Invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression

Abstract

A lattice-valued information system is an important model in the field of artificial intelligence and the notion of homomorphisms between lattice-valued information systems is a kind of tools to study data compression in a lattice-valued information system. This paper investigates invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression. Information structures in a lattice-valued information system is first proposed by using set vectors. Then, dependence and independence between information structures in the same lattice-valued information system is characterized by the inclusion degree. Finally, a complex massive lattice-valued information system can be compressed into a relatively small-scale lattice-valued information system by means of homomorphisms and it is proved that some characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression are invariant, that is, some of the same data structures are obtained.

Keywords

Lattice-valued information system granular computing information granule information structure dependence inclusion degree characterization homomorphism data compression

1 Introduction

An information system based on rough set theory was proposed by Pawlak [25, 26]. Most applications of rough set theory, such as rule extraction [2 , 38], feature selection [7–9 , 24], uncertainty modeling [3 , 32], reasoning with uncertainty [40, 41] are related to information systems.

In reality there are always complex and massive data in information systems. How to remove data redundancy while maintaining the same data structures and to discover potentially useful information is important. Thus, we need to consider “Data compression” in information systems. Data compression is a promising technique for storage and transmission of data. When data compression is used, data is transmitted faster, and file storage requires less space. Many aspects of data compression are described in Leweler et al. [19] and Storer [30]. Data compression in information systems is referred to two aspects of operations on data, one is to reduce stored and transferred data volume, the other is to reduce data dimension. Data volume reduction can be viewed as a many-to-one mapping between information systems in mathematics. Data dimension reduction can be explained as attribute reducts of information systems.

Homomorphisms between information systems as a kind of tool to study data compression in information systems was introduced by Grzymala-Busse in [6]. Homomorphisms are very useful for aggregating sets of objects, attributes, and descriptors of the original system. As explained in [27], homomorphisms allow one to translate the information contained in one granular world into the granularity of another granular world and thus provide a communication mechanism for exchanging information with other granular worlds. Li et al. [22] studied invariant characterizations of information systems under homomorphisms. Wang et al. [35 –37] investigated generalized information systems and covering information systems under homomorphisms, and pointed out that the notion of homomorphisms between information systems can be regarded as a kind of tools to study the compression of data volume in complex massive databases.

Granular computing, introduced by Zadeh [43], is an effective tool in artificial intelligence [43 –46]. Basic notions of granular computing contain information granulation, organization and causation. Zadeh deemed “Granulation involves decomposition of whole into parts, organization involves integration of parts into whole, and causation involves association of causes and effects”. Later, Lin [11, 12] and Yao [41, 42] emphasized the importance of granular computing.

Granules are clumps consisting of objects drawn together by indistinguishability, similarity, and proximity of functionality [12 , 23]. Granulation of objects in an information system leads to information granules. A granular structure is a mathematical structure of the family of information granules, where the inner structure of each information granule is visible, and the interactions between information granules are detected by the visible structures [13 –15].

In granular computing in information systems, the study of information structure is one of important research directions. An equivalence relation is a special kind of similarities between objects from a data set. Given an information system, an attribute subset determines an equivalence relation on the set of objects. This equivalence relation partitions the set of objects into some disjoint subsets, these subsets are called equivalence classes. If two objects belong to the same equivalence class, then we may say that they cannot be distinguished under this equivalence relation. Thus, every equivalence class can be viewed as an information granule consisting of indistinguishable objects [11 , 28]. All information granules constitutes a vector, this vector is called an information structure in this information system induced by this attribute subset. Information structures in an information system are namely granular structures in the meaning of granular computing.

It is known that an unknown target concept can be characterized approximately by existing knowledge structures in a knowledge base, which is one of the strengths of rough set theory. In granular computing in knowledge bases, Qian et al. [28] studied knowledge structures in a knowledge base. Li et al. [20] discussed relationships between knowledge bases and proved that knowledge reductions, coordinate families and necessary relations in a knowledge base are invariant and inverse invariant under homomorphisms. Li et al. [21] studied properties of knowledge structures in a knowledge base by means of condition information amounts, inclusion degrees and lower approximation operators, and gave group, lattice, mapping, and soft characterizations of knowledge structures in a knowledge base. Their results have been shown to be very helpful for knowledge discovery from knowledge bases and significant for establishing a framework of granular computing in knowledge bases [29]. Obviously, knowledge structures in a knowledge base are namely granular structures in the meaning of granular computing.

In an information system, if the domain of every attribute is a lattice according to a decreasing or increasing preference, then this information system is called a lattice-valued information system. The purpose of this paper is to obtain invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression.

The remaining part of this paper is organized as follows. In Section 2, we recall some concepts about binary relations, lattices, lattice-valued information systems and homomorphisms. In Section 3, we describe information structures in a lattice-valued information system through set vectors and define dependence and independence between them. In Sections 4, we give some properties of information structures. In Sections 5, we obtain invariant characterizations of information structures under homomorphisms based on data compression. Sections 6 summarizes this paper.

2 Preliminaries

In this section, we recall some concepts about binary relations, lattices, lattice-valued information systems and homomorphisms.

Throughout this paper, U and V denotes two non-empty finite sets called the universes, 2^U denotes the family of all subsets of U, |X| denotes the cardinality of X ∈ 2^U. All mappings are assumed to be surjective.

Put $\begin{matrix} U & = & {x_{1}, x_{2}, \dots \dots, x_{n}}, \\ V & = & {y_{1}, y_{2}, \dots \dots, y_{m}}, \\ δ & = & U \times U, ▵ = {(x, x) : x \in U} . \end{matrix}$

2.1 Binary relations and lattices

Recall that R is a binary relation on U whenever R ⊆ U × U. If (x, y) ∈ R, then we denote it by xRy.

Let R be a binary relation on U. Then R is called

reflexive, if xRx for any x ∈ U.

symmetric, if xRy implies yRx for any x, y ∈ U.

anti-symmetric, if xRy and yRx imply x = y for any x, y ∈ U.

transitive, if xRy and yRz imply xRz for any x, y, z ∈ U.

Let R be a binary relation on U. Then R is called an equivalence relation on U, if R is reflexive, symmetric and transitive.

In this paper, 2^U×U denotes the family of all binary relations on U.

Given R ∈ 2^U×U. If R = δ, then R is called a universal relation; if R =▵, then R is called an identity relation.

Let L be a non-empty set. Given R is a binary relation on L. Then R is call a partial order on L, if R is reflexive, anti-symmetric and transitive. Moreover, the pair (L, R) or L is called a partial order set (briefly, a poset).

In this paper, the partial order R on L is denoted by ≤.

If a ≤ b and a ≠ b, then we denote it by a < b.

Definition 2.1. Let (L₁, ≤ ₁) and (L₂, ≤ ₂) be two posets. Suppose that f : L₁ → L₂ is a mapping. Then f is called order-preserving, if for any a, b ∈ L₁, $a \leq_{1} b \Leftrightarrow f (a) \leq_{2} f (b) .$

Definition 2.2. [4] Let (L, ≤) be a poset.

a ∈ L is called a top element of L, if a ≤ x for any x ∈ L.

a ∈ L is called a bottom element of L, if x ≤ a for any x ∈ L.

