Algebraic representations for finite-state machines .1. Monoid-ring formulation

Special algebraic structures, which are rings of functions with finite support, are introduced. These structures are used to develop representations for finite-state machines. Three equivalent representations for finite-state machines are presented. The first is given in terms of elements of a monoid ring based on a finite set. The second is given in terms of elements of a monoid ring based on n-tuples. The third is given in terms of the polynomial ring in 2n indeterminates. The representations are shown to be unique, and examples of them are given.