Attributes Reduction on SE-ISI Concept Lattice for an Incomplete Context Using Object Ranking

Abstract

The formal concept of lattice plays a vital role in knowledge discovery. Reduction of the attribute has many applications in machine learning technology and data mining fields. In this paper, we introduce an object ranking concept to define a consistency set and the reduction of the attributes by structural features. An incomplete information system works on the three-way concepts using the SE-ISI Context. The granular was emphasized with join (meet) irreducible sets using the object ranking concepts. A dual operator is defined based on the object ranking concepts and its properties and conditions are verified. Hence, this elaborates on the four kinds of reduction of the attributes. The ordered pairs give the knowledge of the attributes that deal with the interval set of both the approximation of rough set theory concerning the objects. Therefore, the relationship between four kinds of reduction of the attribute was appropriate to access the consistency set using the object ranking concepts by some of the theorems and examples.

1. Introduction

Pawlak [1,2] introduced the Rough set theory for defining two approximation sets and approached the rough set theory as an alternative to the fuzzy set theory. Jianjun [3] analyzed the connection between the three-way concept lattice and hence the structures were generalized for an algorithm. Ting [4] investigated the concept lattice with the isomorphic and anti-isomorphic conditions using the study of 1-dual and 2-dual interactable knowledge. Huilai [5] described a target set of positive and negative formal concepts to determine attribute characterization. Ruisi [6] examined the three-way concept lattice based on attribute reduction and also examined the results based on the reduction and analysis of a formal concept lattice. The attribute reduction for an object-oriented concept lattice was constructed by Jian [7] to define its relation based on attribute ranking. Wei [8] defined the strong and weak consistency for decision concept lattices. The similarity of objects was investigated for an indiscernibility relation by Yan [9]. Yiyu [10] constructed two heuristics reduced for the attribute measure.

The interval sets of three-way formal concepts that represented the intent and extent SE-ISI context were analyzed by Renu [11]. Zhen [12] proposed attribute reduction using the discernibility matrix. Furthermore, this relationship between consistency set and reduction of discernibility matrix or discernibility function is obtained by attribute reduction for an incomplete context. Zhang [13] defined an interval set with the degree and discussed a new method for an optimal approximation for an interval set. He analyzed a new binary relation with an approximation set by developing both intervals set model and granular theory.

Chen [14] represented a formal concept lattice that gives knowledge on graph theory and granular reduction and investigated the result for an optimization problem and numerical methods. Lei-Jun [15] proposed a simple discernibility matrix for the objects and attributes that construct a complex concept lattice and its properties. Jianqin [16] investigated the three types of formal concepts based on rectangular and covering theories. Furthermore, a new algorithm was explicitly used for the bits operators to demonstrate the effectiveness of attribute reduction. Srirekha [17] emphasized an ordered pair of distributive lattices to define the projection. A rough approximation space investigated the lattice homomorphism condition and equivalence relation.

Binghan [18] introduced a relation between the classical formal context and the incomplete fuzzy formal context. A double threshold operator was studied with the knowledge of order pairs consisting of objects and attributes. Qian [19] constructed a three-way concept lattice and transformed the formal context with the complementary context. He introduced Type I and II combinatorial contexts and studied the isomorphic to object and attribute induced respectively. Quintero [20] studied the application in chemistry for the formal concept with molecular structure and radionuclides. Dias [21] analyzed the reduction of concept lattice by three classes of reduction techniques and examined them with an experimental study. Xiao [22] combined a formal concept analysis with the weight attribute concept. The granularity was described using the dimension of weight through the threshold structure. Zhenquan [23] proposed rough set theory and granular theory to determine the weight of the attributes to define their effectiveness from the nonredundant aspect and redundant aspects. Li [24] experimented with the fundamental geographical data for formal concept analysis and hence obtained the feasible result of the complex geo ontological methods.

From the above literature survey, we concluded that the three way concept lattice gives a wide knowledge regarding information systems. Hence the granular has been discussed with various consistency sets to deduct the attribute reduction.

Hence, in this paper our contribution is as follows: In Section 2, the definition of rough set theory and formal concepts were given. In Section 3, we introduced the reduction of formal concept lattice with object ranking. In Section 4, we merged the object ranking concept with SE-ISI context. In Section 5, a real-time observation study was analyzed.

