The iterates of circulant matrices under the max-min product is investigated in this article. It is shown that if the first row of a fuzzy circulant matrix is in decreasing order, then the iterates of the circulant converge and if the first row is in increasing order, then the iterates oscillate. In both cases a complete description of the iterates is obtained. As an application, this yields an O(n) algorithm for computing the transitive closure of a certain type of fuzzy relation; viz. relations whose adjacency matrices are fuzzy circulants with decreasing first row.