Algebraic Structures on a Set of Discrete Dynamical System and a Set of Profile

A discrete dynamical system is represented as a directed graph with graph nodes called states that can be seen on the dynamical map. This discrete dynamical system is symbolized by , where is a finite set of states and the function g is a function from to . In the dynamical map, the discrete dynamical system has a height where the number of states in each height is called a profile. The set of discrete dynamical systems has an addition operation defined as a disjoint union on the graph and a multiplication operation defined as a tensor product on the graph. The set of discrete dynamical systems and the set of profiles are very interesting to observe from the algebraic point of view. Considering operation on the set of discrete dynamical systems and the set of profiles, we can see their algebraic structure. By recognizing the algebraic structure, it will be easy to solve the polynomial equation in the discrete dynamical system and in the profile. In this research, we will investigate the algebraic structure of discrete dynamical systems and the set of profiles. This research shows that the set of discrete dynamical system has an algebraic structure, which is a commutative semiring and the set of profiles has an algebraic structure, which is a commutative semiring and -semimodule. Moreover, both sets have the same property, which is isomorphic to the set of non-negative integers.