2. Preliminaries

In this section, the formal concept of the lattice with the notion of $O$ as a finite universal set as an object and $M$ as an attribute is discussed. Formal concept lattice has the attribute value $“0”$ and $“1”$ where there exists the relation $R_n$ such that if the object is related to the attribute, then the value is defined as $1,$ and if the object is not related to the attributes, then the value is defined as $0$.

2.1. Rough Set Theory

The information system $\left(IS\right)$ introduced by Pawlak [1], consists of an object defined as finite non-empty set as $\left(O\right)$, knowledge obtained as an attribute $\left(M\right)$ where $M$ is described as a combination of condition attributes $\left(C\right)$ and decision attributes $\left(D\right)$, $V$ is a cartesian product of object and attributes $(O\times M),$ and $f$ is defined as a function then $f:M\to V$. Hence, the information system is denoted as $IS=⟨O,M,V,f⟩$.

Let the discernibility relation obtained from the attribute as $B_{\alpha }\subseteq M$, then the approximation space is defined as a pair of $(O,{\left[p_{\alpha }\right]}_B)$ where ${\left[p_{\alpha }\right]}_B$ is an equivalence relation which partition the universal set $O$. Consider $P$ be a subset of object $\left(O\right)$, then the approximation for the corresponding notion can be defined as [1,2].

$$\overline{apr\left(P_{\alpha }\right)}=\{\left(p_{\alpha }\in O \right|{\left[p_{\alpha }\right]}_B\subseteq P_{\alpha }\}$$

$$\underset{\_}{apr\left(P_{\alpha }\right)}=\{\left(p_{\alpha }\in O \right|{\left[p_{\alpha }\right]}_B\cap P_{\alpha }\ne \varnothing \}$$

The set is said to be definable set if $\overline{apr\left(P_{\alpha }\right)}=\underset{\_}{apr\left(P_{\alpha }\right)}$, else the set is said to be rough set.

2.2. Formal Concepts

The triplet $(O,M,R_n)$ is said to be a formal concept where $O \mathrm{and} M$ represents a finite set of object and attribute respectively, and the binary relation $R_n$ represents a cartesian product of object and attribute such that $R_n\in O\times M.$ The relation $R_n$ is denoted as $x_{\alpha }{R}_n{a}_{\alpha } \mathrm{or} \left(x_{\alpha },a_{\alpha }\right)\in R_n$ [16].

3. Reduction of Formal Concept Lattice with Object Ranking

This section views the formal concept of lattice and the reduction of attributes using the object ranking concept. Further, a dual operator is considered to defined the consistency set and its properties.

Definition 1.

Let consider the triplet $(O,M,R_n)$  where object $O$  and attribute $M$  represents the finite universal set and knowledge about the object respectivelyt. The relation $R_n$  is the indiscernibility relation between the object and the attribute and it is denoted as $\left(x_{\alpha },a_{\alpha }\right)\in R_n$. [4,6,15].

A pair of $(X_{\alpha },B_{\alpha })$ such that $P_{\alpha }\subseteq O$ and $B_{\alpha }\subseteq M$ then the operator can be defined as

$$P_{\alpha }^{*}=\{\left(f_{\alpha }\in M \right|p_{\alpha }{R}_n{f}_{\alpha },\forall p_{\alpha }\in P_{\alpha }\}$$

$$B_{\alpha }^{*}=\{\left(p_{\alpha }\in O \right|p_{\alpha }{R}_n{g}_{\alpha },\forall g_{\alpha }\in B_{\alpha }\}$$

If $P_{\alpha }^{*}$ is the greatest collection of set of attributes that associate with all the objects in $P_{\alpha }$ and $B_{\alpha }^{*}$ is the greatest collection of set of an object that associates with all attributes in $B_{\alpha }$ then

$$p_{\alpha }^{*}=\{\left(f_{\alpha }\in M \right|\left(p_{\alpha },f_{\alpha }\right)\in R_n\}$$

$$f_{\alpha }^{*}=\{\left(p_{\alpha }\in O \right|\left(p_{\alpha },f_{\alpha }\right)\in R_n\}$$

Throughout this paper, formal context has the covering of $M$ as $\{\left(p_{\alpha }^{*} \right|p_{\alpha }\in O\}$ and covering of $O$ as $\{\left(f_{\alpha }^{*} \right|f_{\alpha }\in M\}$.

