The operator “ ≤ ” defines a partial order relation on the partition of domain U(the relation ≤ is reflexive, anti-symmetric, and transitive). If π1 ≤ π2, then we say partition π1 is smaller than π2.

Take the partition as a computing object, and define intersection and union operations:

π1 ∧ π2 denotes the largest partition smaller than both π1 and π2;

π1 ∨ π2 denotes the smallest partition larger than both π1 and π2.

The union operation of partitions is equivalent to the supremum in computing, and the intersection operation is equivalent to the infimum operation. At the same time, the thickness of the partition can be defined so that we have a partition lattice. The partition lattice is introduced into the DFDT because the process of constructing a decision tree involves partitioning an instance set through branch attributes. The definition of the partition of the DFDT instance set is given so that we can get the Dynamic Fuzzy Partition Lattice.

Definition 3.3 Suppose $\cup \left\{X_i|1\le i\le m\right\}=U.$ is the training instance set of DFDT. $\left(\overleftarrow{U,}\overrightarrow{U}\right)$$\left(\overleftarrow{\pi ,}\overrightarrow{\pi }\right)=$ is a partition of $\left\{{\overleftarrow{X}}_{i,}\overrightarrow{X}|1\le i\le m\right\}$ satisfies the following conditions:

$\begin{array}{l}({\overleftarrow{X}}_i,{\overrightarrow{X}}_{\text{i}})\text{is not empty;}\\ ({\overleftarrow{X}}_i,{\overrightarrow{X}}_{\text{i}})\cap ({\overleftarrow{X}}_j,{\overrightarrow{X}}_j)=\varphi (i\ne j);\\ \cup \{({\overleftarrow{X}}_{\text{i}},{\overrightarrow{X}}_i)|1\le i\le m\}=(\overleftarrow{U},\overrightarrow{U}).\end{array}$

Define the partial order relationship between partitions (the intersection and union operations) as follows:

$\left(\overleftarrow{{\pi }_1}\overrightarrow{{\pi }_1}\right),\left(\overleftarrow{{\pi }_2},\overrightarrow{{\pi }_2}\right)$ are partitions of $\left(\overleftarrow{U,}\overrightarrow{U}\right),\left(\overleftarrow{{\pi }_1},\overrightarrow{{\pi }_1}\right)\le \left(\overleftarrow{\pi {}_2},\overrightarrow{{\pi }_2}\right)$$\forall \left(\overleftarrow{X_i},\overrightarrow{X_i}\right)\in$ such that $\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)\subseteq \left({\overleftarrow{X}}_j,{\overrightarrow{X}}_j\right).if\left({\overleftarrow{\pi }}_1,{\overrightarrow{\pi }}_1\right)$ is strictly smaller than $\left({\overleftarrow{\pi }}_2,{\overrightarrow{\pi }}_2\right),\text{WeWrite}\left({\overleftarrow{\pi }}_1,{\overrightarrow{\pi }}_1\right)<\left({\overleftarrow{\pi }}_2,{\overrightarrow{\pi }}_2\right).$

At the same time, define the intersection and union operations as follows:

$\begin{array}{l}(\overleftarrow{{\pi }_1},\overrightarrow{{\pi }_1})\wedge (\overleftarrow{{\pi }_2},\overrightarrow{{\pi }_2})=\{\sup ((\overleftarrow{X_i},\overrightarrow{X_i})\wedge (\overleftarrow{X_j},\overrightarrow{X_j}))|(\overleftarrow{X_i},\overrightarrow{X_i})ϵ(\overleftarrow{{\pi }_1},\overrightarrow{{\pi }_1}),\\ (\overleftarrow{X_j},\overrightarrow{X_j})ϵ(\overleftarrow{{\pi }_2},\overrightarrow{{\pi }_2})\}\\ (\overleftarrow{{\pi }_1},\overrightarrow{{\pi }_1})\wedge (\overleftarrow{{\pi }_2},\overrightarrow{{\pi }_2})=\{\inf ((\overleftarrow{X_i},\overrightarrow{X_i})\vee (\overleftarrow{X_j},\overrightarrow{X_j}))|(\overleftarrow{X_i},\overrightarrow{X_i})ϵ(\overleftarrow{{\pi }_1},\overrightarrow{{\pi }_1}),\\ (\overleftarrow{X_j},\overrightarrow{X_j})ϵ(\overleftarrow{{\pi }_2},\overrightarrow{{\pi }_2})\}.\end{array}$

Through the previous analysis of DFDT, the set of instances that fall into each node constitutes a partial order relation. In DFBDT, a set of instances falling into each node of an adjacent layer forms a semi-lattice. After the dynamic fuzzy partition is given, there are some theorems about the relation between the two sets of decision trees:

Theorem 3.1 The set of DFDTs can be divided into a linear ordered set.

Proof: The process of constructing a DFDT uses branch attributes on the instances in the process of division. Thus, the lower set of examples of the division is finer than the upper instance set. $\left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)<\left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right)$ (j is located at a lower level than i in the DFDT) is always established, even if the division is linear, as ≺ constitutes a partial order relationship. Therefore, the division of the set of instances is a linear ordered set.

Theorem 3.2 The set of DFDTs is composed of a set of dynamic fuzzy partitioning grids.

