FINITE-MEMORY SYSTEMS

Let K be a held, k and n positive integers and let A(l),..., A(k) be n x n-matrices with coefficients in K. For any function g: Gamma:= {t : = (t(l),..., t(k)) is an element of N-k \ t(1)t(2)...t(k) = 0} --> K-n there exists a unique solution x: N-k --> K-n of the system of differ ence equations (lozenge) x (t(1) + 1,..., t(k) +1) = =Sigma(j=1)(k) A( j)x(t(1) + 1,..., t(j-1) +1, t(j), t(j+1) + 1,..., t(k) + 1) defined b y the matrix-k-tuple (A(1),..., A(k)) is an element of M(n; K)(k) such that x(\Gamma) = g. The system (lozenge) is called ''finite-memory sy stem'' iff for every function g with finite support the values x(t(1), ...,t(k)) are 0 for sufficiently big t(1) +... + t(k). In the case k = 2, K = R, these systems and the corresponding matrix-k-tuples have be en studied in [1], [3], [4], [5], [6], [7] In this paper I generalize these results to an arbitrary positive integer k and to an arbitrary f ield K.