Some Studies Of Finite Semirings And Semiring Varieties

In this thesis we study some varieties generated by finite semirings and theirs subvarieties lattices,and obtain some meaningful results.The results can be divided into the following four parts.Firstly,we give a decompo-sition theorem of semirings in HSP(L2,R2,D2,Z2)generated by the two-element additively completely regular and multiplicatively idempotent semir-ings L2,R2,D2 and Z2,and prove that the subdirectly irreducible ring in HSP(L2,R2,D2,Z2)only has Z2.Using the above results,we character-ize the subvarieties lattice L(HSP(L2,R2,D2,Z2))of the semiring variety HSP(L2,R2,D2,Z2),and show that every member of this lattice is finite based.Based on the above results,we obtain that the subvarieties lattice L(HSP(L2,R2,D2,Z2,M2))of the semiring variety HSP(L2,R2,D2,Z2,M2)is a Boolean algebra of order 32.Secondly,we give a decomposition theo-rem of semirings in the Mal'cev product Rn O(N?S+l)of the semiring va-riety Rn and the additively idempotent semiring variety N?S+l,using this result,we prove that the Mal'cev product Rn O(N? S+l)is a semiring vari-ety determined by additional identities xn ? x and x +(2n-2)xyx ? x,and prove that Rn O(N? S+l)= Rn ?(N? S+l),also characterize the subdirectly irreducible members in Rn O(N? S+l).Based on the above results,we obtain that each subvariety of Rn O(N? S+l)is finite based.Thirdly,using the notion of p + 1-closed subsets of semigroups and the free objects of the semigroup variety SAp+1,we give the free object mod-el of the semiring variety SAp+1 O.By means of the complete epimorphis-m?:L(ROBAp+1 O)?L(Sr(2,1)),V ? V ? Sr(2,1),we divide the sub-varieties lattice L(ROBAp+1 O)of the additively idempotent semiring variety ROBAp+1 O into 78 intervals,using the five intervals of the subvarieties lattice L(Sr(2,1))of the idempotent semiring variety Sr(2,1):[T,N ? P ? Sr(2,1)],[M,P ? Sr(2,1)],[D,N ? Sr(2,1)],[D ? M,K ? Sr(2,1)]and[Bi,Sr(2,1)],we discuss the above 78 intervals,and prove that L(ROBAp+1 O)is a 179-element distributive lattice,and show that each member of this lattice is finite based and finite generated.Lastly,we introduce the notion of(m,n,1)-closed subsets of semigroups,using this notion and the free objects of the semigroup variety Sg(m,n,1),we construct a free object modal of the semiring variety Sr(m,n,1).