If the poset L has top elements a₁, a₂ (resp. bottom elements b₁, b₂), then a₁ = a₂ (resp. b₁ = b₂). This unique top element (resp. bottom element) is denoted by 1_L or 1 (resp. 0_L or 0).

Definition 2.3. [4] Let (L, ≤) be a poset. Given S ⊆ L and a, b ∈ L.

a ∈ L is called a above boundary in S, if a ≤ x for any x ∈ S.

a ∈ L is called a under boundary in S, if x ≤ a for any x ∈ S.

a = ∨ S, if a is a minimal above boundary in S.

a = ∧ S, if a is a maximal under boundary in S.

If S = {a, b}, then we denote ∨S = a ∨ b and ∧S = a ∧ b.

Definition 2.4. [4] Let (L, ≤) be a poset. Then L is called a lattice, if a ∨ b ∈ L, a ∧ b ∈ L for any a, b ∈ L.

2.2 Lattice-valued information systems

Definition 2.5. [26] Let U be a set of objects and A a set of attributes. Suppose that U and A are finite sets. Then the pair (U, A) is called an information system, if each attribute a ∈ A determines an information function a : U → V_a, where V_a is the information function values set of the attribute a.

If P ⊆ A, then (U, P) is called a subsystem of (U, A).

If (U, A) is an information system and P ⊆ A, then an equivalence relation (or indiscernibility relation) ind (P) can be defined by $(x, y) \in ind (P) \Leftrightarrow \forall a \in P, a (x) = a (y) .$

Obviously, ind (P) = ⋂ _a∈Pind ({a}).

Definition 2.6. [50] Let (U, A) be an information system. Then (U, A) is called a lattice-valued information system, if for each attribute a ∈ A, V_a = {a (x) : x ∈ U} is a lattice. In this paper, a lattice-valued information system is denoted by L^≼ = (U, A), the partial order on the poset V_a is denoted by ≤_a.

Let L^≼ = (U, A) be a lattice-valued information system. If for each attribute a ∈ A, V_a = ({1, 2, ⋯, k}, ∨, ∧), then L^≼ = (U, A) is a classic information system; If for each attribute a ∈ A, V_a = ([0, 1], ∨, ∧), then L^≼ = (U, A) is a fuzzy information system; If for each attribute a ∈ A, V_a = (2^W, ∪, ∩), then L^≼ = (U, A) is a set-valued information system; If for each attribute a ∈ A, V_a = (I ([0, 1]), ∨, ∧), then L^≼ = (U, A) is a interval value information system.

Given a ⊆ A. Define x ≼ _ay ⇔ a (x) ≤ _aa (y). Then ≼_a is called the dominance relation on U with respect to the attribute a ([39]). And x ≼ _ay means that “y is at least as good as x with respect to the criterion a” and then we may say that y dominates x or x is dominated by y with respect to the attribute a.

Example 2.7. A lattice-valued information system is showed in Table 1, where U = {x₁, x₂, x₃, x₄, x₅, x₆} and A = {a₁, a₂, a₃}.

Table 1
A lattice-valued information system (U,A)

a ₁ a ₂ a ₃

x ₁ 1 0.6 {0}

x ₂ 3 0.6 {0,1}

x ₃ 1 0.3 {0,1}

x ₄ 2 0.3 {0,1,2}

x ₅ 3 0.9 {0,1}

x ₆ 3 0.6 {0,1,2}

	a ₁	a ₂	a ₃
x ₁	1	0.6	{0}
x ₂	3	0.6	{0,1}
x ₃	1	0.3	{0,1}
x ₄	2	0.3	{0,1,2}
x ₅	3	0.9	{0,1}
x ₆	3	0.6	{0,1,2}

According to above expression, we can find that V_{a
₁} = {1, 2, 3} is a lattice with real numbers, where ≤_{a
₁} is “≤” between two real numbers. So the dominance relation on U with respect to the attribute a₁ can be defined as $x ≼_{a_{1}} y \Leftrightarrow a_{1} (x) \leq a_{1} (y) .$

V_{a
₂} = {0.3, 0.6, 0.9} is a lattice with numerical numbers located in the unit interval [0, 1], from which ≤_{a
₂} is “≤” between two fuzzy elements. And the dominance relation on U with respect to the attribute a₂ can be defined as $x ≼_{a_{2}} y \Leftrightarrow a_{2} (x) \leq a_{2} (y) .$

V_{a
₃} = {{0}, {0, 1}, {0, 1, 2}} is a lattice with set-valued elements, where ≤_{a
₃} is “⊆” between two two sets. Then the dominance relation on U with respect to the attribute a₃ can be defined as $x ≼_{a_{3}} y \Leftrightarrow a_{3} (x) \subseteq a_{3} (y) .$

Given P ⊆ A. Define $x ≼_{P} y \Leftrightarrow \forall a \in P, x ≼_{a} y .$

Then ≼_P is called the dominance relation on U with respect to the attribute subset P [39]. And x ≼ _Py means that “y is at least as good as x with respect to the criterion set P” and then we may say that y dominates x or x is dominated by y with respect to the attribute subset P.

Obviously, ≼_P = ⋂ _a∈P ≼ _a.

Denote $[x]_{P}^{≼} = {y \in U : x ≼_{P} y},$ $U / ≼_{P} = {_{P}^{≼} : x \in U} .$

Then, $[x]_{P}^{≼}$ is called a dominance class x ∈ U with respect to P, U/≼ _P is referred to as the quotient set of U with respect to P.

Obviously,

≼_P is reflexive and transitive, U/≼ _P constitutes a covering of U.

2.3 Homomorphisms between lattice-valued information systems

In mathematics, homomorphisms between two information systems can be explained as a mapping. Li et al. [22] studied invariant characterizations of information systems under homomorphisms. Wang et al. [37] pointed out that this homomorphism can be regarded as a kind of tools to study data compression in information system.

Definition 2.8. [22] Let (U, A) and (V, B) be two information systems. Given that h₁ : U → V, h₂ : A → B and h₃ : U^* → V^* are three mappings where $\begin{matrix} U^{*} & = & {a (u) : u \in U, a \in A}, \\ V^{*} & = & {b (v) : v \in V, b \in B} . \end{matrix}$

The triple h = (h₁, h₂, h₃) is called a homomorphism from (U, A) to (V, B), if for all u ∈ U and a ∈ A, $h_{3} (a (u)) = h_{2} (a) (h_{1} (u)) .$

The above definition may also applies to lattice-valued information systems.

