The basic question addressed in this paper is one of expressing the dimension of the intersection of kernels of linear operators that arise naturally in multivariate approximation theory in terms of the more easily computable dimensions of some basic blocks. The approach used by authors connects concepts arising in multivariate approximation theory with ideas from general algebra and algebraic geometry. The setting involves a semigroup \(G\) of commuting linear operators on a vector space \(S\) over a field \(k\) with the group operation taken as composition of linear operators. The notion of \(s\)-dimensional additivity of such a semigroup \(G\) plays a major role in this study. The paper extends and partially unites recent studies by W. Dahmen and C. A. Micchelli, C. de Boor, N. Dyn and A. Ron, Z. Shen, and others. The authors also observe the \(s\)-dimensional additivity for partial differential operators given by polynomials, and the \(s\)-dimensional additivity for difference operators given by polynomials. In both cases they give some examples where simple explicit formulas exist for the dimension of the associated set of linear operators. This is a fundamental paper with profound implications that extend beyond multivariate approximation theory, the topic which motivated this development.