8.2.1The conception dynamic fuzzy relations

In the real word, there are a large number of relations that are dynamic and fuzzy. In some way, it reflects whether the relationship is close or apart and whether the relations are dynamic. “Wang Fang grows more and more like her mother” and “The relationship between the students and teachers is more and more harmonious” – these are dynamic fuzzy relations.

Definition 8.6 Assume $\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)$ is a common DFD set and $\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)$ is dynamic fuzzy lattice; we call the mapping $\left(\stackrel{\leftharpoonup }{A},\stackrel{\rightharpoonup }{A}\right):\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\to \left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)as\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)$ type dynamic fuzzy set, and we note that $DF\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)=\left\{\left(\stackrel{\leftharpoonup }{A},\stackrel{\rightharpoonup }{A}\right):\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\to \left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)\right\}.$

Assume $\left(\stackrel{\leftharpoonup }{A},\stackrel{\rightharpoonup }{A}\right),\left(\stackrel{\leftharpoonup }{A},\stackrel{\rightharpoonup }{B}\right)\in DF\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right),\text{if}\left(\stackrel{\leftharpoonup }{A},\stackrel{\rightharpoonup }{A}\right)\left(\stackrel{\leftharpoonup }{x},\stackrel{\rightharpoonup }{x}\right)\le \left(\stackrel{\leftharpoonup }{B},\stackrel{\rightharpoonup }{B}\right)\left(\stackrel{\leftharpoonup }{x},\stackrel{\rightharpoonup }{x}\right),$$\left(\forall \left(\stackrel{\leftharpoonup }{x},\stackrel{\rightharpoonup }{x}\right)\in \left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\right).$ Namely, contained in $\left(\stackrel{\leftharpoonup }{B},\stackrel{\rightharpoonup }{B}\right),$ we note that $\left(\stackrel{\leftharpoonup }{A},\stackrel{\rightharpoonup }{A}\right)\subset$$\left(\stackrel{\leftharpoonup }{B},\stackrel{\rightharpoonup }{B}\right),\text{then}\left(DF\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right),c\right)$ is dynamic fuzzy poset.

Definition 8.7 If $\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right)\in DF\left(\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\times \left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right)\right),$ we note that $\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right)$ is the L-type dynamic fuzzy relations among $\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\text{to}\left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right);\text{if}\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)=\left[\left(\stackrel{\leftharpoonup }{0},\stackrel{\rightharpoonup }{0}\right),\left(\stackrel{\leftharpoonup }{1},\stackrel{\rightharpoonup }{1}\right)\right],$$\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right)\in DF\left(\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\times \left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right)\right),$ then we note that $\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right)$ is the dynamic fuzzy relations among $\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\text{to}\left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right).$

Definition 8.8 As mentioned, binary dynamic fuzzy relations can be promoted to n set dynamic fuzzy relations.

Definition 8.9 Assume $\left({\stackrel{\leftharpoonup }{X}}_1,{\stackrel{\rightharpoonup }{X}}_1\right),\left({\stackrel{\leftharpoonup }{X}}_2,{\stackrel{\rightharpoonup }{X}}_2\right),\mathrm{...},\left({\stackrel{\leftharpoonup }{X}}_n,{\stackrel{\rightharpoonup }{X}}_n\right)$ is n nonempty set, then

$\left(\overleftarrow{R},\overrightarrow{R}\right):\left({\overleftarrow{X}}_1,{\overrightarrow{X}}_1\right)\times \left({\overleftarrow{X}}_2{\overrightarrow{X}}_2\right)\times \mathrm{...}\times \left({\overleftarrow{X}}_n,{\overrightarrow{X}}_n\right)\to \left[\left(\overleftarrow{0},\overrightarrow{0}\right),\left(\overleftarrow{1},\overrightarrow{1}\right)\right].$

We note that $\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)={\stackrel{\leftharpoonup }{X}}_1,{\stackrel{\rightharpoonup }{X}}_1)\times \left({\stackrel{\leftharpoonup }{X}}_2,{\stackrel{\rightharpoonup }{X}}_2\right)\times \mathrm{...}\times \left({\stackrel{\leftharpoonup }{X}}_n,{\stackrel{\rightharpoonup }{X}}_n\right)$ as the n set dynamic fuzzy relations.