Definition 2.9. Let (U, A) and (V, B) be two lattice-valued information systems. Given that h₁ : U → V, h₂ : A → B and h₃ : U^* → V^* are three mappings where $\begin{matrix} U^{*} & = & ⋃_{a \in A} U_{a}, U_{a} = {a (u) : u \in U}, \\ V^{*} & = & ⋃_{b \in B} V_{b}, V_{b} = {b (v) : v \in V} . \end{matrix}$

Then the triple h = (h₁, h₂, h₃) is called a homomorphism from (U, A) to (V, B), if for each a ∈ A, h₃ ↾ _{U
_a} : U_a → V_h₂(a) is order-preserving, and for any u ∈ U, a ∈ A, $h_{3} (a (u)) = h_{2} (a) (h_{1} (u)) .$

We write (U, A) ∼ _h (V, B).

3 Some concepts of information structures in a lattice-valued information system

In this section, we propose some concepts of information structures in the same lattice-valued information system.

Let L^≼ = (U, A) be a lattice-valued information system. Given P ⊆ A. Then, $[x]_{P}^{≼}$ is the dominance class of x ∈ U under the dominance relation ≼_P. A dominance relation reflects a special kind of similarities between objects from a data set. If two objects x₁, x₂ of U belong to the same dominance class, then we may say that x₁, x₂ cannot be distinguished under the dominance relation ≼_P. Every dominance class may be viewed as an information granule consisting of indistinguishable objects. All these dominance classes or information granules constitutes a vector, this vector can be seen as the information structure of the subsystem (U, P). The research of this kind of information structures will be significative for establishing a framework of granular computing in information systems. The following definition give the concept of information structures in a lattice-valued information system.

Definition 3.1. Let L^≼ = (U, A) be a lattice-valued information system. Then for P ⊆ A, $S (P) = ([x_{1}]_{P}^{≼}, [x_{2}]_{P}^{≼}, \dots \dots, [x_{n}]_{P}^{≼})$ is called the information structure of the subsystem (U, P).

If ≼_P is an identity relation on U, then S (P) = ([x₁], [x₂], ⋯⋯, [x_n]).

Example 3.2.S (∅) = (U, U, ⋯, U).

Definition 3.3. Let L^≼ = (U, A) be a lattice-valued information system. Given that S (P), S (Q) are the information structures of the subsystems (U, P) and (U, Q), respectively. Then S (P) and S (Q) are called to be the same, if for each i, $[x_{i}]_{P}^{≼} = [x_{i}]_{Q}^{≼}$ . We still write S (P) = S (Q).

Definition 3.4. Let L^≼ = (U, A) be a lattice-valued information system. Then $S (U, A) = {S (P) : P \in 2^{A}}$ is called the information structure base of (U, A).

Example 3.5. [Continued from Example 2.7] Pick P₁ = {a₁}, P₂ = {a₂}, P₃ = {a₃}, P₄ = {a₁, a₂}. P₅ = {a₁, a₃}, P₆ = {a₂, a₃}.

Note that $\begin{matrix} [x_{1}]_{P_{1}}^{≼} & = & [x_{3}]_{P_{1}}^{≼} = U, [x_{2}]_{P_{1}}^{≼} \\ = & [x_{5}]_{P_{1}}^{≼} = [x_{6}]_{P_{1}}^{≼} = {x_{2}, x_{5}, x_{6}}, \\ [x_{4}]_{P_{1}}^{≼} & = & {x_{2}, x_{4}, x_{5}, x_{6}}; \\ [x_{1}]_{P_{2}}^{≼} & = & [x_{2}]_{P_{2}}^{≼} = [x_{6}]_{P_{2}}^{≼} \\ = & {x_{1}, x_{2}, x_{5}, x_{6}}, [x_{3}]_{P_{2}}^{≼} = [x_{4}]_{P_{2}}^{≼} = U, \\ [x_{5}]_{P_{2}}^{≼} & = & {x_{5}}; [x_{1}]_{P_{3}}^{≼} = U, \\ [x_{2}]_{P_{3}}^{≼} & = & [x_{3}]_{P_{3}}^{≼} = [x_{5}]_{P_{3}}^{≼} \\ = & {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, [x_{4}]_{P_{3}}^{≼} \\ = & [x_{6}]_{P_{3}}^{≼} = {x_{4}, x_{6}}; \\ [x_{1}]_{P_{4}}^{≼} & = & {x_{1}, x_{2}, x_{5}, x_{6}}, [x_{2}]_{P_{4}}^{≼} = [x_{6}]_{P_{4}}^{≼} \\ = & {x_{2}, x_{5}, x_{6}}, [x_{3}]_{P_{4}}^{≼} = U, \\ [x_{4}]_{P_{4}}^{≼} & = & {x_{2}, x_{4}, x_{5}, x_{6}}, [x_{5}]_{P_{4}}^{≼} = {x_{5}}; \\ [x_{1}]_{P_{5}}^{≼} & = & U, [x_{2}]_{P_{5}}^{≼} = [x_{5}]_{P_{5}}^{≼} \\ = & {x_{2}, x_{5}, x_{6}}, \\ [x_{3}]_{P_{5}}^{≼} & = & {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, \\ [x_{4}]_{P_{5}}^{≼} & = & {x_{4}, x_{6}}, [x_{6}]_{P_{5}}^{≼} = {x_{6}}; \\ [x_{1}]_{P_{6}}^{≼} & = & {x_{1}, x_{2}, x_{5}, x_{6}}, [x_{2}]_{P_{6}}^{≼} \\ = & {x_{2}, x_{5}, x_{6}}, [x_{3}]_{P_{6}}^{≼} \\ = & {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, \\ [x_{4}]_{P_{6}}^{≼} & = & {x_{4}, x_{6}}, [x_{5}]_{P_{6}}^{≼} = {x_{5}}, \\ [x_{6}]_{P_{6}}^{≼} & = & {x_{6}}; [x_{1}]_{A}^{≼} = {x_{1}, x_{2}, x_{5}, x_{6}}, \\ [x_{2}]_{A}^{≼} & = & {x_{2}, x_{5}, x_{6}}, [x_{3}]_{A}^{≼} \\ = & {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, [x_{4}]_{A}^{≼} = {x_{4}, x_{6}}, \\ [x_{5}]_{A}^{≼} & = & {x_{5}}, [x_{6}]_{A}^{≼} = {x_{6}} . \end{matrix}$

Then $\begin{matrix} S (\emptyset) & = & (U, U, \dots, U), \\ S (P_{1}) & = & (U, {x_{2}, x_{5}, x_{6}}, U, {x_{2}, x_{4}, x_{5}, x_{6}}, \\ {x_{2}, x_{5}, x_{6}}, {x_{2}, x_{5}, x_{6}}), \\ S (P_{2}) & = & ({x_{1}, x_{2}, x_{5}, x_{6}}, {x_{1}, x_{2}, x_{5}, x_{6}}, \\ U, U, {x_{5}}, {x_{1}, x_{2}, x_{5}, x_{6}}), \\ S (P_{3}) & = & (U, {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, \\ {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, {x_{4}, x_{6}}, \\ {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, {x_{4}, x_{6}}), \\ S (P_{4}) & = & ({x_{1}, x_{2}, x_{5}, x_{6}}, {x_{2}, x_{5}, x_{6}}, \\ U, {x_{2}, x_{4}, x_{5}, x_{6}}, {x_{5}}, \\ {x_{2}, x_{5}, x_{6}}), \\ S (P_{5}) & = & (U, {x_{2}, x_{5}, x_{6}}, {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, \\ {x_{4}, x_{6}}, {x_{2}, x_{5}, x_{6}}, {x_{6}}), \\ S (A) & = & S (P_{6}) = ({x_{1}, x_{2}, x_{5}, x_{6}}, {x_{2}, x_{5}, x_{6}}, \\ {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, {x_{4}, x_{6}}, \\ {x_{5}}, {x_{6}}) . \end{matrix}$