Definition 2.

Let a triple of $(O,M,R_n)$  then $P_{\alpha }\subseteq O,B_{\alpha }\subseteq M$  defined as a pair of dual approximation $\oplus ,⨀:2^O\to 2^M$  for a formal context as [7].

$$P_{\alpha }^{\oplus }=\{\left(f_{\alpha }\in M \right|f_{\alpha }^{*}\subseteq P_{\alpha }\}$$

$$P_{\alpha }^{⨀}=\{f_{\alpha }\in M|f_{\alpha }^{*}\cap P_{\alpha }\ne \varnothing \}$$

Similarly, $⨀,\oplus :2^M\to 2^O$ as

$$B_{\alpha }^{\oplus }=\{\left(p_{\alpha }\in O \right|p_{\alpha }^{*}\subseteq B_{\alpha }\}$$

$$B_{\alpha }^{⨀}=\{p_{\alpha }\in O|p_{\alpha }^{*}\cap B_{\alpha }\ne \varnothing \}$$

Properties 1.

Consider ${P_{\alpha }}_1,{P_{\alpha }}_2,P_{\alpha }\subseteq O$  and ${F_{\alpha }}_1,{F_{\alpha }}_2,F_{\alpha }\subseteq M$  then the approximation operators satisfy.

(i)

${P_{\alpha }}_1\subseteq {P_{\alpha }}_2\Rightarrow {P_{\alpha }}_1^{\oplus }\subseteq {P_{\alpha }}_2^{\oplus },{P_{\alpha }}_1^{⨀}\subseteq {P_{\alpha }}_2^{⨀}$ and ${F_{\alpha }}_1\subseteq {F_{\alpha }}_2\Rightarrow {F_{\alpha }}_1^{\oplus }\subseteq {F_{\alpha }}_2^{\oplus },{F_{\alpha }}_1^{⨀}\subseteq {F_{\alpha }}_2^{⨀}$

(ii)

$P_{\alpha }^{\oplus ⨀}\subseteq P_{\alpha }\subseteq P_{\alpha }^{⨀\oplus } and F_{\alpha }^{\oplus ⨀}\subseteq F_{\alpha }\subseteq F_{\alpha }^{⨀\oplus }$

(iii)

${P_{\alpha }}^{⨀\oplus ⨀}=P_{\alpha }^{⨀},F_{\alpha }^{⨀\oplus ⨀}=F_{\alpha }^{⨀} and P_{\alpha }^{\oplus ⨀\oplus }=P_{\alpha }^{\oplus },F_{\alpha }^{\oplus ⨀\oplus }=F_{\alpha }^{\oplus }$

(iv)

${\left({P_{\alpha }}_1\cap {P_{\alpha }}_2\right)}^{\oplus }={P_{\alpha }}_1^{\oplus }\cap {P_{\alpha }}_2^{\oplus },{\left({P_{\alpha }}_1\cup {P_{\alpha }}_2\right)}^{⨀}={P_{\alpha }}_1^{⨀}\cup {P_{\alpha }}_2^{⨀} and$

$${\left({F_{\alpha }}_1\cap {F_{\alpha }}_2\right)}^{\oplus }={F_{\alpha }}_1^{\oplus }\cap {F_{\alpha }}_2^{\oplus },{\left({F_{\alpha }}_1\cup {F_{\alpha }}_2\right)}^{⨀}={F_{\alpha }}_1^{⨀}\cup {F_{\alpha }}_2^{⨀}$$

A Pair of $(P_{\alpha },B_{\alpha })$ where $P_{\alpha }\subseteq O$ and $B_{\alpha }\subseteq M$ is called context if $P_{\alpha }=B_{\alpha }^{⨀}$ and $B_{\alpha }=P_{\alpha }^{\oplus }$, then $P_{\alpha }$ are called extent and $B_{\alpha }$ is called intent of $(P_{\alpha },B_{\alpha })$. Then $(P_{\alpha }^{⨀\oplus },P_{\alpha }^{\oplus })$ if $P_{\alpha }\subseteq O$ and $(B_{\alpha }^{⨀},B_{\alpha }^{⨀\oplus })$ if $B_{\alpha }\subseteq M$ is called concept.

