| Al-semiring is regarded as an significant part of semiring. In this disserta-tion, we mainly study Al-semiring variety. The results are as follows:First, the variety of Al-semirings satisfying (xy)2≈xy is studied. The notion of (2,2,1)-closed subset of a semigroup is introduced, then a model of the free object in this variety is given by using the (2,2,1)-closed subsets of a free semigroup satisfying(xy)2≈xy.Second,the variety of Al-semirings satisfying{x1x2…xn)2≈x1x2…xn is studied.The notion of (n,2,1)-closed subset of a semigroup is introduced, then a model of the free object in this variety is given by using the (n,2,1)-closed subsets of a free semigroup satisfying [x1x2…xn)2≈x1x2…xn.Third, the varieties generated by Al-semirings of order two is studied. Some congruences on a semiring in S2 is given by using the Green’s relation of a semiring. And we establish Mal’cev product decompositions of S2 into its some subvarieties. |