Then $\begin{matrix} S (U, A) & = & {S (\emptyset), S (P_{1}), S (P_{2}), \\ S (P_{3}), S (P_{4}), S (P_{5}), S (A)} . \end{matrix}$

The following definition depicts dependence and independence between information structures in a lattice-valued information system.

Definition 3.6. Let S^≼ = (U, A) be a lattice-valued information system. Given that S (P), S (Q) are the information structures of the subsystems (U, P) and (U, Q), respectively.

a) S (Q) is called to depend on S (P), if for each i, $[x_{i}]_{P}^{≼} \subseteq [x_{i}]_{Q}^{≼}$ , we write S (P) ⪯ S (Q);

b) S (Q) is called to depend strictly on S (P), if S (P) ⪯ S (Q) and S (P) ≠ S (Q), we write S (P) ≺ S (Q).

a) S (Q) is called to depend partially on S (P), if for exists i, $[x_{i}]_{P}^{≼} \subseteq [x_{i}]_{Q}^{≼}$ , we write S (P) ⊑ S (Q);

b) S (Q) is called to depend partially strictly on S (P), if S (P) ⊑ S (Q) and S (P) ≠ S (Q), we write S (P) ⊏ S (Q).

S (Q) is called to be independent of S (P), if for each i, $[x_{i}]_{P}^{≼} ⊈ [x_{i}]_{Q}^{≼}$ , we write S (P) ⋈ S (Q).

Fig.1-4

S(P) and S(Q)

Figure 1 means S (P) ≺ S (Q), Fig. 2 means S (P) ⊏ S (Q) and S (Q) S (P), Fig. 3 means S (P) ⊏ S (Q) and S (Q) ⊏ S (P), Fig. 4 means S (P) ⋈ S (Q) and S (Q) ⋈ S (P).

Obviously, $\begin{matrix} S (P) & = & S (Q) \Leftrightarrow S (P) \\ ⪯ & S (Q) and S (Q) ⪯ S (P); \\ S (P) & ⪯ & S (Q) \Rightarrow S (P) ⊑ S (Q), S (P) \\ ≺ & S (Q) \Rightarrow S (P) ⊏ S (Q); \\ S (P) & ⋢ & S (Q) \Leftrightarrow S (P) ⋈ S (Q) . \end{matrix}$

Example 3.7. (Continued from Example 3.5) We have

S (A) ≺ S (P₄) ≺ S (P₁) ≺ S (∅), S (A) ≺ S (P₅) ≺ S (P₃) ≺ S (∅), S (A) ≺ S (P₂) ≺ S (∅), S (P₄) ≺ S (P₂), S (P₄) ≺ S (P₃), S (P₅) ≺ S (P₁).

S (P₁) ⊏ S (P₂), S (P₂) ⊏ S (P₁); S (P₁) ⊏ S (P₃), S (P₃) ⊏ S (P₁); S (P₂) ⊏ S (P₃), S (P₃) ⊏ S (P₂); S (P₂) ⊏ S (P₅), S (P₅) ⊏ S (P₂); S (P₄) ⊏ S (P₅), S (P₅) ⊏ S (P₄) (see Fig. 5).

Fig.5

S(U,A)

4 Properties of information structures in a lattice-valued information system

Theorem 4.1. Let S^≼ = (U, A) be a lattice-valued information system. Then for P, Q ⊆ A,

S (P) = S (Q) ⇔ ≼ _P = ≼ _Q ;

S (P)⪯ S (Q) ⇔ ≼ _P ⊆ ≼ _Q ;

Proof. This is obvious. □

Proposition 4.2. Let S^≼ = (U, A) be a lattice-valued information system. Given P, Q ⊆ A. If P ⊆ Q, then S (Q) ⪯ S (P).

Proof. Since P ⊆ Q, we have ≼_Q ⊆ ≼ _P. By Theorem 4.1, S (Q) ⪯ S (P). □

Proposition 4.3. Let S^≼ = (U, A) be a lattice-valued information system. Given P ⊆ A. Then S (A) ⪯ S (P) ⪯ S (∅).

Proof. This holds by Proposition 4.2. □

Definition 4.4. [50] Let S^≼ = (U, A) be a lattice-valued information system. Given that S (U, A) is the information structure base of (U, A). Then a mapping D : S (U, A) × S (U, A) → [0, 1] is called the inclusion degree on S (U, A), if

0 ≤ D (S (Q)/S (P)) ≤1;

S (P) ⪯ S (Q) implies D (S (Q)/S (P)) =1;

S (P) ⊑ S (Q) ⊑ S (L) implies D (S (P)/S (L)) ≤ D (S (P)/S (Q)).

Definition 4.5. Let S^≼ = (U, A) be a lattice-valued information system. For any P, Q ⊂ A, define $D (S (Q) / S (P)) = \sum_{l = 1}^{n} \frac{| [x_{l}]_{Q}^{≼} |}{\sum_{i = 1}^{n} | [x_{i}]_{Q}^{≼} |} χ_{[x_{l}]_{Q}^{≼}} ([x_{l}]_{P}^{≼}),$ where $χ_{[x_{l}]_{Q}^{≼}} ([x_{l}]_{P}^{≼}) = {\begin{matrix} 1, & if [x_{l}]_{P}^{≼} \subseteq [x_{l}]_{Q}^{≼}, \\ 0, & if [x_{l}]_{P}^{≼} ⊈ [x_{l}]_{Q}^{≼} . \end{matrix}$

Proposition 4.6. D in Definition 4.5 is the inclusion degree under Definition 4.4.