8.2.2Property of dynamic fuzzy relations

Proposition 8.1 Assume $\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right),\left(\stackrel{\leftharpoonup }{S},\stackrel{\rightharpoonup }{S}\right),\left(\stackrel{\leftharpoonup }{P},\stackrel{\rightharpoonup }{P}\right),{\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right)}_{\left(\stackrel{\leftharpoonup }{t},\stackrel{\rightharpoonup }{t}\right)}\in DF\left(\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\times \left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{y}\right)\right),$$\left(\stackrel{\leftharpoonup }{t},\stackrel{\rightharpoonup }{t}\right)\in \left(\stackrel{\leftharpoonup }{T},\stackrel{\rightharpoonup }{T}\right),$ then we have the following properties:

$\begin{array}{l}\left(6\right)\left(\overleftarrow{R},\overrightarrow{R}\right)\vee \left(\overleftarrow{I},\overrightarrow{I}\right)=\left(\overleftarrow{I},\overrightarrow{I}\right),\left(\overleftarrow{R},\overrightarrow{R}\right)\Lambda \left(\overleftarrow{I},\overrightarrow{I}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right)\\ \left(\overleftarrow{R},\overrightarrow{R}\right)\vee \left(\overrightarrow{0},\overleftarrow{0}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right),\left(\overleftarrow{R},\overrightarrow{R}\right)\Lambda \left(\overrightarrow{0},\overleftarrow{0}\right)=\left(\overleftarrow{0},\overrightarrow{0}\right),\end{array}$

where $\left(\stackrel{\leftharpoonup }{I},\stackrel{\rightharpoonup }{I}\right),\left(\stackrel{\leftharpoonup }{O},\stackrel{\rightharpoonup }{O}\right)\in DF\left(\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\times \left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right)\right),\left(\stackrel{\leftharpoonup }{I},\stackrel{\rightharpoonup }{I}\right)\left(\left(\stackrel{\leftharpoonup }{x},\stackrel{\rightharpoonup }{x}\right),\left(\stackrel{\leftharpoonup }{y},\stackrel{\rightharpoonup }{y}\right)\right)=\left(\stackrel{\leftharpoonup }{1},\stackrel{\rightharpoonup }{1}\right),$$\left(\stackrel{\leftharpoonup }{0},\stackrel{\rightharpoonup }{0}\right)\left((\stackrel{\leftharpoonup }{x},\stackrel{\rightharpoonup }{x}),(\stackrel{\leftharpoonup }{y},\stackrel{\rightharpoonup }{y})\right)=\left(\stackrel{\leftharpoonup }{0},\stackrel{\rightharpoonup }{0}\right)\left(\forall \left(\stackrel{\leftharpoonup }{x},\stackrel{\rightharpoonup }{x}\right),\left(\stackrel{\leftharpoonup }{y},\stackrel{\rightharpoonup }{y}\right)\right)\in \left(\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right),\left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right)\right)).$ We note respectively that dynamic fuzzy universal relation and dynamic fuzzy zero relation among $\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\text{to}\left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right).$

Definition 8.10 If $\left({\stackrel{\leftharpoonup }{R}}_1,{\stackrel{\rightharpoonup }{R}}_1\right),\left({\stackrel{\leftharpoonup }{R}}_2,{\stackrel{\rightharpoonup }{R}}_2\right)\in D{F}_{\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)}\left(\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right),\left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right)\right)\forall \left(\left(\stackrel{\leftharpoonup }{x},\stackrel{\rightharpoonup }{x}\right),\left(\stackrel{\leftharpoonup }{y},\stackrel{\rightharpoonup }{y}\right)\right)\in$$\left((\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}),(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y})\right),$ we define that:

Theorem 8.9 ${\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right)}^{-1}$ have the following properties:

$\begin{array}{l}1)\left({\overleftarrow{R}}_1,{\overrightarrow{R}}_1\right)\subset \left({\overleftarrow{R}}_2,{\overrightarrow{R}}_2\right)\Rightarrow {\left({\overleftarrow{R}}_1,{\overrightarrow{R}}_1\right)}^{-1}\subset {\left({\overleftarrow{R}}_2,{\overrightarrow{R}}_2\right)}^{-1}\\ 2){\left({\left(\overleftarrow{R},\overrightarrow{R}\right)}^{-1}\right)}^{-1}=\left(\overleftarrow{R},\overrightarrow{R}\right)\\ 3){\left(\left({\overleftarrow{R}}_1,{\overrightarrow{R}}_1\right)\cup \left({\overleftarrow{R}}_2,{\overrightarrow{R}}_2\right)\right)}^{-1}={\left({\overleftarrow{R}}_1,{\overrightarrow{R}}_1\right)}^{-1}\cup {\left({\overleftarrow{R}}_2,{\overrightarrow{R}}_2\right)}^{-1}\\ 4){\left(\left({\overleftarrow{R}}_1,{\overrightarrow{R}}_1\right)\cap \left({\overleftarrow{R}}_2,{\overrightarrow{R}}_2\right)\right)}^{-1}={\left({\overleftarrow{R}}_1,{\overrightarrow{R}}_1\right)}^{-1}\cap {\left({\overleftarrow{R}}_2,{\overrightarrow{R}}_2\right)}^{-1}\end{array}$

