Word problem for ringoids of numerical functions
Awad A. Iskander · Transactions of the American Mathematical Society · 1971
A. The composition ringoid of functions on (i) the positive integers, (ii) all integers, (iii) the reals and (iv) the complex numbers, do not satisfy any identities other than those satisfied by all composition ringoids. B. Given two words u , υ u,\upsilon of the free ringoid, specific functions on the positive integers, f 1 , … , f k {f_1}, \ldots ,{f_k} can be described such that u ( f 1 , … , f k ) u({f_1}, \ldots ,{f_k}) and υ ( f 1 , … , f k ) \upsilon ({f_1}, \ldots ,{f_k}) , evaluated at 1, are equal iff u = υ u = \upsilon is an identity of the free ringoid.