Since $(O,M,R_n)$ is regular if $(O,M)$ and $(\mathrm{\varnothing },\mathrm{\varnothing })$ are formal concept denoted as $L(O,M,R_n)=\left\{\right(P_{\alpha },B_{\alpha }\left)\right|P_{\alpha }=B_{\alpha }^{⨀},B_{\alpha }=P_{\alpha }^{\oplus }\}$.

Hence a binary relation $\le$ is defined as $\left({P_{\alpha }}_1,{B_{\alpha }}_1\right)\le \left({P_{\alpha }}_2,{B_{\alpha }}_2\right)\iff {P_{\alpha }}_1\subseteq {P_{\alpha }}_2\iff {B_{\alpha }}_1\subseteq {B_{\alpha }}_2$.

Therefore, the hierarchy of partial relation $\left(L\right(O,M,R_n),\le )$ is a formal concept lattice if $\left({P_{\alpha }}_1,{B_{\alpha }}_1\right)\le \left({P_{\alpha }}_3,{B_{\alpha }}_3\right)\le ({P_{\alpha }}_2,{B_{\alpha }}_2) then ({P_{\alpha }}_1,{B_{\alpha }}_1)$ is called child concept and $({P_{\alpha }}_2,{B_{\alpha }}_2)$ is a parent concept of $\left({P_{\alpha }}_1,{B_{\alpha }}_1\right)\Rightarrow \left({P_{\alpha }}_1,{B_{\alpha }}_1\right)\le ({P_{\alpha }}_2,{B_{\alpha }}_2)$. [6,9]

Since a partially ordered relation defined in a finite ordered set as a lattice approximation $L\left(O,M,R_n\right)$ with the operator meet $\wedge$ and join $\vee$ as

$$\left({P_{\alpha }}_1,{B_{\alpha }}_1\right)\wedge ({P_{\alpha }}_2,{B_{\alpha }}_2)=({\left({P_{\alpha }}_1\cap {P_{\alpha }}_2\right)}^{\oplus ⨀},{B_{\alpha }}_1\cap {B_{\alpha }}_2)$$

$$\left({P_{\alpha }}_1,{B_{\alpha }}_1\right)\vee ({P_{\alpha }}_2,{B_{\alpha }}_2)=({P_{\alpha }}_1\cup {P_{\alpha }}_2,{\left({B_{\alpha }}_1\cup {B_{\alpha }}_2\right)}^{⨀\oplus })$$

Definition 3.

Let $(O,M,R_n)$  be a formal concept, the number of attributes possessing the object $p_{\alpha }\in O$  then the object ranking is defined as the cardinality of $p_{\alpha }^{⨀}$  i.e., $rank\left(p_{\alpha }\right)=\left|p_{\alpha }^{⨀}\right|$.

Hence, from the above definition and properties the object ranking concept can be examined with some lemma and theorems that emerged with the join irreducible concepts.

Lemma 1.

Let a formal context $(O,M,R_n)$, consider $p_{\alpha }\in O$  then $rank\left(p_{\alpha }\right)=n$  then there exist $\left(P_{\alpha },B_{\alpha }\right)\in L(O,M,R\_n)$  such that $\left|B_{\alpha }\right|=n$.

Proof. 

Since $rank\left(p_{\alpha }\right)=n$  then $\left|p_{\alpha }^{⨀}\right|=n$, Let $P_{\alpha }=p_{\alpha }^{⨀}$  and $B_{\alpha }=p_{\alpha }^{\oplus }=p_{\alpha }^{⨀\oplus }$  by Properties 1 and Definition 2, Therefore, $(P_{\alpha },B_{\alpha })=(p_{\alpha }^{⨀},p_{\alpha }^{⨀\oplus })$. $\left|B_{\alpha }\right|=\left|p_{\alpha }^{⨀}\right|=n$. □

Theorem 1.

