Obviously, a dense complete lattice has the following property: $\vee \{\overleftarrow{\alpha },\overrightarrow{\alpha });(\overleftarrow{\alpha },\overrightarrow{\alpha })<$$\left(\overleftarrow{\beta },\overrightarrow{\beta }\right)\}=\left(\overleftarrow{\beta },\overrightarrow{\beta }\right);\wedge \left\{\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right);\left(\overleftarrow{\beta },\overrightarrow{\beta }\right)\right\}=\left(\overleftarrow{\beta },\overrightarrow{\beta }\right).$

Definition 8.24 Assume $\left(\left(\overleftarrow{K},\overrightarrow{K}\right),\le \right)\text{is}\text{a}\text{sun-ordered}\text{set;}\text{mapping}N;\left(\overleftarrow{K},\overrightarrow{K}\right)\to$$\left(\left(\overleftarrow{K},\overrightarrow{K}\right),\le \right)\text{is}\text{a}\text{sub-odered}\text{set;}\text{mapping}N:\left(\overleftarrow{K},\overrightarrow{K}\right)\to$ is order preserving mapping. When $\left(\overleftarrow{a},\overrightarrow{a}\right)\ge \left(\overleftarrow{b},\overrightarrow{b}\right)(\forall \left(\overleftarrow{a},\overrightarrow{a}\right),\left(\overleftarrow{b},\overrightarrow{b}\right)\in$$\left(\overleftarrow{K},\overrightarrow{K}\right),\text{there}\text{is}N\left(\overleftarrow{b},\overrightarrow{b}\right)\le N\left(\overleftarrow{a},\overrightarrow{a}\right),$$N\left(N\left(\overleftarrow{a},\overrightarrow{a}\right)\right)=\left(\overleftarrow{a},\overrightarrow{a}\right),$, and we call N as reverse mapping. If i reverse mapping N satisfies the law of unity, namely, $N\left(N\left(\overleftarrow{a},\overrightarrow{a}\right)\right)=\left(\overleftarrow{a},\overrightarrow{a}\right),$ then we call N as pseudo-complement.

Theorem 8.17 Assume N is the pseudo-complement on lattice $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right),$ then

$\text{N}\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overleftarrow{b}\right))=\text{N}\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overleftarrow{b}\right);\text{N}\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overleftarrow{b}\right))=\text{N}\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \text{N}\left(\overleftarrow{b},\overleftarrow{b}\right).$

Proof: By $\left(\overleftarrow{a},\overrightarrow{a}\right)\le \left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right)\right),\left(\overleftarrow{b},\overrightarrow{b}\right)\le \left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right)\right),\text{N}\left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right)\right)\le$$\left(\text{N}\left(\overleftarrow{a},\overrightarrow{a}\right)\wedge \text{N}\left(\overleftarrow{b},\overrightarrow{b}\right)\right).$

Corollaries: $\text{N}\left(\left(\overleftarrow{a},\overrightarrow{a}\right)\wedge \left(\overleftarrow{b},\overrightarrow{b}\right)\right)\ge \left(\text{N}\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \text{N}\left(\overleftarrow{b},\overrightarrow{b}\right)\right),\text{then}\text{N(N}\left(\overleftarrow{a},\overrightarrow{a}\right)\wedge$$\text{N}\left(\overleftarrow{b},\overrightarrow{b}\right))\ge \left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right)\right);\text{thus,}\text{N}\left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right)\right)$ 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 $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)\text{is}\text{lattice;}\text{if}\forall \left(\text{<mover accent="true"> H <mo>←</mo> </mover> ,<mover accent="true"> H <mo>→</mo> </mover>}\right)\subset \left(\overleftarrow{L},\overrightarrow{L}\right),\left(\overleftarrow{H},\overrightarrow{H}\right)$ has the maximum lower bound and the minimum upper bound, we call $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)$ as complete lattice. If $\forall \left(\overleftarrow{H},\overrightarrow{H}\right)\subset \left(\overleftarrow{L},\overrightarrow{L}\right),$ there is

$\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \underset{\left(\overleftarrow{h},\overrightarrow{h}\right)\in \left(\overleftarrow{H},\overrightarrow{H}\right)}{\wedge }\left(\overleftarrow{h},\overrightarrow{h}\right)=\underset{\left(\overleftarrow{h},\overrightarrow{h}\right)\in \left(\overleftarrow{H},\overrightarrow{H}\right)}{\wedge }\left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{h},\overrightarrow{h}\right)\right)\left(8.1\right)$

$\left(\overleftarrow{a},\overrightarrow{a}\right)\wedge (\underset{\left(\overleftarrow{h},\overrightarrow{h}\right)\in \left(\overleftarrow{H},\overrightarrow{H}\right)}{\vee }\left(\overleftarrow{h},\overrightarrow{h}\right)=\underset{\left(\overleftarrow{h},\overrightarrow{h}\right)\in \left(\overleftarrow{H},\overrightarrow{H}\right)}{\vee }\left(\left(\overleftarrow{a},\overrightarrow{a}\right)\wedge \left(\overleftarrow{h},\overrightarrow{h}\right)\right)\left(8.2\right)$

And we call $\left(\overleftarrow{L},\overrightarrow{L}\right)$ as complete distributive lattice.

