International Journal of Research in Advent Technology, Vol.7, No.2, February 2019 E-ISSN: 2321-9637 Available online at www.ijrat.org 867 Classes of Inverse Semirings and its Ordering G. Rajeswari 1 , T. Vasanthi 2 Department of Applied Mathematics 1,2 , Yogi Vemana University 1,2 , Kadapa, Andhra Pradesh, India Email:grajeswariyvu@gmail.com 1 ,vasanthi_thatimakula@yahoo.co.in 2 Abstract- In this paper we study the conditions under which an additively inverse semiring is a band and commutative. Also in the case of totally ordered additive inverse semiring in which (, ) S  is positively totally ordered, we prove the additive structure is max( , ) a b b a ab     for all , ab in . S We have also framed an example for this result by considering 4 elements. Index Terms - Additive inverse semiring; zero-square semiring; positively totally ordered semiring. 1. INTRODUCTION Additive and multiplicative structures play an important role in determining the structure of semiring. If the multiplicative structure of a semiring is a rectangular band, then its additive structure is a band. Karvellas [4] proved that if the additively inverse semiring contains right multiplicative identity, then its additive structure is commutative. In [9] Zeleznekow studied additive inverse semirings and examined the conditions under which (, ) S  is a semilattice. In this paper we study the conditions under which (, ) S  is a commutative band and determine the structure of additively inverse semiring [4], [8] if (, ) S  is positively totally ordered [1],[2], [5],[6]. 2. PRELIMINARIES In this section, we have given some important definitions that are used in the theorems which are stated and proved in the following sections. Definition 2.1 A triple   ,, S  is called a semiring if   , S  is a semigroup;   , S  is a semigroup; ( ) ab c ab ac    and ( ) b ca ba ca    for every ,, abc in S . The two operations + and  are called additive and multiplicative operations respectively. Definition 2.2 A semiring   ,, S  is called an additive inverse semiring if (, ) S  is an inverse semigroup that is for each a S  there exists a unique element a S  such that a a a a     and a a a a       . Definition 2.3 In a totally ordered semiring   , ,, S  (i)   , , S  is positively totally ordered (p.t.o), if , a x ax   for all , ax in . S (ii)   ,, S  is positively totally ordered (p.t.o), if , ax ax  for all , ax in . S (iii)   , , S  is negatively totally ordered (n.t.o), if , a x ax   for all , ax in . S (iv)   ,, S  is negatively totally ordered (n.t.o) if , ax ax  for all , ax in . S Definition 2.4 A semiring S is Positive Rational Domain (PRD) if   , S  is an abelian group. Definition 2.5 Zeroid of a semiring   ,, S  is the set of all x in S such that x y y   or y x y   for some y in S. we may also term this as the zeroid of   , S  and it is denoted by Zd. Definition 2.6 A semiring S with multiplicative zero is said to be a zero square semiring if 2 0 a  for all a in S . Definition 2.7 A semigroup ( ) is commutative if  for all , ax in S . A semigroup () is commutative if    for all , ax in S . Definition 2.8