Let us consider an ordered pair $\left(P_{\alpha },B_{\alpha }\right)$  that belongs to the lattice $L_0(O,M,R_n)$  and $rank\left(p_{\alpha }\right)=\left|B_{\alpha }\right|$  then $(P_{\alpha },B_{\alpha })=(p_{\alpha }^{⨀},p_{\alpha }^{⨀\oplus })$  is a join irreductable concept $\forall p_{\alpha }\in P_{\alpha }$.

Proof. 

By the above Lemma 1, Let $\left|B_{\alpha }\right|=n$, If there exist $p_{\alpha }\in P_{\alpha }$  such that $rank\left(p_{\alpha }\right)=\left|p_{\alpha }^{⨀}\right|=n$. By Definition 3, therefore, $\left(p_{\alpha }^{⨀},p_{\alpha }^{⨀\oplus }\right)\le (P_{\alpha },B_{\alpha }) hence p_{\alpha }^{⨀}=P_{\alpha }$  and $p_{\alpha }^{⨀}=\left|B_{\alpha }\right|=n$, thus $B_{\alpha }=p_{\alpha }^{⨀}$  and that $(P_{\alpha },B_{\alpha })=\left(p_{\alpha }^{⨀\oplus }\right)$. Hence the result. □

Definition 4.

Let $(O,M,R_n)$  be a context and $L_0(O,M,R_n)=\left\{P_{\alpha }\right|\left(P_{\alpha },B_{\alpha }\right)\in L(O,M,R_n)\}$. If an attribute set $B_{\alpha }\subseteq M$  then  $L_0(O,S,{R_n}_{B_{\alpha }})=L_0(O,M,R_n)$  where  ${R_n}_{B_{\alpha }}=R_n\cap O\times B_{\alpha }$, is said to be consistent set as $B_{\alpha }$. If $f_{\alpha }\in B_{\alpha }$  then  $L_0\left(O,B_{\alpha }-\left\{f_{\alpha }\right\},{R_n}_{B_{\alpha }}-\left\{f_{\alpha }\right\}\right)\ne L_0(O,M,R_n)$  then  $B_{\alpha }$  is called reduct.

Definition 5.

Let $(O,M_1,{R_n}_1)$  and $(O,M_2,{R_n}_2)$  be the two formal context if $\left(P_{\alpha },B_{\alpha }\right)\in L_0(O,M_2,{R_n}_2)$  then there exist $\left(P_{\alpha }^{\prime },B_{\alpha }^{\prime }\right)\in L_0(O,M_2,{R_n}_2)$  such that $P_{\alpha }^{\prime }=P_{\alpha }$. Hence it is denoted as $L_0\left(O,M_1,{R_n}_1\right)\le L_0(O,M_2,{R_n}_2)$.

Definition 6.

Let us consider a lattice $L$  with the element $p_{\alpha }\in L$  is called join irreducible if  $p_{\alpha }\ne 0$  and  $p_{\alpha }=f_{\alpha }\vee g_{\alpha }\Rightarrow p_{\alpha }=f_{\alpha }$  or  $p_{\alpha }=g_{\alpha }\forall f_{\alpha },g_{\alpha }\in L$  and dually the element can be defined as meet irreducible.

Example 1.

From Table 1: the formal context $(O,M,R_n)$ where $O=\{\mathrm{1,2},\mathrm{3,4}\}$,$M=\{a,b,c,d,e\}$  and the relation is defined as$(p_{\alpha },{f_{\alpha }}_i)=1$  if  $p_{\alpha }{R}_n{f_{\alpha }}_i$  and  $(p_{\alpha },{f_{\alpha }}_i)=0$  if

$$p_{\alpha }\neg R_n{f_{\alpha }}_i\forall p_{\alpha }\in O, {f_{\alpha }}_i\in M.$$

Table 1. Information Table.

U	a	b	c	d	e
1	1	1	0	1	1
2	1	1	1	0	0
3	0	0	0	1	0
4	1	1	1	0	0

Object Rank: $Rank\left(1\right)=4$, $Rank\left(2\right)=3, Rank\left(3\right)=1, Rank\left(4\right)=3$ Then $O_1=\left\{3\right\},O_2=\left\{\mathrm{2,4}\right\},O_3=\left\{1\right\}$.

(i)

The universal set $O$ and the attruites set $M$ can be represented as a set itself in concept of the set.

