Neutrosophic Sets and Systems, Vol. 4, 2014 Mumtaz Ali, Florentin Smarandache, Muhammad Shabir and Munazza Naz, Neutrosophic Bi-LA-Semigroup and Neu- trosophic N-LA-Semigroup Neutrosophic Bi-LA-Semigroup and Neutrosophic N-LA- Semigroup Mumtaz Ali 1* , Florentin Smarandache 2 , Muhammad Shabir 3 and Munazza Naz 4 1,3 Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000,Pakistan. E-mail: mumtazali770@yahoo.com, mshabirbhatti@yahoo.co.uk 2 University of New Mexico, 705 Gurley Ave., Gallup, New Mexico 87301, USA E-mail: fsmarandache@gmail.com 4 Department of Mathematical Sciences, Fatima Jinnah Women University, The Mall, Rawalpindi, 46000, Pakistan. E-mail: munazzanaz@yahoo.com Abstract. In this paper we define neutrosophic bi-LA- semigroup and neutrosophic N-LA-semigroup. Infact this paper is an extension of our previous paper neutrosophic left almost semigroup shortly neutrosophic LA- semigroup. We also extend the neutrosophic ideal to neu- trosophic biideal and neutrosophic N-ideal. We also find some new type of neutrosophic ideal which is related to the strong or pure part of neutrosophy. We have given sufficient amount of examples to illustrate the theory of neutrosophic bi-LA-semigroup, neutrosophic N-LA- semigroup and display many properties of them this pa- per. Keywords: Neutrosophic LA-semigroup, neutrosophic ideal, neutrosophic bi-LA-semigroup, neutrosophic biideal, neutrosophic N-LA-semigroup, neutrosophic N-ideal. 1 Introduction Neutrosophy is a new branch of philosophy which studies the origin and features of neutralities in the nature. Floren- tin Smarandache in 1980 firstly introduced the concept of neutrosophic logic where each proposition in neutrosophic logic is approximated to have the percentage of truth in a subset T, the percentage of indeterminacy in a subset I, and the percentage of falsity in a subset F so that this neutro- sophic logic is called an extension of fuzzy logic. In fact neutrosophic set is the generalization of classical sets, con- ventional fuzzy set  1 , intuitionistic fuzzy set  2 and in- terval valued fuzzy set  3 . This mathematical tool is used to handle problems like imprecise, indeterminacy and in- consistent data etc. By utilizing neutrosophic theory, Vasantha Kandasamy and Florentin Smarandache dig out neutrosophic algebraic structures in   11 . Some of them are neutrosophic fields, neutrosophic vector spaces, neu- trosophic groups, neutrosophic bigroups, neutrosophic N- groups, neutrosophic semigroups, neutrosophic bisemi- groups, neutrosophic N-semigroup, neutrosophic loops, neutrosophic biloops, neutrosophic N-loop, neutrosophic groupoids, and neutrosophic bigroupoids and so on. A left almost semigroup abbreviated as LA-semigroup is an algebraic structure which was introduced by M .A. Kazim and M. Naseeruddin  3 in 1972. This structure is basically a midway structure between a groupoid and a commutative semigroup. This structure is also termed as Able-Grassmann’s groupoid abbreviated as AG -groupoid  6 . This is a non associative and non commutative algebraic structure which closely resemble to commutative semigroup. The generalization of semigroup theory is an LA-semigroup and this structure has wide applications in collaboration with semigroup. We have tried to develop the ideal theory of LA- semigroups in a logical manner. Firstly, preliminaries and basic concepts are given for neutrosophic LA-semigroup. Then we presented the newly defined notions and results in neutrosophic bi-LA-semigroups and neutrosophic N- LA-semigroups. Various types of neutrosophic biideals and neutrosophic N-ideal are defined and elaborated with the help of examples. 2 Preliminaries Definition 1. Let   , S  be an LA-semigroup and let   : , S I a bI ab S     . The neutrosophic LA- semigroup is generated by S and I under  denoted as     , NS S I    , where I is called the neutrosophic element with property 2 I I  . For an integer n , n I  and nI are neutrosophic elements and 19