Dynamic Fuzzy Machine Learning

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, $N(N(\overleftarrow{a},\overrightarrow{a}$