebraic (sub) structures on them and studied some of their elementary properties. In 2010, Ali-Shabir-Shum[1] introduced the notions of soft semigroup (left ideal, right ideal, ideal, quasi-ideal, bi-ideal) over a semigroup and studied some of their properties. On the other side, Murthy-Maheswari[11], showed that for any soft set over a universal set, there is a crisp set in such a way that the complete lattice of all soft subsets of the given soft set is complete epimorphic to a complete lattice of certain subsets of the crisp set, where the join in the later complete lattice is the meet induced join, and there is a crisp set in such a way that the complete lattice of all regular soft subsets of the given soft set is complete isomorphic to a complete lattice of certain subsets of the crisp set, where the meet in the former complete lattice is the join induced meet and the join in the later complete lattice is the meet induced join. JASC: Journal of Applied Science and Computations Volume VI, Issue IV, April/ 2019 ISSN NO: 1076-5131 Page No:3278