Assume $\left(\overleftarrow{X},\overrightarrow{X}\right)=\left\{\left(\overleftarrow{x},\overrightarrow{x}\right),\mathrm{...},{\left({\overleftarrow{x}}_n,{\overrightarrow{x}}_n\right)}_t\right\},\left(\overleftarrow{Y},\overrightarrow{Y}\right)=\left\{\left({\overleftarrow{y}}_1,{\overrightarrow{y}}_1\right),\mathrm{...}\left({\overleftarrow{y}}_m,{\overrightarrow{y}}_m\right)\right\},\left((\overleftarrow{L},\overrightarrow{L}\right),$ ≤) is a DF grid, and that 0 and 1 respectively represent the minimum and maximum elements of $\left(\overleftarrow{L},\overrightarrow{L}\right).$

Definition 6.7 If $\left(\overleftarrow{R},\overrightarrow{R}\right):\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\to \left(\overleftarrow{L},\overrightarrow{L}\right),$ we say that $\left(\overleftarrow{R},\overrightarrow{R}\right)$ is an $\left(\overleftarrow{L},\overrightarrow{L}\right)$-type DF matrix, written as

$\left(\overleftarrow{R},\overrightarrow{R}\right)=\left(\begin{array}{cccc}\left({\overleftarrow{r}}_{11},{\overrightarrow{r}}_{11}\right)& \left({\overleftarrow{r}}_{12},{\overrightarrow{r}}_{12}\right)& \cdots & \left({\overleftarrow{r}}_{1m},{\overrightarrow{r}}_{1m}\right)\\ \left({\overleftarrow{r}}_{21},{\overrightarrow{r}}_{21}\right)& \left({\overleftarrow{r}}_{22},{\overrightarrow{r}}_{22}\right)& ⋮& \left({\overleftarrow{r}}_{2m},{\overrightarrow{r}}_{2m}\right)\\ ⋮& ⋮& ⋮& ⋮\\ \left({\overleftarrow{r}}_{n1},{\overrightarrow{r}}_{n1}\right)& \left({\overleftarrow{r}}_{n2},{\overrightarrow{r}}_{n2}\right)& \dots & \left({\overleftarrow{r}}_{nm},{\overrightarrow{r}}_{nm}\right)\end{array}\right),$

where $\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right)\left(\left({\overleftarrow{x}}_i,{\overrightarrow{x}}_i\right),\left({\overleftarrow{y}}_j,{\overrightarrow{y}}_j\right)\right)\left(i\le n,j\le m\right).$

In particular, $\left(\overleftarrow{R},\overrightarrow{R}\right):\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\to \left[0,1\right]\times \left[\leftarrow ,\to \right]$ is a DF matrix.

Obviously, an L-type DF matrix is an L-type DF relationship, and a DF matrix is a DF relationship.

Write $D{F}_L\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right)=\left\{\left(\overleftarrow{R},\overrightarrow{R}\right);\left(\overleftarrow{R},\overrightarrow{R}\right):\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\to \left(\overleftarrow{L},\overrightarrow{L}\right)\right\}.$ If $\left(\overleftarrow{R},\overrightarrow{R}\right),\left(\overleftarrow{S},\overrightarrow{S}\right)\in D{F}_L\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right),\left(\overleftarrow{R},\overrightarrow{R}\right)=\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right),\left(\overleftarrow{S},\overrightarrow{S}\right)=\left({\overleftarrow{s}}_{ij},{\overrightarrow{s}}_{ij}\right).$ If

$\forall i\le n,j\le m,\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)\le \left({\overleftarrow{s}}_{ij},{\overrightarrow{s}}_{ij}\right),\left(\overleftarrow{R},\overrightarrow{R}\right)$ is contained in $\left(\overleftarrow{S},\overrightarrow{S}\right),$ which we write as $\left(\overleftarrow{R},\overrightarrow{R}\right)\subset \left(\overleftarrow{S},\overrightarrow{S}\right).$

Obviously, $D{F}_L\left(\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right),\subset \right)$ is a poset.

