1 BOOLEAN EMBEDDING ━ From a Partially Ordered Structure to a Boolean Algebra ━

We prove the following theorem. Theorem Let（Ｐ,≦）be an arbitrary partially ordered structure. Then, there is a canonical embedding (1:1, order-preserving map) ｅ：（Ｐ,≦）→ such that (i). is a complete Boolean algebra. (ii). (∀p,q∈Ｐ)(p and q are compatible →ｅ(p)∧ｅ(q) ≠) (iii). {ｅ(p)｜p∈Ｐ} satisfies “a small property ” in the sense that; (a) If Ｐ is finite, then is finite. (b) If Ｐ is infinite, then for any infinite branch l in － {}, ( ∀ ∈l)(∃q∈Ｐ)(ｅ(q) ≦). (c) for any minimal element a in Ｐ, ｅ(a) becomes a minimal element in － {}. □ Especially, when（Ｐ,≦）is finite, we give an algorithm called “the Normal Separativization”which realize the embedding ｅ. As the byproduct, we obtain an interesting insight that the set of all finite partially ordered structures can be classified to categories called“Boolean complexity of the type (n,m)”, where n and m are natural numbers