Binary Operations on Finite Sequences
Wojciech A. Trybulec · 1990
The papers [9], [4], [5], [2], [3], [8], [6], [7], and [1] provide the notation and terminology for this paper. For simplicity we adopt the following convention: D denotes a non-empty set, d, d1, d2, d3 denote elements of D, F , G, H denote finite sequences of elements of D, f denotes a function from into D, g denotes a binary operation on D, k, n, l denote natural numbers, and P denotes a permutation of Seg(lenF ). Let us consider D, n, d. Then n 7−→ d is a finite sequence of elements of D. Let us consider D, F , g. Let us assume that g has a unity or lenF ≥ 1. The functor g ⊙ F yields an element of D and is defined by: (Def.1) g ⊙ F = 1g if g has a unity and lenF = 0, there exists f such that f(1) = F (1) and for every n such that 0 6= n and n < lenF holds f(n + 1) = g(f(n), F (n + 1)) and g ⊙ F = f(lenF ), otherwise.