Assume (L,S,T,N) is a module system, and write

$\begin{array}{l}\left(\overleftarrow{R},\overrightarrow{R}\right)\cup \left(\overleftarrow{S},\overrightarrow{S}\right)=\left(\left(\overleftarrow{S},\overrightarrow{S}\right)\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right),\left(\overleftarrow{s_{ij}},{\overrightarrow{s}}_{ij}\right)\right);i\le n,j\le m\\ \left(\overleftarrow{R},\overrightarrow{R}\right)\cup \left(\overleftarrow{S},\overrightarrow{S}\right)=\left(\left(\overleftarrow{S},\overrightarrow{S}\right)\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right),\left(\overleftarrow{s_{ij}},{\overrightarrow{s}}_{ij}\right)\right);i\le n,j\le m\\ {\left(\overleftarrow{R,}\overrightarrow{R}\right)}^c=\left(\overleftarrow{N},\overrightarrow{N}\right)\left(\left({\overrightarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right);l\le n,j\le m\right).\end{array}$

Then, $D{F}_L\left(\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right),\cap ,\cup ,c\right)$ is also a module system, $\left(\overleftarrow{0},\overrightarrow{0}\right)=\left(\left({\overleftarrow{o}}_{ij},{\overrightarrow{o}}_{ij}\right)\right)=$$\left(\overleftarrow{0},\overrightarrow{0}\right);i\le n,j\le m)\mathrm{and}\left(\overleftarrow{E},\overrightarrow{E}\right)=\left(\left({\overleftarrow{e}}_{ij},{\overrightarrow{e}}_{ij}\right)=\left(\overleftarrow{1},\overrightarrow{1}\right);i\le n,j\le m\right)$ are the minimum element and maximum element of $D{F}_L\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right).$

Definition 6.8 Assume $\left(\overleftarrow{R},\overrightarrow{R}\right)$ is an L-type DF matrix and $\left(\overleftarrow{R},\overrightarrow{R}\right)=(\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right);$$i\le n,j\le m).$ If there exists an L-type DF matrix $\left(\overleftarrow{Q},\overrightarrow{Q}\right)=\left(\left({\overleftarrow{q}}_{ij},{\overrightarrow{q}}_{ij}\right);j\le m,i\le n\right)$ such that $\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right),\text{we}\text{call}\left(\overleftarrow{R},\overrightarrow{R}\right)$ regular and say that $\left(\overleftarrow{Q},\overrightarrow{Q}\right)$ is the general type-T matrix of

If there is some $\left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)=\left(\overleftarrow{Q},\overrightarrow{Q}\right),\text{then }\left(\overleftarrow{Q},\overrightarrow{Q}\right)$ is the T inverse matrix of $\left(\overleftarrow{R},\overrightarrow{R}\right)$

Theorem 6.1 Assume $\left(\overleftarrow{Q},\overrightarrow{Q}\right)$ is the general T inverse matrix of $\left(\overleftarrow{R},\overrightarrow{R}\right).\text{Then},\left(\overleftarrow{S},\overrightarrow{S}\right)=$$\left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)$ is the T inverse matrix of $\left(\overleftarrow{R},\overrightarrow{R}\right)$ where T is a distribution grid.

Proof: According to Definition 6.8,

$\begin{array}{l}\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{S},\overrightarrow{S}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\\ =\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right).\end{array}$

Thus,

$\begin{array}{l}\left(\overleftarrow{S},\overrightarrow{S}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{S},\overrightarrow{S}\right)=\left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\\ =\left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)=\left(\overleftarrow{S},\overrightarrow{S}\right),\end{array}$

which completes the proof.

Theorem 6.2 Assume $\left(\overleftarrow{R},\overrightarrow{R}\right)$ is a DF matrix, and the sufficient and necessary condition for $\left(\overleftarrow{R},\overrightarrow{R}\right)$ to be regular is

$\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{X},\overrightarrow{X}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right).$

Now, $\left(\overleftarrow{X},\overrightarrow{X}\right)$ is the biggest general DF inverse matrix. When $\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)=$$\left(\overleftarrow{R},\overrightarrow{R}\right),\text{then}\left(\overleftarrow{Q},\overrightarrow{Q}\right)\subset \left(\overleftarrow{X},\overrightarrow{X}\right).$

Proof: The sufficient condition is obviously true, so we need to prove the necessary condition.

