Regular and Intra-Regular Neutrosophic Left Almost Semihypergroups

1. Introduction

Zadeh in 1965 has introduced the fundamental concept of a fuzzy set which provide a natural frame-work for generalizing several basic notions of algebra (Zadeh, 1965). As a generalization of fuzzy set the intuitionistic fuzzy set was introduced by Atanassov (Atanassov, 1986). Inspired from the reallities of natural phenomena like from sport games (winning/tie/defeating), from votes decision (yes/NA/no), from decision making (conform decision/ hesitating/not making), from statement (accepted /pending /rejected) etc. conducted by the fact that the law of excluded middle did not work any longer in the modern logics. Smarandache in 1995 for the first time introduced the concept of philosophy (Smarandache, 2005) and combined the non-standard analysis (Broumi, Smarandache, & Maji. 2014) with a tri-component logic/set/probability theory which represents the main distinction between fuzzy and intuitionistic fuzzy set. (Broumi, & Smarandachi. 2014). Here he included the middle component i.e. the indeterminate besides the membership and non-membership components that both seem in fuzzy logic to separate between 'absolute membership and relative membership' or 'absolute non-membership and relative non-membership' (Maji., 2013). There are also some authors who have enhanced the theory of neutrosophic sets to another extended form namely soft set (Cutland, 1988).

At instant Kazim et al. introduced the concept of left almost semigroups and right almost semigroups (Golan, 1999). Mushtaq et al. explored this structure and added various useful results to the theory of LA-semigroups (Clifford, & Preston. 1961). Later, Madad et al. work on intra-regular LA-semigroups characterized by their anti-fuzzy ideals (Khan, & Ahmad., 2010) and an ideal in intra-regular LA-semigroup (Protic, & Stevanovic., 1995) etc.

Hyper structure theory was introduced in 1934 when Marty defined hyper groups, began to analyze their properties and applied them to groups (Mushtaq, & Yusuf., 1979). A lot of papers and several books have been written on hyper structure theory (Kazim, Naseeruddin, Mushtaq, Khan., et al. 1972). Many authors studied on different aspects of hyper structure theory, for instance, (Davvaz, et al., 2007), (Aslam, & Howie., 1976), (Hasankhani, 1999) and (Yaqoob, et al., 2013), (Hilla, et al., 2011) etc introduced a semihypergroups and studied the quasi hyper ideals in semihypergroups.

Recently Hilla, et al. in 2011 introduced the notion of LA-semihypergroups as a generalization of semi groups, semihypergroups, and LA-semigroups. Yousafzai et al. works on some characterization problems in LA-semihypergroup (Yousafzai, & Corsini., 2013). Naveed et al. display a solid result on intra-regular LA-semihypergroup with pure left identity (Yaqoob, Corsini, &Yousafzai., 2013) also they present a partially ordered LA-semihypergroup.

Using neutrosophic theory, Kandasamy, et al. introduced the concept of neutrosophic algebraic structures. Soon M. Khan et al. write a book on neutrosophic set approaches to algebraic structure and a theory of Abel Grossmann’s groupoids. Agboola, et al. studied neutrosophic groups and subgroups and also the structure of neutrosophic polynomial.

Let (H,*) be an LA-semihypergroup and The neutrosophic left almost semihypergroup is generated by H and I under the binary hyper operation ‘o’ represented by the part k is non-neutrosophic and the part l is called neutrosophic part of generating set. Then the couple becomes a neutrosophic hypergroupoid if hold for all A neutrosophic hyper groupoid satisfying the neutrosophic left invertive law i.e