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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, ≤) - B such that (i) B 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 B is finite. (b) If P is infinite, then for any infinite branch l in B - {0}, (∀p ∈ l) (∃q ∈ P)(e(q)≤ p). (c) for any minimal element a in P, e (a) becomes a minimal element in B - {0}. ■ 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 compexity of the type (n,m), where n and m are natural numbers.