Definition 8.26 Assume $\left(\overleftarrow{L},\overrightarrow{L}\right)$ is complete lattice, $\left(\overleftarrow{a},\overrightarrow{a}\right)\in \left(\overleftarrow{L},\overrightarrow{L}\right),\left(\overleftarrow{A},\overrightarrow{A}\right)\subset \left(\overleftarrow{L},\overrightarrow{L}\right);$ we call $\left(\overleftarrow{A},\overrightarrow{A}\right)$ as a minimal family of $\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right).\text{When}\left(\overleftarrow{A},\overrightarrow{A}\right)\text{meets:}\left(1\right)sup\left(\overleftarrow{A},\overrightarrow{A}\right)=$$\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right),\left(2\right)\text{if}\left(\overleftarrow{B},\overrightarrow{B}\right)\subset \left(\overleftarrow{L},\overrightarrow{L}\right),sup\left(\overleftarrow{B},\overrightarrow{B}\right)=\left(\overleftarrow{a},\overrightarrow{a}\right),\text{then}\forall \left(\overleftarrow{x},\overrightarrow{x}\right)\in \left(\overleftarrow{A},\overrightarrow{A}\right),\exists \left(\overleftarrow{y},\overrightarrow{y}\right)\in$$\left(\overleftarrow{B},\overrightarrow{B}\right),\left(\overleftarrow{y},\overrightarrow{y}\right)\ge \left(\overleftarrow{x},\overrightarrow{x}\right),\text{we}\text{call}\left(\overleftarrow{A},\overrightarrow{A}\right)$ as a very big family of $\left(\overleftarrow{a},\overrightarrow{a}\right).$ When $\left(\overleftarrow{A},\overrightarrow{A}\right)\text{meets:}\left(1\right)\inf \left(\overleftarrow{A},\overrightarrow{A}\right)=\left(\overleftarrow{a},\overrightarrow{a}\right),\left(2\right)\text{if}\left(\overleftarrow{B},\overrightarrow{B}\right)\subset \left(\overleftarrow{L},\overrightarrow{L}\right),inf\left(\overleftarrow{B},\overrightarrow{B}\right)-\left(\overleftarrow{a},\overrightarrow{a}\right),\text{then}$$\forall \left(\overleftarrow{x},\overrightarrow{x}\right)\in \left(\overleftarrow{A},\overrightarrow{A}\right)\exists \left(\overleftarrow{y},\overrightarrow{y}\right)\in \left(\overleftarrow{B},\overrightarrow{B}\right),\left(\overleftarrow{y},\overrightarrow{y}\right)\le \left(\overleftarrow{x},\overrightarrow{x}\right).$

Definition 8.27 Assume lattice $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)\text{has}\left(\stackrel{\leftharpoonup }{O},\stackrel{\rightharpoonup }{O}\right),\left(\stackrel{\leftharpoonup }{I},\stackrel{\rightharpoonup }{I}\right)[\text{where}\left(\stackrel{\leftharpoonup }{O},\stackrel{\rightharpoonup }{O}\right)$ is zero element and $\left(\stackrel{\leftharpoonup }{I},\stackrel{\rightharpoonup }{I}\right)\text{is}\text{identity];}\text{if}\left(\stackrel{\leftharpoonup }{\alpha },\stackrel{\rightharpoonup }{\alpha }\right)\vee \left(\stackrel{\leftharpoonup }{\beta },\stackrel{\rightharpoonup }{\beta }\right)=\left(\stackrel{\leftharpoonup }{I},\stackrel{\rightharpoonup }{I}\right),\left(\stackrel{\leftharpoonup }{\alpha },\overrightarrow{\alpha }\right)\wedge \left(\stackrel{\leftharpoonup }{\beta },\stackrel{\rightharpoonup }{\beta }\right)=$$\left(\stackrel{\leftharpoonup }{O},\stackrel{\rightharpoonup }{O}\right),$ then we note $\left(\overleftarrow{\beta },\overrightarrow{\beta }\right)$ as a complement of $\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right).$

Definition 8.28 Lattice $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)$ is a complemented lattice when $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)$ has $\left(\overleftarrow{O},\overrightarrow{O}\right),\left(\overleftarrow{I},\overrightarrow{I}\right)$ and the other element has a complement.

Definition 8.29 Lattice $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)$ is a complemented distributive lattice when lattice $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)$ is a complemented lattice and meets ((8.1)) and ((8.2)). We call complemented distributive lattice as Boolean lattice.