β Scribed by Wolfgang Rautenberg
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Algebras d = (A, F), d'= (A, F') are polynomial equivalent if for each (m-place) f~ F there is a term ~Ps in the function from F' such that in A holds the defining equationfxl ... x m = ~Pl (xl, "", xm), and conversely, to eachf'~ F' there is a defining equation forf' in functions from F. Matrices (d, D), (d', D') are polynomial equivalent if ,~r d' are polynomial equivalent and D = D' PROBLEM. Let (d, D), (d', D) be polynomial equivalent finite matrices. Is the consequence defined by the matrix (d, D)finitely based (on sequential rules) iff the consequence of the matrix (~r D) is finitely based?
It can be shown that there is no 2-element matrix with this property so that the consequence of a 2-element matrix is finitely based in any finite language, not only in the standard languages used in Post's classification of 2-element algebras.
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