The Classification Of Some Congruence-Simple Finite Semirings With Commutative And Idempotent Addition

The notion of a semiring was introduced by Vandiver in 1934. A semiring is a universal algebra with two associative binary operations, where one of them distributes over the other. Semirings have already found their full place, e.g., various applications in theoretical computer science and algorithm theory.Congruenc-simple semirings (i.e., those possessing just two congruence relations) serve a basic construction material for many algebraic structure. In spite of the fact and in contrast to the enormous and opulent supply of information on worldwide popular simple groups and rings, not much is known on congruence-simple semirings.If both (5,+) and (s,.) are commutative, S is simply called commutative. Bashir, Hurt, Jancarek, and Kepka proved the following theorem about congruence-simple, commutative finite semirings.Theorem Let S be a congruence-simple, finite commutative semiring. Then just one of the following cases takes place:1. |5| = 2.2. 5 is a finite field.3. 5 is a zero-multiplication ring of prime order.4. 5 = V(G), where G is a finite Abelian group, V(G) = G{J{} andFor general semirings, Monico proved the following result:Theorem Let 5 be a additively commutative, congruence-simlpe finite semiring, then one of the following cases takes place:1. |S|=2.2. 5 is isomorphic to the matrix ring of finite field.3. 5 is a zero-multiplication ring of prime order.4. S is additively idempotent.Thus the classification of additively commutative, congruence-simple finite semirings is attributed to the classification of additively idempotent ones.The purpose of this dissertation is to give the classification of some additively commutative and idempotent, congruence-simple finite semirings. The dissertation is divided into three parts.In ?, we give some basic definitions about semirings. For example, S is called additively idempotent, if a + a ?a for any a 5.In ?, we give the main results of this thesis.Theorem 3.1 Let 5 be a congruence-simple finite semiring with commutative and idempotent addition. If S has an such that G = S \ {00} is a group, then S = V(G).Theorem 3.2 Let S be a congruence-simple finite semiring with commutative and idempotent addition. If (S, ? is an inversive semigroup, thenS has an infinity or a zero.Theorem 3.3 Let S be a congruence-simple finite semiring with commutative and idempotent addition. If abed - acbd, for any a,6, c, d 5, then one of the following cases takes place.(1) 5 is commative;(2) S is isomorphic to the semiring with the following addition andmultiplication tables.0 10 11 10 10 1 0 1(3) S is anti-isomorphic to the semiring in (2).