Proof: Let $\left({\overleftarrow{U}}_{\pi },{\overrightarrow{U}}_{\pi }\right)$ be the set of partitions of the set of instances in the DFDT. From Theorem 3.1, we know that $\left({\overleftarrow{U}}_{\pi },{\overrightarrow{U}}_{\pi }\right)$ is a poset. Let $\left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right),\left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)$ be an arbitrary division of the set of instances of DFDT in layers i and j(j < i). When $\left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right)=$$\left\{\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)|1\le i\le m\right\},\left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)=\left\{\left({\overleftarrow{X}}_j,{\overrightarrow{X}}_j\right)|1\le jm\right\},$ there are two cases:

(1)Layer j is directly below layer i. Then, there two cases for $\forall \left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)\in \left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right):$

If $\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)$ is defined by the leaf node, then we should stop the division in the process of constructing the DFDT. Thus, $\exists \left({\overleftarrow{X}}_j,{\overrightarrow{X}}_j\right)\in \left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)$ that makes $\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)=\left({\overleftarrow{X}}_j,{\overrightarrow{X}}_j\right),$ that is, $\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)\supseteq \left({\overleftarrow{X}}_j,{\overrightarrow{X}}_j\right).$

If $\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)$ is not a terminal node, in the process of creating a tree, they are divided by selecting attributes, i.e. $\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)=\left({\overleftarrow{X}}_j^1,{\overrightarrow{X}}_j^1\right)\cup \left({\overleftarrow{X}}_j^2,{\overrightarrow{X}}_j^2\right)\cup \mathrm{...}$$\cup \left({\overleftarrow{X}}_j^{\left|\left({\overleftarrow{V}}_{{\overleftarrow{A}}^{.}}{\overrightarrow{V}}_{\overrightarrow{A}}\right)\right|},{\overrightarrow{X}}_j^{\left|\left({\overleftarrow{V}}_{\overleftarrow{A}},{\overrightarrow{V}}_{\overrightarrow{A}}\right)\right|}\right),\text{Where}\left({\overleftarrow{V}}_{{\overleftarrow{A}}^{,}}{\overrightarrow{V}}_{\overrightarrow{A}}\right)$ is the range of branch attribute $\left(\overleftarrow{A},\overrightarrow{A}\right).$ As j is directly below layer i, the choice of attribute $\left({\overleftarrow{V}}_{{\overleftarrow{A}}^{,}}{\overrightarrow{V}}_{\overrightarrow{A}}\right)$ for division $\left({\overleftarrow{X}}_i,{\overrightarrow{X}}_i\right)$ falls into $\left({\overleftarrow{\pi }}_{j^{,}}{\overrightarrow{\pi }}_j\right),\text{that}\text{is},\left({\overleftarrow{X}}_j^k,{\overrightarrow{X}}_j^k\right)\in \left({\overleftarrow{\pi }}_{j^{,}}{\overrightarrow{\pi }}_j\right),\text{Where}k=1,2,\mathrm{...},\left|{\overleftarrow{V}}_{{\overleftarrow{A}}^{,}}{\overrightarrow{V}}_{\overrightarrow{A}}\right|.$

It can be seen from the above that $\left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)$ consists of a partition defined by the leaf node and a direct upper layer by selecting a partition of attributes. Thus, $\left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)\prec$$\left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right)$ is established, that is:

$\begin{array}{l}(\overleftarrow{{\pi }_j},\overrightarrow{{\pi }_j})\wedge (\overleftarrow{{\pi }_{\text{i}}},\overrightarrow{{\pi }_{\text{i}}})=(\overleftarrow{{\pi }_j},\overrightarrow{{\pi }_j})ϵ(\overleftarrow{U_{\pi }},\overrightarrow{U_{\pi }})\\ (\overleftarrow{{\pi }_j},\overrightarrow{{\pi }_j})\vee (\overleftarrow{{\pi }_{\text{i}}},\overrightarrow{{\pi }_{\text{i}}})=(\overleftarrow{{\pi }_i},\overrightarrow{{\pi }_i})ϵ(\overleftarrow{U_{\pi }},\overrightarrow{U_{\pi }}).\end{array}$

(2) Layer j is not directly below layer i.

In this case, there is a sequence i, i + 1, ..., k, k + 1, ..., j between i and j. From Theorem 3.1, we know that i, i + 1, ..., k, k + 1, ..., j is linear and orderly. Thus,

$\begin{array}{l}(\overleftarrow{{\pi }_j},\overrightarrow{{\pi }_j})\wedge (\overleftarrow{{\pi }_{\text{i}}},\overrightarrow{{\pi }_{\text{i}}})=(\overleftarrow{{\pi }_j},\overrightarrow{{\pi }_j})ϵ(\overleftarrow{U_{\pi }},\overrightarrow{U_{\pi }})\\ (\overleftarrow{{\pi }_j},\overrightarrow{{\pi }_j})\vee (\overleftarrow{{\pi }_{\text{i}}},\overrightarrow{{\pi }_{\text{i}}})=(\overleftarrow{{\pi }_i},\overrightarrow{{\pi }_i})ϵ(\overleftarrow{U_{\pi }},\overrightarrow{U_{\pi }}).\end{array}$

From cases (1) and (2), we can determine that for $\forall \left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right),\left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)\in \left({\overleftarrow{U}}_{\pi },{\overrightarrow{U}}_{\pi }\right),$ both $\left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right)\wedge \left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)\in \left({\overleftarrow{U}}_{\pi },{\overrightarrow{U}}_{\pi }\right)\text{and}\left({\overleftarrow{\pi }}_i,{\overrightarrow{\pi }}_i\right)\vee \left({\overleftarrow{\pi }}_j,{\overrightarrow{\pi }}_j\right)\in \left({\overleftarrow{U}}_{\pi },{\overrightarrow{U}}_{\pi }\right)$ hold. From Theorem 3.1, we know that $\left({\overleftarrow{U}}_{\pi },{\overrightarrow{U}}_{\pi }\right)$ is a poset, so $\left({\overleftarrow{U}}_{\pi },{\overrightarrow{U}}_{\pi }\right)$ is a lattice. This is called a Dynamic Fuzzy Lattice, written as Π(U).