Powers of Circulants in Bottleneck Algebra

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 d/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 consequerntly all of these powers have a “dovetailing” property that in each row, the elements of each size are equally spaced between the larger elements. 1 This author gratefully thanks the Office of Naval Research (Grants N00014-92-J1375 and N00014-92-J-4083) for the support. Preprint submitted to Elsevier Preprint June 30, 1997