Powers of circulants in bottleneck algebra Original Research Article

Consider the powers of a square matrix A of order n in bottleneck algebra, where addition and multiplication are replaced by the max and min operations. The powers are periodic, starting from a certain power AK. The smallest such K is called the exponent of A, and the length of the period is called the index of A. Cechlárová has characterized the matrices of index 1. Here we consider the case where A is a circulant matrix. We show that a circulant A is idempotent (exponent and period equal to 1) if and only if the set of positions of those entries of the first row that exceed any constant forms a group under addition modulo n (positions are indexed from 0 to n - 1). The exponent of a circulant of order n does not exceed n - 1, and this bound is best possible. The index of a circulant of order n is frsold/d′, where d = gcd(n, j2 − j1, …, jt - j1), d′ = gcd(d, j1), and j1, …, jt is the set of positions of the maximal elements in the first row. When the index is 1, we say that the circulant is strongly stable; this happens if and only if d divides j1, and this fact is shown to be equivalent to the result of Cechlárová for the case of circulant matrices. One of the powers Ak, k ≥ K, is idempotent, and consequently all of these powers have the “dovetailing” property that in each row, the elements of each size are equally spaced between the larger elements.