On The Finite Basis Problem For An Ai-semiring Variety

The finite basis problem is one of the fundamental problems in the theory of varieties.In this thesis we study the finite basis problem for subvarieties of the ai-semiring variety W,which is the join of the ai-semiring variety defined by X2 ? x and the ai-semiring variety defined by xy?zt.The main results are as follows:Firstly,we prove that every identity that is satisfied by the ai-semiringvariety W can be derivable from 8 specific identities that hold in W.This shows that W is finitely based.Secondly,we study the lattice L(W)of subvarieties of W.By constructingthe complete epimorphism from L(W)to L(Sr(2,1))of subvarieties of Sr(2,1),the characterization of L(W)is given and L(W)is proved to be an distributive lattice of order 312.This implies that all subvarieties of W are finitely based.