Proof. This is obvious. □

Example 4.7. (Continued from Example 3.5) Pick P = P₁, Q = P₂. $\begin{matrix} S (P) & = & (U, {x_{2}, x_{5}, x_{6}}, U, {x_{2}, x_{4}, x_{5}, x_{6}}, {x_{2}, x_{5}, x_{6}}, {x_{2}, x_{5}, x_{6}}), \\ S (Q) & = & ({x_{1}, x_{2}, x_{5}, x_{6}}, {x_{1}, x_{2}, x_{5}, x_{6}}, U, U, {x_{5}}, {x_{1}, x_{2}, x_{5}, x_{6}}), \end{matrix}$

We have $\begin{matrix} D (S (Q) / S (P)) \\ = \frac{| [x_{2}]_{Q}^{≼} |}{\sum_{i = 1}^{6} | [x_{i}]_{Q}^{≼} |} + \frac{| [x_{3}]_{Q}^{≼} |}{\sum_{i = 1}^{6} | [x_{i}]_{Q}^{≼} |} + \frac{| [x_{6}]_{Q}^{≼} |}{\sum_{i = 1}^{6} | [x_{i}]_{Q}^{≼} |} \\ = \frac{4}{25} + \frac{6}{25} + \frac{4}{25} = \frac{14}{25}, \\ D (S (P) / S (Q)) \\ = \frac{| [x_{1}]_{P}^{≼} |}{\sum_{i = 1}^{6} | [x_{i}]_{P}^{≼} |} + \frac{| [x_{3}]_{P}^{≼} |}{\sum_{i = 1}^{6} | [x_{i}]_{P}^{≼} |} + \frac{| [x_{5}]_{P}^{≼} |}{\sum_{i = 1}^{6} | [x_{i}]_{P}^{≼} |} \\ = \frac{6}{25} + \frac{6}{25} + \frac{3}{25} = \frac{15}{25} . \end{matrix}$

This example illustrates that $D (S (Q) / S (P)) + D (S (P) / S (Q)) \neq 1 .$

The following theorem show that dependence and independence between information structures in a lattice-valued information system can be quantitatively described by the inclusion degree.

Theorem 4.8. Let S^≼ = (U, A) be a lattice-valued information system. Given P, Q ⊂ A. Then

S (P) ⪯ S (Q) ⇔ D (S (Q)/S (P)) =1.

S (P) ⋈ S (Q) ⇔ D (S (Q)/S (P)) =0.

S (P) ⊑ S (Q) ⇔0 < D (S (Q)/S (P)) ≤1.

Proof. (1) “⇒”. This is obvious.

“⟸”. Put $| [x_{i}]_{Q}^{≼} | = q_{i}, \sum_{i = 1}^{n} | [x_{i}]_{Q}^{≼} | = q .$

Then $q = \sum_{i = 1}^{n} q_{i}$ . Since D (S (Q)/S (P)) =1, we have $\sum_{i = 1}^{n} q_{i} χ_{[x_{i}]_{Q}^{≼}} ([x_{i}]_{P}^{≼}) = q = \sum_{i = 1}^{n} q_{i} .$

Then $\sum_{i = 1}^{n} q_{i} (1 - χ_{[x_{i}]_{Q}^{≼}} ([x_{i}]_{P}^{≼})) = 0 .$

Thus for each i, $1 - χ_{[x_{i}]_{Q}^{≼}} ([x_{i}]_{P}^{≼}) = 0 .$

It follows that for each i, $[x_{i}]_{P}^{≼} \subseteq [x_{i}]_{Q}^{≼}$ .

Hence S (P) ⪯ S (Q).

(2) “⇒”. Since S (P) ⋈ S (Q), we have ∀ i, $[x_{i}]_{P}^{≼} ⊈ [x_{i}]_{Q}^{≼}$ . Then for each i, $χ_{[x_{i}]_{Q}^{≼}} ([x_{i}]_{P}^{≼}) = 0 .$

Thus D (S (Q)/S (P)) =0.

“⟸”. Since D (S (Q)/S (P)) =0, we obtain that for each i, $χ_{[x_{i}]_{Q}^{≼}} ([x_{i}]_{P}^{≼}) = 0 .$

Then, ∀ i, $[x_{i}]_{P}^{≼} ⊈ [x_{i}]_{Q}^{≼}$ . Thus S (P) ⋈ S (Q).

(3) This holds by (2). □

5 Invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression

Homomorphisms between two lattice-valued information systems may be a tool for studying data compression in lattice-valued information systems. Then, a complex massive lattice-valued information system can be compressed into a relatively small-scale information system by means of homomorphisms. Meanwhile, some same data structures in lattice-valued information systems are invariant.

Lemma 5.1. Let (U, A) and (V, B) be two lattice-valued information systems. If (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃), then for any P ⊆ A and u^* ∈ U, $h_{1} (u) \in [h_{1} (u^{*})]_{h_{2} (P)}^{≼} \Leftrightarrow u \in g_{P} (u^{*}),$ where g_a = h₃ ∘ a, g_P = {g_a : a ∈ P}, g_P (u^*) = {u ∈ U : ∀ a ∈ P, g_a (u) = g_a (u^*)}.

Proof. “⇒”. Suppose $h_{1} (u) \in [h_{1} (u^{*})]_{h_{2} (P)}^{≼}$ . Then (h₁ (u^*), h₁ (u)) ∈ ≼ _h₂(P). So ∀ a ∈ P, $h_{2} (a) (h_{1} (u)) = h_{2} (a) (h_{1} (u^{*})) .$

Note that (U, A) ∼ _h (V, B). Then, ∀ a ∈ P, $\begin{matrix} h_{2} (a) (h_{1} (u)) & = & h_{3} (a (u)) = g_{a} (u), h_{2} (a) (h_{1} (u^{*})) \\ = & h_{3} (a (u^{*})) = g_{a} (u^{*}) . \end{matrix}$

So ∀ a ∈ P, g_a (u) = g_a (u^*).

Thus u ∈ g_P (u^*).

“⟸”. Suppose u ∈ g_P (u^*). Then, ∀ a ∈ P, g_a (u) = g_a (u^*), i.e., ∀ a ∈ P, h₃ (a (u)) = (h₃ (a (u^*)). Since (U, A) ∼ _h (V, B), we have ∀ a ∈ P, h₂ (a) (h₁ (u)) = h₃ (a (u)), h₃ (a (u^*)) = h₂ (a) (h₁ (u^*). Then $\forall a \in P, h_{2} (a) (h_{1} (u)) = h_{2} (a) (h_{1} (u^{*}) .$

Thus $h_{1} (u) \in [h_{1} (u^{*})]_{h_{2} (P)}^{≼}$ .

Hence $h_{1} (u) \in [h_{1} (u^{*})]_{h_{2} (P)}^{≼} \Leftrightarrow u \in g_{P} (u^{*}) . □$

Theorem 5.2. Let (U, A) and (V, B) be two lattice-valued information systems. Suppose (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃). Denote v_j = h₁ (u_{k
_j}) (j = 1, 2, ⋯, m). Then for any P ∈ 2^A, $\begin{matrix} S (h_{2} (P)) & = & (h_{1} (g_{P} (u_{k_{1}})), h_{1} (g_{P} (u_{k_{2}})) \\ \dots, h_{1} (g_{P} (u_{k_{m}}))), \end{matrix}$ where g_a = h₃ ∘ a, g_P = {g_a : a ∈ P}, g_P (u_{k
_j}) = {u ∈ U : ∀ a ∈ P, g_a (u) = g_a (u_{k
_j})} (j = 1, 2, ⋯, m).

