Undecidable problems concerning generalized pascal triangles of commutative algebras

Abstract

To every algebra a = (A; K, 1, r, .) of signature (0, 1, 1, 2) the generalized Pascal triangle GPT(a) is associated. GPT(a) is constructed analogously as the classical Pascal triangle but the operation "·." on the set A is used instead of the addition on the set N of nonnegative integers. Moreover, the element K is put into the top of GPT(a), and the functions 1, r are used to construct the left and the right margin of GPT(a). Further, for every word w ∈ A⁺ the generalized Pascal triangle GPT(a, w) is associated. It is constructed similarly as GPT(a), we only use the word w in the initial line instead of K. The algebra a is said to be commutative if 1=r and the operation "·" is commutative. It is known that many algorithmic problems (concerning occurences of symbols in GPT(a) or GPT(a, w) etc.) are undecidable for the class of all finite algebras. It will be shown here that some of these problems remain unsolvable for the class of all finite commutative algebras.