The Algebra in Natural Structures

Natural structures are assembled with the formation of NE-connections which are a generalization of attraction forces. Here, we construct two nested axiomatic systems in which NE-connections can be understood as an algebraic operation. Then, we define a partial order ≤ that represents the relation ``being part of'' between the elements of the space of natural entities. We prove that the collection of elements of an arbitrary natural structure n which is denoted as Xₙ is a commutative semigroup (Xₙ,•) in which every element is idempotent. The operator • represents the act of NE-connecting. We follow this by proving the codefinition of ≤ and • in finite NEs as well as their compatibility. Finally, we propose a mathematical definition of the collection of families of NE-connections and point out its relationship with Xₙ.