A new algebraic structure, the R monoid, is associated with a discrete-time, timeinvariant linear dynamical system over a commutative ring R, and its properties are investigated.
R. E. Kalman [1] has discovered that the state space of a (discrete-time, constant) linear dynamical system over a field K admits the structure of a K[z] module, where K[z] is the ring of polynomials in one variable over K. Kalman's construction generalizes verbatim to linear systems over art arbitrary commutative ring R. In this paper we exhibit a new algebraic structure associated with such systems, which we call an R monoid. A basic task of an algebraic theory of linear systems is to tie the linear structure with the dynamics. Kalrnan accomplishes this by defining a new operation-convolution on the state space which is compatible with the R-module structure. We, on the other hand, work with the monoid of transformations induced on the state space by the dynamical action of the inputs. As has been observed by Kalman [5,6], the dynamical action is not compatible, in a classical sense, with the structure of the state space as an R module. We find, however, a new compatibility condition. This new condition does not seem to have been investigated in the mathematical literature. We call monoids satisfying this condition R monoids, and investigate their basic properties. Future papers will discuss deeper applications.