Assume $\left(\overleftarrow{R},\overrightarrow{R}\right)$ is regular and that there exists some $\left(\overleftarrow{Q},\overrightarrow{Q}\right)$ satisfying $\left(\overleftarrow{R},\overrightarrow{R}\right)\circ$$\left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right).$ Thus, there is

$\underset{k=1}{\overset{n}{\vee }}(\underset{j=1}{\overset{m}{\vee }}\left(\left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\wedge \left({\overleftarrow{q}}_{ik},{\overrightarrow{q}}_{ik}\right)\right)\wedge \left({\overleftarrow{r}}_{kj},{\overrightarrow{r}}_{kj}\right)=\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)\left(i\le n,j\le m\right),$

that is,

$\underset{k=1}{\overset{n}{\vee }}(\underset{e=1}{\overset{m}{\vee }}\left(\left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\wedge \left({\overleftarrow{q}}_{ek},{\overrightarrow{q}}_{ek}\right)\right)\wedge \left({\overleftarrow{r}}_{kj},{\overrightarrow{r}}_{kj}\right)=\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)\left(i\le n,j\le m\right).$

Thus, for $\forall k\le n,e\le m,$ there is $\left(\left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\wedge \left({\overleftarrow{q}}_{ik},{\overleftarrow{q}}_{ik}\right)\right)\wedge \left({\overleftarrow{r}}_{kj},{\overrightarrow{r}}_{kj}\right)\le \left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)(i\le n,$ j ≧ m).

We can conclude the following:

$\begin{array}{l}1.\left(\left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\wedge \left({\overleftarrow{r}}_{kj},{\overrightarrow{r}}_{kj}\right)\right)\le \left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)\Rightarrow 0\le \left({\overleftarrow{q}}_{ek},{\overrightarrow{q}}_{el}\right)\le 1;\\ 2.\left(\left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\wedge \left({\overleftarrow{r}}_{kj},{\overrightarrow{r}}_{kj}\right)\right)>\left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)\Rightarrow 0\le \left({\overleftarrow{q}}_{ek},{\overrightarrow{q}}_{el}\right)\le \left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)\end{array}$

Hence, $0\le \left({\overleftarrow{q}}_{ek},{\overrightarrow{q}}_{ek}\right)\le \left\{\left({\overleftarrow{r}}_{st},{\overrightarrow{r}}_{st}\right);\left({\overleftarrow{r}}_{st},{\overrightarrow{r}}_{st}\right)<\left(\left({\overleftarrow{r}}_{st},{\overrightarrow{r}}_{st}\right)\wedge \left({\overleftarrow{r}}_{nt},{\overrightarrow{r}}_{nt}\right)\right)\right\}=$$\left({\overleftarrow{x}}_{ek},{\overrightarrow{x}}_{ek}\right);$ that is, $\left(\overleftarrow{Q},\overrightarrow{Q}\right)\subset \left(\overleftarrow{X},\overrightarrow{X}\right)\mathrm{and}\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{Q},\overrightarrow{Q}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)\subset \left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{X},\overrightarrow{X}\right)\circ$$\left(\overleftarrow{R},\overrightarrow{R}\right)$

Assuming that $\left(\overleftarrow{U},\overrightarrow{U}\right)=\left(\overleftarrow{R},\overrightarrow{R}\right)\circ \left(\overleftarrow{X},\overrightarrow{X}\right)\circ \left(\overleftarrow{R},\overrightarrow{R}\right)$ and

$u_{ij}=\underset{k=1}{\overset{n}{\vee }}\underset{e=1}{\overset{n}{\vee }}\left(\left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\wedge \left({\overleftarrow{x}}_{ek},{\overrightarrow{x}}_{ek}\right)\wedge \left({\overleftarrow{r}}_{ke},{\overrightarrow{r}}_{ke}\right)\right)\left(i\le n,j\le m\right),$

Thus, we can prove that uij ≤ rij(i ≤ n, ≤ j ≤ m) is true.

In fact, when $\left(\left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\wedge \left({\overleftarrow{r}}_{ie},{\overrightarrow{r}}_{ie}\right)\right)\le \left({\overleftarrow{r}}_{ij},{\overrightarrow{r}}_{ij}\right)$ is true, we have