(ii)

We represent $\left(\right\{\mathrm{2,4},3\},\{a,b,c,d\left\}\right)$ as $(234,abcd)$.

(iii)

From Figure 1, it is clearly defined $(234,abcd),(3,d)$ are meet irreducible and join irreducible respectively.

Then $L\left[O_0\right]=(\mathrm{\varnothing },\mathrm{\varnothing }),L\left[O_1\right]=(3,d),L\left[O_2\right]=(2,abc),(4,abc),L\left[O_3\right]=(1,abde)$.

Hence, this generates the set $(\mathrm{\varnothing },\mathrm{\varnothing }),(3,d),(24,abc),(13,abde),(234,abcd) and (O,M)$. Consider a reduct $S=\{a,c,d\}$ whose corresponding formal concept lattice is $L(O,M,{R_n}_S)$.

Figure 1. Object Ranking Concepts with Reduction.

Figure 1. Object Ranking Concepts with Reduction.

Theorem 2.

Let $(O,M,R_n)$  be a formal context then $L_0\left(O,M,R_n\right)\le L_0(O,M,{R_n}_s)$.

Proof. 

Consider $\left(P_{\alpha },B_{\alpha }\right)\subseteq L_0(O,M,{R_n}_s)$  which implies $P_{\alpha }^{{\oplus }_s}=B_{\alpha }$  and $B_{\alpha }^{⨀}=P_{\alpha }$, for $P_{\alpha }\subseteq O$, hence $\left(P_{\alpha }^{\oplus ⨀},P_{\alpha }^{\oplus }\right)\in L_0(O,M,R_n)$  thus, by the Properties 1 $P_{\alpha }^{\oplus ⨀}\subseteq P_{\alpha }$  and $B_{\alpha }=P_{\alpha }^{{\oplus }_s}=P_{\alpha }^{\oplus }\cap S_{\alpha }\subseteq P_{\alpha }^{\oplus }$, hence $P_{\alpha }=B_{\alpha }^{⨀}\subseteq P_{\alpha }^{\oplus ⨀}$, therefore $P_{\alpha }=P_{\alpha }^{\oplus ⨀}$, by Definition 5, $L_0\left(O,M,R_n\right)\le L_0(O,M,{R_n}_s)$. □

4. The 3-Ways Formal Concept Using Object Ranking in SE-ISI Context

This section we discuss about the pair of objects and an interval set of attributes which is defined as SE-ISI [Set Extent-an Interval Set Intent] Context. This interval set gives the knowledge access from lower and upper approximation. The dual operator elaborates the features of lattice theory with the rough set theory, which induce to define the formal concept lattice for reduction of attributes.

Definition 7.

A finite set of an object $O$  is defined using an interval set such that $[\underset{\_}{P_{\alpha }},\overline{P_{\alpha }}]=\{\overline{P_{\alpha }}\subseteq P_{\alpha }\subseteq \underset{\_}{P_{\alpha }}\}=\{P_{\alpha }\in 2^O|\overline{P_{\alpha }}\subseteq P_{\alpha }\subseteq \underset{\_}{P_{\alpha }}\}$where  $\underset{\_}{P_{\alpha }} and \overline{P_{\alpha }}$  are LUB and GLB on interval set respectively.

Let $2^O=\{\left(\left[\underset{\_}{P_{\alpha }},\overline{P_{\alpha }} \right]\right|\underset{\_}{P_{\alpha }},\overline{P_{\alpha }}\subseteq O,\underset{\_}{P_{\alpha }}\subseteq \overline{P_{\alpha }}\}$ be a set of all interval set. A partially order $\le$ between $\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]$, $\left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]$ is defined as $\left[\underset{\_}{{P_{\alpha }}_1},{\overline{P_{\alpha }}}_1\right]$ $\le \left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]$ which implies $\underset{\_}{{P_{\alpha }}_1}\subseteq \underset{\_}{{P_{\alpha }}_2}$ and $\overline{{P_{\alpha }}_1}\subseteq \overline{{P_{\alpha }}_2}$.

An interval set $\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]=\left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]$ then $\underset{\_}{{P_{\alpha }}_1}=\underset{\_}{{P_{\alpha }}_2}$ and $\overline{{P_{\alpha }}_1}=\overline{{P_{\alpha }}_2}$. $\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]<\left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]$ then $\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]\le \left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]$ and $\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]\ne \left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]$.

