Obviously, a dense complete lattice has the following property: ∨{α←,α→); (α←,α→)<(β←,β→)}=(β←,β→);∧{(α←,α→);(β←,β→)}=(β←,β→).
Definition 8.24 Assume ((K←,K→),≤) is a sun-ordered set; mapping N;(K←,K→)→((K←,K→),≤) is a sub-odered set; mapping N:(K←,K→)→ is order preserving mapping. When (a←,a→)≥(b←,b→) (∀(a←,a→), (b←,b→)∈(K←,K→), there is N(b←,b→)≤N(a←,a→),N(N(a←,a→))=(a←,a→),, and we call N as reverse mapping. If i reverse mapping N satisfies the law of unity, namely, N(N(a←,a→))=(a←,a→), then we call N as pseudo-complement.
Theorem 8.17 Assume N is the pseudo-complement on lattice ((L←,L→),≤), then
N(a←,a→)∨(b←,b←))=N(a←,a→)∨(b←,b←);N(a←,a→)∨(b←,b←))=N(a←,a→)∨N(b←,b←).
Proof: By (a←,a→)≤((a←,a→)∨(b←,b→)), (b←,b→)≤((a←,a→)∨(b←,b→)), N((a←,a→)∨(b←,b→))≤(N(a←,a→)∧N(b←,b→)).
Corollaries: N((a←,a→)∧(b←,b→))≥(N(a←,a→)∨N(b←,b→)), then N(N(a←,a→) ∧N(b←,b→))≥((a←,a→)∨(b←,b→)); thus, N((a←,a→)∨(b←,b→)) Namely, the first part of theorem has been proved. A similar process is used to prove the other part.
[Prove up].
Definition 8.25 Assume Unexpected text node: ' H ' has the maximum lower bound and the minimum upper bound, we call ((L←,L→),≤) as complete lattice. If ∀(H←,H→)⊂(L←,L→), there is
(a←,a→)∨∧(h←,h→)∈(H←,H→)(h←,h→)=∧(h←,h→)∈(H←,H→)((a←,a→)∨(h←,h→)) (8.1)
(a←,a→)∧(∨(h←,h→)∈(H←,H→)(h←,h→)=∨(h←,h→)∈(H←,H→)((a←,a→)∧(h←,h→)) (8.2)
And we call (L←,L→) as complete distributive lattice.
Definition 8.26 Assume (L←,L→) is complete lattice, (a←,a→)∈(L←,L→),(A←,A→)⊂(L←,L→); we call (A←,A→) as a minimal family of (α←,α→). When (A←,A→) meets: (1) sup(A←,A→)=(α←,α→),(2) if (B←,B→)⊂ (L←,L→), sup(B←,B→)=(a←,a→), then ∀(x←,x→)∈(A←,A→), ∃(y←,y→)∈(B←,B→),(y←,y→)≥(x←,x→), we call (A←,A→) as a very big family of (a←,a→). When (A←,A→)meets:(1) inf(A←,A→)=(a←,a→),(2) if(B←,B→)⊂(L←,L→), inf(B←,B→)−(a←,a→), then∀(x←,x→)∈(A←,A→)∃(y←,y→)∈(B←,B→),(y←,y→)≤(x←,x→).
Definition 8.27 Assume lattice ((L←,L→),≤)has(O↼,O⇀),(I↼,I⇀) [where (O↼,O⇀) is zero element and (I↼,I⇀)is identity]; if(α↼,α⇀)∨(β↼,β⇀)=(I↼,I⇀),(α↼,α→)∧(β↼,β⇀)=(O↼,O⇀), then we note (β←,β→) as a complement of (α←,α→).
Definition 8.28 Lattice ((L←,L→),≤) is a complemented lattice when ((L←,L→),≤) has (O←,O→),(I←,I→) and the other element has a complement.
Definition 8.29 Lattice ((L←,L→),≤) is a complemented distributive lattice when lattice ((L←,L→),≤) is a complemented lattice and meets ((8.1)) and ((8.2)). We call complemented distributive lattice as Boolean lattice.