Proof. By Lemma 5.1, $h_{1} (u) \in [h_{1} (u_{k_{j}})]_{h_{2} (P)}^{≼} \Leftrightarrow u \in g_{P} (u_{k_{j}}) (j = 1, 2, \dots, m) .$

Then $h_{1} (g_{P} (u_{k_{j}})) = [h_{1} (u_{k_{j}})]_{h_{2} (P)}^{≼} (j = 1, 2, \dots, m) .$

Thus $\begin{matrix} S (h_{2} (P)) \\ = ([v_{1}]_{h_{2} (P)}^{≼}, [v_{2}]_{h_{2} (P)}^{≼}, \dots, [v_{m}]_{h_{2} (P)}^{≼}) \\ = ([h_{1} (u_{k_{1}})]_{h_{2} (P)}^{≼}, [h_{1} (u_{k_{2}})]_{h_{2} (P)}^{≼}, \dots, \\ [h_{1} (u_{k_{m}})]_{h_{2} (P)}^{≼}) \\ = (h_{1} (g_{P} (u_{k_{1}})), h_{1} (g_{P} (u_{k_{2}})) \dots, h_{1} (g_{P} (u_{k_{m}}))) . □ \end{matrix}$

Proposition 5.3. Let (U, A) and (V, B) be two lattice-valued information systems. Suppose (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃). Then for any P₁, P₂ ⊆ A, $≼_{P_{1}} \subseteq ≼_{P_{2}} \Leftrightarrow ≼_{h_{2} (P_{1})} \subseteq ≼_{h_{2} (P_{2})} .$

Proof. Suppose ≼_{P
₁} ⊆ ≼ _{P
₂}. ∀ (v, y) ∈ ≼ _h₂(P₁), we have v ≼ _h₂(P₁)y. Then $\forall a \in P_{1}, h_{2} (a) (v) \leq_{h_{2} (a)} h_{2} (a) (y) .$

Let y = h₁ (x), v = h₁ (u). Then, $\forall a \in P_{1}, h_{2} (a) (h_{1} (u)) \leq_{h_{2} (a)} h_{2} (a) (h_{1} (x)) .$

Note that (U, A) ∼ _h (V, B). Then, ∀ a ∈ A, $h_{2} (a) (h_{1} (x)) = h_{3} (a (x)), h_{2} (a) (h_{1} (u)) = h_{3} (a (u)) .$

So ∀ a ∈ P₁, h₃ (a (u)) ≤ _h₂(a)h₃ (a (x)).

Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₁, a (u) ≤ _aa (x). This implies (u, x) ∈ ≼ _{P
₁}. By hypothesis, (u, x) ∈ ≼ _{P
₂}, i.e., ∀ a ∈ P₂, a (u) ≤ _aa (x). Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₂, $h_{3} (a (u)) \leq_{h_{2} (a)} h_{3} (a (x)) .$

Then, ∀ a ∈ P₂, $\begin{matrix} h_{2} (a) (v) & = & h_{2} (a) (h_{1} (u)) = h_{3} (a (u)) \\ \leq_{h_{2} (a)} h_{3} (a (x)) = h_{2} (a) (h_{1} (x)) \\ = h_{2} (a) (y) . \end{matrix}$

This implies v ≼ _h₂(P₂)y. So (v, y) ∈ ≼ _h₂(P₂).

Thus ≼_h₂(P₁) ⊆ ≼ _h₂(P₂).

Conversely, suppose ≼_h₂(P₁) ⊆ ≼ _h₂(P₂), ∀ (u, x) ∈ ≼ _{P
₁}, we have u ≼ _{P
₁}x. Then $\forall a \in P_{1}, a (u) \leq_{a} a (x) .$

Note that (U, A) ∼ _h (V, B). Then, ∀ a ∈ A, $h_{2} (a) (h_{1} (x)) = h_{3} (a (x)), h_{3} (a (x)) = h_{2} (a) (h_{1} (u)) .$

Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₁, h₃ (a (u)) ≤ _h₂(a)h₃ (a (x)).

Then, ∀ a ∈ P₁, h₂ (a) (h₁ (u)) ≤ _h₂(a)h₂ (a) (h₁ (x)). This implies (h₁ (x), h₁ (u)) ∈ ≼ _h₂(P₁).

By hypothesis, (h₁ (x), h₁ (u)) ∈ ≼ _h₂(P₂), i.e., ∀ a ∈ P₂, h₂ (a) (h₁ (u)) ≤ _h₂(a)h₂ (a) (h₁ (x)). Then, ∀ a ∈ P₂, h₃ (a (u)) ≤ _h₂(a)h₃ (a (x)).

Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₂, a (u) ≤ _aa (x). This implies u ≼ _{P
₂}x. Then (u, x) ∈ ≼ _{P
₂}.

Thus ≼_{P
₁} ⊆ ≼ _{P
₂}. □

Corollary 5.4. Let (U, A) and (V, B) be two lattice-valued information systems. Suppose (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃). Then for any P₁, P₂ ⊆ A, $≼_{P_{1}} = ≼_{P_{2}} \Leftrightarrow ≼_{h_{2} (P_{1})} = ≼_{h_{2} (P_{2})} .$

Proof. This holds by Proposition 5.3. □

Theorem 5.5. Let (U, A) and (V, B) be two lattice-valued information systems. Suppose (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃). Then for any P₁, P₂ ⊆ A,

S (P₁) = S (P₂) ⇔ S (h₂ (P₁)) = S (h₂ (P₂)) ;

S (P₁) ⪯ S (P₂) ⇔ S (h₂ (P₁)) ⪯ S (h₂ (P₂)).

Proof. (1) This holds by Theorem 4.1 and Corollary 5.4.

(1) This follows from Theorem 4.1 and Proposition 5.3. □

Corollary 5.6. Let (U, A) and (V, B) be two lattice-valued information systems. Suppose (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃). Then for any P₁, P₂ ⊆ A, $S (P_{1}) ≺ S (P_{2}) \Leftrightarrow S (h_{2} (P_{1})) ≺ S (h_{2} (P_{2})) .$

Proof. This holds by Theorem 5.5. □

Theorem 5.7. Let (U, A) and (V, B) be two lattice-valued information systems. Suppose (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃). Then for any P₁, P₂ ⊆ A, $S (P_{1}) ⊑ S (P_{2}) \Leftrightarrow S (h_{2} (P_{1})) ⊑ S (h_{2} (P_{2})) .$

Proof. Suppose S (P₁) ⊑ S (P₂). Then ∃ i, $[x_{i}]_{P_{1}}^{≼} \subseteq [x_{i}]_{P_{2}}^{≼}$ . Let y_j = h₁ (x_i).

$\forall v \in [y_{j}]_{h_{2} (P_{1})}^{≼}$ , we have (v, y_j) ∈ ≼ _h₂(P₁). This implies v ≼ _h₂(P₁)y_j. Then $\forall a \in P_{1}, h_{2} (a) (v) \leq_{h_{2} (a)} h_{2} (a) (y_{j}) .$

Let v = h₁ (u). Then, ∀ a ∈ P₁, h₂ (a) (h₁ (u)) ≤ _h₂(a)h₂ (a) (h₁ (x_i)).