8.2.3Dynamic fuzzy matrix

Definition 8.11 We name the matrix like

$\left(\stackrel{\leftharpoonup }{E},\stackrel{\rightharpoonup }{E}\right)=\left[\begin{array}{l}({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{11},\stackrel{\rightharpoonup }{{\text{e}}_{11}})({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{12},\stackrel{\rightharpoonup }{{\text{e}}_{12}})\cdot \cdot \cdot \left({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{1n}},\stackrel{\rightharpoonup }{{\text{e}}_{1n}}\right)\\ ({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{21},\stackrel{\rightharpoonup }{{\text{e}}_{21}})({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{22},\stackrel{\rightharpoonup }{{\text{e}}_{22}})\cdot \cdot \cdot \left({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{2n}},\stackrel{\rightharpoonup }{{\text{e}}_{2n}}\right)\\ \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \\ ({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{m}1},\stackrel{\rightharpoonup }{{\text{e}}_{\text{m}1}})({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{m}2},\stackrel{\rightharpoonup }{{\text{e}}_{\text{m}2}})\cdot \cdot \cdot \left({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{mn}},\stackrel{\rightharpoonup }{{\text{e}}_{\text{m}n}}\right)\end{array}\right]$ as DF matrix with m × n order, where $\left({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{ij}},\stackrel{\rightharpoonup }{{\text{e}}_{\text{ij}}}\right)\in \left[\left(\stackrel{\leftharpoonup }{0},\stackrel{\rightharpoonup }{0}\right),\left(\stackrel{\leftharpoonup }{1},\stackrel{\rightharpoonup }{1}\right)\right],\left(\text{I}\text{=}\text{1,2,}\mathrm{...}\text{,m;}\text{j}=1,2,\mathrm{...},\text{n}\right),$ we name $\left(\stackrel{\leftharpoonup }{E},\stackrel{\rightharpoonup }{E}\right)={\left({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{ij}},\stackrel{\rightharpoonup }{{\text{e}}_{\text{ij}}}\right)}_{m\times n,}\left({\text{<mover accent="true"> e <mo>↼</mo> </mover>}}_{\text{ij}},{\stackrel{\rightharpoonup }{\text{e}}}_{\text{ij}}\right)$ as the elements in row i and column j of DF matrix $\left(\stackrel{\leftharpoonup }{E},\stackrel{\rightharpoonup }{E}\right).$

Assume $\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)=\left\{\left({\stackrel{\leftharpoonup }{x}}_1,{\stackrel{\rightharpoonup }{x}}_1\right),\left({\stackrel{\leftharpoonup }{x}}_{2,},{\stackrel{\rightharpoonup }{x}}_2\right),\mathrm{...},\left({\stackrel{\leftharpoonup }{x}}_n,{\stackrel{\rightharpoonup }{x}}_n\right)\right\},\left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right)=\{\left({\stackrel{\leftharpoonup }{y}}_1,{\stackrel{\rightharpoonup }{y}}_1\right),\left({\stackrel{\leftharpoonup }{y}}_2,{\stackrel{\rightharpoonup }{y}}_2\right)$$\mathrm{...},\left({\stackrel{\leftharpoonup }{y}}_m,{\stackrel{\rightharpoonup }{y}}_m\right)\},\left(\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right),\le \right)$ is DF lattice, $\left(\stackrel{\leftharpoonup }{0},\stackrel{\rightharpoonup }{0}\right),\left(\stackrel{\leftharpoonup }{1},\stackrel{\rightharpoonup }{1}\right)$ are the maximum an minimum element in $\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right).$

Definition 8.12 If $\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right):\left(\stackrel{\leftharpoonup }{X},\stackrel{\rightharpoonup }{X}\right)\times \left(\stackrel{\leftharpoonup }{Y},\stackrel{\rightharpoonup }{Y}\right)\to \left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right),$ we name that $\left(\stackrel{\leftharpoonup }{R},\stackrel{\rightharpoonup }{R}\right)$ is $\left(\stackrel{\leftharpoonup }{L},\stackrel{\rightharpoonup }{L}\right)$-type DF matrix; we note that