By Seidenberg A.

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Motivation In graph models of combinatory algebras we can represent algebraic, relational, and algorithmic structures as inner algebras (cf. contributions of Engeler, Weibel, and Amrhein). Problems of the original domain are thus transferred into the combinatory algebra and transformed into equations. In this way, the solvability of these problems is reformulated as solvability questions for the corresponding equations. If there is no solution of an equation in a certain inner algebra, we might be able to extend this algebra with an abstract solution.

Xi : Uk'\(U\T)k, - t T. (r'3) For each equation t=s E r, we consider its transformation according to Definition 2, in Section 4. LkX(L~\Tt*) = LkX(L~\TS*) t* = s* if var(t,s) = {Vb ... ,vd if there are no variables. Remark. The first equation is equivalent to \lVb ... ,Vk : L~\Tt*(XV1)··· (XVk) = L~\T S*(XV1)··· (XVk). To verify the first part (i) of the Representation Theorem r we assume the sequence (h; IiEl) to be a solution of r' in the universal combinatory model M. Notice that r' indeed has a matching set of variables X,XiEl.

E. r. between functions, can be expressed in the model. r. g holds if and only if the representative F of f is in a certain subalgebra that is generated by the representative G of g. Through the embedding of the functions as elements (combinators) of the graph model VN each degree d becomes a set of combinators, namely, the representatives of all functions of degree d. Since our aim is to represent degrees by subalgebras (which are closed under partial recursive derivations), we will include in the subalgebra representing a degree d not only the representatives (graphs) of the functions of degree d, but those of all functions of degree d or lower.

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A new decision method for elementary algebra by Seidenberg A.

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