Note that (U, A) ∼ _h (V, B). Then, ∀ a ∈ A, $h_{2} (a) (h_{1} (x_{i})) = h_{3} (a (x_{i})), h_{2} (a) (h_{1} (u)) = h_{3} (a (u)) .$

So ∀ a ∈ P₁, h₃ (a (u)) ≤ _h₂(a)h₃ (a (x_i)).

Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₁, a (u) ≤ _aa (x_i). This implies u ≼ _{P
₁}x_i, i.e., (u, x_i) ∈ ≼ _{P
₁}. Then $u \in [x_{i}]_{P_{1}}^{≼}$ . Since $[x_{i}]_{P_{1}}^{≼} \subseteq [x_{i}]_{P_{2}}^{≼}$ , we have $u \in [x_{i}]_{P_{2}}^{≼}$ . This follows from (u, x_i) ∈ ≼ _{P
₂}, i.e., ∀ a ∈ P₂, a (u) ≤ _aa (x_i). Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₂, $h_{3} (a (u)) \leq_{h_{2} (a)} h_{3} (a (x_{i})) .$

Then, ∀ a ∈ P₂, $\begin{matrix} h_{2} (a) (v) & = & h_{2} (a) (h_{1} (u)) = h_{3} (a (u)) \\ \leq_{h_{2} (a)} h_{3} (a (x_{i})) = h_{2} (a) (h_{1} (x_{i})) \\ = h_{2} (a) (y_{j}) . \end{matrix}$

So (v, y_j) ∈ ≼ _h₂(P₂). This implies $v \in [y_{j}]_{h_{2} (P_{2})}^{≼}$

Thus S (h₂ (P₁)) ⊑ S (h₂ (P₂)).

Conversely, suppose S (h₂ (P₁)) ⊑ S (h₂ (P₂)). Then ∃ j, $[y_{j}]_{h_{2} (P_{1})}^{≼} \subseteq [y_{j}]_{h_{2} (P_{2})}^{≼}$ . Let h₁ (x_i) = y_j.

$\forall u \in [x_{i}]_{P_{1}}^{≼}$ , we have (u, x_i) ∈ ≼ _{P
₁}. Then $\forall a \in P_{1}, a (u) \leq_{a} a (x_{i}) .$

Note that (U, A) ∼ _h (V, B). Then, ∀ a ∈ A, $h_{2} (a) (h_{1} (x_{i})) = h_{3} (a (x_{i})), h_{3} (a (u)) = h_{2} (a) (h_{1} (u)) .$

Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₁, h₃ (a (u)) ≤ _h₂(a)h₃ (a (x_i)).

Then, ∀ a ∈ P₁, h₂ (a) (h₁ (u)) ≤ _h₂(a)h₂ (a) (h₁ (x_i)). This implies (h₁ (u), h₁ (x_i)) ∈ ≼ _h₂(P₁). Then $h_{1} (u) \in [y_{j}]_{h_{2} (P_{1})}^{≼}$ . So $h_{1} (u) \in [y_{j}]_{h_{2} (P_{2})}^{≼}$ .

This implies (h₁ (u), h₁ (x_i)) ∈ ≼ _h₂(P₂), i.e., ∀ a ∈ P₂, h₂ (a) (h₁ (u)) ≤ _h₂(a)h₂ (a) (h₁ (x_i)). Then, ∀ a ∈ P₂, h₃ (a (u)) ≤ _h₂(a)h₃ (a (x_i)).

Since h₃ ↾ _{U
_a} is order-preserving, we have ∀ a ∈ P₂, a (u) ≤ _aa (x_i). Then (u, x_i) ∈ ≼ _{P
₂}. So $u \in [x_{i}]_{P_{2}}^{≼}$ .

Thus S (P₁) ⊑ S (P₂). □

Corollary 5.8. Let (U, A) and (V, B) be two lattice-valued information systems. Suppose (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃). Then for any P₁, P₂ ⊆ A,

S (P₁)⊏ S (P₂) ⇔ S (h₂ (P₁)) ⊏ S (h₂ (P₂)) ;

S (P₁) ⋈ S (P₂) ⇔ S (h₂ (P₁)) ⋈ S (h₂ (P₂)).

Proof. (1) This holds by Theorems 5.5 and 5.7.

(1) This follows from Theorem 5.7. □

Example 5.9. Let (U, A) and (V, B) be two lattice-valued information systems (see Tables 2 and 3).

Table 2
The lattice-valued information system (U,A)

