一类乘法幂等半环的格林关系

Green’s relations on a class of semiring which multiplicative reduct is an idempotent semigroup

研究了满足附加恒等式x+x+x≈x,2x+2y≈2(x+y)的乘法幂等半环,给出了与该半环的乘法半群(加法半群)的格林关系相关的二元关系^(·)L∧^(+)D,^(·)L∧^(+)L,^(·)L∧^(+)R,^(+)L∧^(·)D的刻画,得到了这些二元关系是同余的充要条件,进而证明了由这些同余确定的半环类是簇。

The multiplicatively idempotent semirings satisfying the identities x+x+x=x,2x+2y=2(x+y)are studied.The characterizations of the binary relations ^(·)L∧^(+)D,^(·)L∧^(+)L,^(·)L∧^(+)R,^(+)L∧^(·)D related to the Greens relation of the multiplicative semigroups(additive semigroups)of the semirings are given,and the sufficient and necessary conditions which make these binary relations be congruences are obtained.Moreover,the classes of semirings which are determined by these congruences are proved to be semiring varieties.