Consider ${P_{\alpha }}_1,{P_{\alpha }}_2,P_{\alpha }\subseteq O$ and ${F_{\alpha }}_1,{F_{\alpha }}_2,F_{\alpha }\subseteq M$ then the intersection, union, and difference were defined as

$$\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]\cap \left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]=[\underset{\_}{{P_{\alpha }}_1}\cap \underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_1}\cap \overline{{P_{\alpha }}_2}]$$

$$\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]\cup \left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]=[\underset{\_}{{P_{\alpha }}_1}\cup \underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_1}\cup \overline{{P_{\alpha }}_2}]$$

$$\left[\underset{\_}{{P_{\alpha }}_1},\overline{{P_{\alpha }}_1}\right]-\left[\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_2}\right]=[\underset{\_}{{P_{\alpha }}_1}-\underset{\_}{{P_{\alpha }}_2},\overline{{P_{\alpha }}_1}-\overline{{P_{\alpha }}_2}]$$

Definition 8.

Let $(O,M,V_a,R_n)$  be a formal concept for $P_{\alpha }\in O$  the number of attribute value $\{1,?\}$  possessing the object $P_{\alpha }$, then the cardinality of ${P_{\alpha }}^{\oplus }$  is called object ranking of interval set of $P_{\alpha }$.

Definition 9.

Let $(O,M,\{\mathrm{0,1},?\},R_n)$  be the incomplete information system, where $O$  is an object,  $M$  is an attribute and  $\{\mathrm{0,1},?\}$  are the 3 possible entries of the corresponding table. A relation  $R_n\subseteq O\times M\times \{\mathrm{0,1},?\}$  and a mapping is defined as  $R_n\to R_n:O\times M\to \{\mathrm{0,1},?\}$ Hence the relation is defined as

(i)

$\left(p_{\alpha },f_{\alpha },1\right)\in R_n$ this suggests the object ($p_{\alpha })$  that pairs with the attribute ( $f_{\alpha })$.

(ii)

$\left(p_{\alpha },f_{\alpha },0\right)\in R_n$  this suggests the object ( $p_{\alpha })$  that does not pair with the attribute $\left(a_{\alpha }\right)$.

(iii)

$\left(p_{\alpha },f_{\alpha },?\right)\in R_n$  this suggests the object which is unknown or not in ( $p_{\alpha })$  that pairs with the attribute $\left(a_{\alpha }\right)$.

For an incomplete information system $(O,M,V_a=\{\mathrm{0,1},?\},R_n)$, then $\underset{\_}{B_{\alpha }}$ and $\overline{B_{\alpha }}$ are defined on $P_{\alpha }\in 2^O$ and $B_{\alpha }\in 2^M$ by

$$\underset{\_}{\pi \left(P_{\alpha }\right)}=\{f_{\alpha }\in M \forall p_{\alpha }\in P_{\alpha },\left(p_{\alpha },f_{\alpha },1\right)\in R_n\}$$

$$\overline{\pi \left(P_{\alpha }\right)}=\{f_{\alpha }\in M \forall p_{\alpha }\in P_{\alpha },\left(p_{\alpha },f_{\alpha },1\right)\in R_n\vee \left(p_{\alpha },a_{\alpha },?\right)\in R_n\}$$

$$\underset{\_}{\pi B_{\alpha }}=\{\left(p_{\alpha }\in O \right|\forall g_{\alpha }\in B_{\alpha },\left(p_{\alpha },f_{\alpha },1\right)\in R_n\}$$

$$\overline{\pi \left(B_{\alpha }\right)}=\{p_{\alpha }\in O \forall g_{\alpha }\in B_{\alpha },\left(p_{\alpha },f_{\alpha },1\right)\in R_n\vee \left(p_{\alpha },b_{\alpha },?\right)\in R_n\}$$

Definition 10.

