| In this thesis we study Burnside AI-semiring variety Sr(n, m) which is determined by xn≈xm. Firstly, we give a model of the free object in Sr(n,m). Secondly, we show that that variety generated by all AI-semirings of order two is hereditarily finitely based, each subvariety of this variety is finitely generated, and the lattice of subvarieties of it is a Boolean algebra of order 64; we show that the multiplicative semigroup of each member of Sr(n,1) is a regular orthocryptogroup, and give models of free objects of some subvarieties of Sr(n,1); we study some subvarieties of the variety SGn° which consists of all members of Sr(n,1) whose multiplicative semigroups are Clifford semigroups, and show that SG3°is hereditarily finitely based, each subvariety of this variety is finitely based, and the lattice of subvarieties of it is a distributive lattice of order 9. Finally, we show that in addition to one semiring, other AI-semirings of order three are finitely based. |