a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ a ₈ a ₉ a ₁₀

u ₁ 0.1 4 {1} 2 {1} 0.1 2 {1,2} 0.1 4

u ₂ 0.3 4 {1,2} 6 {1,2} 0.2 4 {1,2} 0.2 4

u ₃ 0.1 2 {1,2} 2 {1,2} 0.2 4 {1} 0.2 2

u ₄ 0.2 2 {1,2,3} 4 {1,2,3} 0.3 6 {1} 0.3 2

u ₅ 0.3 6 {1,2} 6 {1,2} 0.2 4 {1,2,3} 0.2 6

u ₆ 0.1 2 {1,2} 2 {1,2} 0.2 4 {1} 0.2 2

u ₇ 0.1 2 {1,2} 2 {1,2} 0.2 4 {1} 0.2 2

u ₈ 0.3 4 {1,2} 6 {1,2} 0.2 4 {1,2} 0.2 4

u ₉ 0.3 6 {1,2} 6 {1,2} 0.2 4 {1,2,3} 0.2 6

u ₁₀ 0.1 4 {1} 2 {1} 0.1 2 {1,2} 0.1 4

u ₁₁ 0.3 4 {1,2} 6 {1,2} 0.2 4 {1,2} 0.2 4

u ₁₂ 0.3 6 {1,2} 6 {1,2} 0.2 4 {1,2,3} 0.2 6

u ₁₃ 0.3 4 {1,2,3} 6 {1,2,3} 0.3 6 {1,2} 0.3 4

u ₁₄ 0.1 2 {1,2} 2 {1,2} 0.2 4 {1} 0.2 2

u ₁₅ 0.3 4 {1,2,3} 6 {1,2,3} 0.3 6 {1,2} 0.3 4

	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	a ₈	a ₉	a ₁₀
u ₁	0.1	4	{1}	2	{1}	0.1	2	{1,2}	0.1	4
u ₂	0.3	4	{1,2}	6	{1,2}	0.2	4	{1,2}	0.2	4
u ₃	0.1	2	{1,2}	2	{1,2}	0.2	4	{1}	0.2	2
u ₄	0.2	2	{1,2,3}	4	{1,2,3}	0.3	6	{1}	0.3	2
u ₅	0.3	6	{1,2}	6	{1,2}	0.2	4	{1,2,3}	0.2	6
u ₆	0.1	2	{1,2}	2	{1,2}	0.2	4	{1}	0.2	2
u ₇	0.1	2	{1,2}	2	{1,2}	0.2	4	{1}	0.2	2
u ₈	0.3	4	{1,2}	6	{1,2}	0.2	4	{1,2}	0.2	4
u ₉	0.3	6	{1,2}	6	{1,2}	0.2	4	{1,2,3}	0.2	6
u ₁₀	0.1	4	{1}	2	{1}	0.1	2	{1,2}	0.1	4
u ₁₁	0.3	4	{1,2}	6	{1,2}	0.2	4	{1,2}	0.2	4
u ₁₂	0.3	6	{1,2}	6	{1,2}	0.2	4	{1,2,3}	0.2	6
u ₁₃	0.3	4	{1,2,3}	6	{1,2,3}	0.3	6	{1,2}	0.3	4
u ₁₄	0.1	2	{1,2}	2	{1,2}	0.2	4	{1}	0.2	2
u ₁₅	0.3	4	{1,2,3}	6	{1,2,3}	0.3	6	{1,2}	0.3	4

Table 3

A lattice-valued information system (U,A)

	a ₁	a ₂	a ₃
x ₁	1	0.6	{0}
x ₂	3	0.6	{0,1}
x ₃	1	0.3	{0,1}
x ₄	2	0.3	{0,1,2}
x ₅	3	0.9	{0,1}
x ₆	3	0.6	{0,1,2}

A mapping h₁ : U → V is defined as follows: $\begin{matrix} \frac{u_{1}, u_{10} u_{2}, u_{8}, u_{11} u_{3}, u_{6}, u_{7}, u_{14} u_{4}}{v_{1} v_{2} v_{3} v_{4}} . \\ \frac{u_{5}, u_{9}, u_{12} u_{13}, u_{15}}{v_{5} v_{6}} . \end{matrix}$

Define a mapping h₂ : A → B as follows: $\begin{matrix} \frac{a_{1}, a_{4} a_{2}, a_{8}, a_{10} a_{3}, a_{5}, a_{6}, a_{7}, a_{9}}{b_{1} b_{2} b_{3}} . \end{matrix}$

Put U_{a
_i} = {a_i (u_s) :1 ≤ s ≤ 15}, V_{b
_j} = {b_j (v_t) :1 ≤ t ≤ 6}. Then $U^{*} = ⋃_{i = 1}^{10} U_{a_{i}}, V^{*} = ⋃_{j = 1}^{3} V_{b_{j}} .$

A mapping h₃ : U^* → V^* is defined as follows: ∀ w ∈ U_a $h_{3} ↾_{U_{a}} (w) = {\begin{matrix} 3 w, a = a_{1}, a_{6} or a_{9} \\ \frac{1}{2} w, a = a_{2}, a_{4}, a_{7} or a_{10} \end{matrix},$ and for a = a₃, a₅ or a₈,

h₃ ↾ _{U
_a}: $\begin{matrix} \frac{{1} {1, 2} {1, 2, 3}}{{0} {0, 1} {0, 1, 2}} . \end{matrix}$

It is easy to verify that for any u ∈ U and a ∈ A, $h_{3} (a (u)) = h_{2} (a) (h_{1} (u)) .$

Thus (U, A) ∼ _h (V, B) with h = (h₁, h₂, h₃).

It is clear that the original lattice-valued information system (U, A) is relatively complex and its image lattice-valued information system (V, B) is simpler. Theorem 5.5, Corollary 5.6, Theorem 5.7, Corollary 5.8 illustrate the fact that equality, dependence, partial dependence and independence between information structures in (U, A) and (V, B) are equivalent to each other.

Pick Q₁ = {b₁}, Q₂ = {b₂}, Q₃ = {b₃}, Q₄ = {b₁, b₂}. Q₅ = {b₁, b₃}, Q₆ = {b₂, b₃}.

It is easy to verify that S (B) ≺ S (Q₄) ≺ S (Q₁) ≺ S (∅), S (B) ≺ S (Q₅) ≺ S (Q₃) ≺ S (∅), S (B) ≺ S (Q₂) ≺ S (∅), S (Q₄) ≺ S (Q₂), S (Q₄) ≺ S (Q₃), S (Q₅) ≺ S (Q₁).

S (Q₁) ⊏ S (Q₂), S (Q₂) ⊏ S (Q₁) ; S (Q₁) ⊏ S (Q₃), S (Q₃) ⊏ S (Q₁); S (Q₂) ⊏ S (Q₃), S (Q₃) ⊏ S (Q₂); S (Q₂) ⊏ S (Q₅), S (Q₅) ⊏ S (Q₂); S (Q₄) ⊏ S (Q₅), S (Q₅) ⊏ S (Q₄).

Now, equality, dependence, partial dependence and independence between information structures of the image information system (V, B) are obtained. Thus, we may obtain equality, dependence, partial dependence and independence between information structures of the original lattice-valued information system (U, A).

6 Conclusions

In this paper, information structures in a lattice-valued information system have been considered as set vectors. Dependence and independence between information structures have been depicted. Properties of information structures are given by using the inclusion degree. Invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression have been obtained (see Table 4).

Table 4

Invariant characterizations of information structures in lattice-valued information systems under homomorphisms based on data compression

Characterizations	Invariant
Equality between information structures in lattice-valued information systems (=)	√
Dependence between information structures in lattice-valued information systems (⪯)	√
Strict dependence between information structures in lattice-valued information systems (≺)	√
Partial dependence between information structures in lattice-valued information systems (⊑)	√
Strictly partial dependence between information structures in lattice-valued information systems (⊏)	√
Independence between information structures in lattice-valued information systems (⋈)	√

These results will be significant for establishing a framework of granular computing in information systems. In the future, we will give some applications for dealing with knowledge discovery in lattice-valued information systems.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable suggestions which have helped immensely in improving the quality of this paper. This work is supported by National Natural Science Foundation of China (11461005), Natural Science Foundation of Guangxi (2016GXNSFAA380045, 2016GXNSFAA380282, 2016GXNSFAA380286), The excellent Youth Foundation of Hunan Provincial Department of Education (16B141).

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Invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Binary relations and lattices

2.2 Lattice-valued information systems

Table 1 A lattice-valued information system (U,A) a 1 a 2 a 3 x 1 1 0.6 {0} x 2 3 0.6 {0,1} x 3 1 0.3 {0,1} x 4 2 0.3 {0,1,2} x 5 3 0.9 {0,1} x 6 3 0.6 {0,1,2}

3 Some concepts of information structures in a lattice-valued information system

5 Invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression

Footnotes

Acknowledgments

References

Table 1
A lattice-valued information system (U,A)

a ₁ a ₂ a ₃

x ₁ 1 0.6 {0}

x ₂ 3 0.6 {0,1}

x ₃ 1 0.3 {0,1}

x ₄ 2 0.3 {0,1,2}

x ₅ 3 0.9 {0,1}

x ₆ 3 0.6 {0,1,2}