The Green Relations On A Class Of Half Rings

Green’s relation is one of the main tools which to study the algebraic theory of semiring. In this paper, Green’s-relations of a semiring whose additive reduct is band and multiplicative reduct is completely regular semigroup arc studied, obtaining some conclusions of this class of semiring variety. The main results are as follows:1. Green’s L-relation of a semiring whose additive reduct is band and mul-tiplicative reduct is completely regular semigroup arc studied. We will describe the relation of L∧D,L∧L,L∧R,L∧D and give the sufficient and nec-essary conditions which make L∧D,L∧L,L∧R,L∧D are congruenc relation, and we also obtain the classes of semiring which are determined by these Green’s-relations are semiring varieties.2. Green’s D-relation of a semiring whose additive reduct is band and multiplicative reduct is completely regular semigroup are studied, we will describ the relations of D∩D,D∩L,D∩R and obtain the sufficient and necessary condition of P n X) is congruence relation.3. The L-subvarieties and D-subvarieties of the semiring variety whose additive reduct is band and multiplicative reduct is completely regular semigroup are studied, obtaining the decomposition of Mal’cev product of some subvarieties into other subvarieties.