Abstract
We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 + log ( 2 ) log ( α ) where α = 1 + 5 2 .
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