On The Varieties Generated By Semirings Of Order Two

The finite basis problem of varieties is one of the important contents in the study of Algebra. In this dissertation, we mainly study varieties generated by some semirings of order two. The results are as follows:First, the variety generated by all two-element Al-semirings and Z₁,Z₂ is studied. We solve the word problem for Z₁ and Z₂, then we solve the word problem of S8, and prove that it is finitely based. Moreover, we give the equational basis for it.Second, all the commutative additively non-idempotent semirings of order two are studied. We solve the word problem for them, and give equational basis for it.Third, the variety A₄ generated by two-element commutative semirings with non-idempotent additive reduct is studied. We solve the word problem for A₄, and prove that it is finitely based. Moreover, we give equational basis for it.