Math. Systems Theory 11, 169-175 (1977) Mathematical Systems Theory 01977 by Sp~ag~-V~rlag New York Inc. The Lattice of Minimal Realizations of Response Maps Over Rings Eduardo D. Sontag* Department of Mathematics, Rutgers University, New Brunswick, N.J. 08903 Abstract. A lattice characterization is .given for the class of minimal-rank realizations of a linear response map defined over a (commutative) Noetherian integral domain. As a corollary, it is proved that there are only finitely many nonisomorphic minimal-rank realizations of a response map over the integers, while for delay-differential systems these are classified by a lattice of subspaces of a finite-dimensional real vector space. 1. Definitionsand Notations The following notational conventions hold throughout the paper: R is a fixed (commutative) Noetherian integral domain, Q its quotient field. "Module" means R-module, "linear" means R-linear. For any module M, M' is the module HomR(M,R); rank M: = dimo(M ® Q ). Definition 1.1. Let m,p be positive integers. A response map (over R) is an infinite sequence f=(At,AE, A3,...) of p X m matrices over R. The rank off is the Q-rank of the (block) "behavior" or Hankel matrix -At A2 L-/(f): = A 3 A2 A3 A3 Aa Aa A5 .°. °°. .°° °°. Let f be an arbitrary but fixed response map of finite rank. *This research was supported in part by US Army Research Grant DA-ARO-D-31-124- 7243114 and by US Air Force Grant 72-2268 through the Center for Mathematical System Theory, University of Florida, Gainesville, FL 32611, USA.