The Tat Algebra And Its Irreducible Ideal

At first, we give the forms of the meet irreducible ideals and ideals of the algebraT_n(?)C(S~1)respectively in this paper. There is a one-to-one correspondence betweenmeet irreducible ideals in T_n(?)C(S~1)and {(e, x)|e∈T_n, x∈S~1}. Next we give the meetirreducible ideals' form of the algebra sum from i=1 to n⊕(T_i(?)C(S~1)) and the equivalent relationshipsbetween its meet irreducible ideals and other kinds of ideals. Then we introduce a hull-kernel topology in this algebra. Afterwards we give the definition of the TAT algebra,also define the MI-chain of this algebra and its equivalent forms. Then we introduce theconcept of the matrix unit and the infinite chain by which we give the meet irreducibleideals' form of strongly maximal TAT algebra and point out that its ideal Z may beexpressed as the cap of all the meet irreducible ideals those contain Z. Finally, we pointout the relationships between the strongly maximal TAT algebra and the MI-chain aswell as the CMI-chain.