Consider an incomplete context with two dual operator $\oplus :2^O\to 2^M and ⨀:2^M\to 2^O$  are defined on the object ranking interval set $p_{\alpha }\in 2^O and \left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right]\in 2^M$ is ${P_{\alpha }}^{\oplus }=\left[\underset{\_}{\pi \left(P_{\alpha }\right)},\overline{\pi \left(P_{\alpha }\right)}\right]$ and ${\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right]}^{⨀}=\underset{\_}{\pi \left(B_{\alpha }\right)}\cap \overline{\pi \left(B_{\alpha }\right)}$, for an incomplete formal context  $(O,M,V_a,R_n)$ then for any $P_{\alpha },{P_{\alpha }}_i,{P_{\alpha }}_j\in 2^O$,  $\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right],\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right],\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]\in 2^M$ then the following condition statisfies,

$${P_{\alpha }}_i\subseteq {P_{\alpha }}_j\Rightarrow {P_{\alpha }}_j^{⨀}\le {P_{\alpha }}_i^{⨀},\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]\le \left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]\Rightarrow {\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]}^{\oplus }\subseteq {\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]}^{\oplus }$$

$$P_{\alpha }\subseteq {P_{\alpha }}^{\oplus ⨀},\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right]\le {\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right]}^{\oplus ⨀}.$$

$${P_{\alpha }}^{⨀}={P_{\alpha }}^{⨀\oplus ⨀},{\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right]}^{\oplus }={\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right]}^{\oplus ⨀\oplus }.$$

$${\left({P_{\alpha }}_i\cup {P_{\alpha }}_j\right)}^{⨀}={P_{\alpha }}_i^{⨀}\cap {P_{\alpha }}_j^{⨀},\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]\cap {\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]}^{\oplus }={\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]}^{\oplus }\cup {\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]}^{\oplus }.$$

Definition 11.

Let $(O,M,V_a,R_n)$  be an incomplete context then the set is extended to an interval set which is defined as $(O,\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right])$ using the object ranking concept. Hence, the set is formally called SE-ISI formal concept with the condition

$${P_{\alpha }}^{⨀}-\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right],{\left[\underset{\_}{B_{\alpha }},\overline{B_{\alpha }}\right]}^{\oplus }-O$$

For any two SE-ISI concept $({P_{\alpha }}_i,\left[\underset{\_}{B_i},\overline{B_i}\right])$ and $({P_{\alpha }}_j,\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right])$ are ordered by $\left({P_{\alpha }}_i,\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]\right)\le \left({P_{\alpha }}_j,\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]\right)\iff {P_{\alpha }}_i\subseteq {P_{\alpha }}_j\iff \left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]\le \left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right].$.

Hence, it describes the $LUB$ and $GLB$ using SE-ISI formal concept lattice and it is defined as

$$\left({P_{\alpha }}_i,\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]\right)\wedge ({P_{\alpha }}_j,\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right])=({P_{\alpha }}_i\cap {P_{\alpha }}_j,{\left(\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]\cup \left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]\right)}^{⨀\oplus })$$

$$\left({P_{\alpha }}_i,\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]\right)\vee ({P_{\alpha }}_j,\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]=({\left({P_{\alpha }}_i\cup {P_{\alpha }}_j\right)}^{\oplus ⨀},\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right]\cap \left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right])$$

Therefore, the SE-ISI context is a complete lattice. Additionally, in the SE-ISI context it is possible to define the parent and son context lattice as $({P_{\alpha }}_i,\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right])<({P_{\alpha }}_k,\left[\underset{\_}{{B_{\alpha }}_k},\overline{{B_{\alpha }}_k}\right])<({P_{\alpha }}_j,\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right]), then ({P_{\alpha }}_i,\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right])$  is said to be son concept of $({P_{\alpha }}_j,\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right])$  and $({P_{\alpha }}_j,\left[\underset{\_}{{B_{\alpha }}_j},\overline{{B_{\alpha }}_j}\right])$  is said to be parent concept of $({P_{\alpha }}_i,\left[\underset{\_}{{B_{\alpha }}_i},\overline{{B_{\alpha }}_i}\right])$.

Definition 12.

Let $(O,M,V_a,R_n)$  be a SE-ISI formal context and consider a subset of attribute Reduct as $S\subseteq M$  then it is said to be a consistent set of $(O,M,V_a,R_n)$  if $ex{t}_{ORL}(O,M,R_n)=ex{t}_{ORL}(O,M,R_{n_{s_{\alpha }}})$  then $R_{n_{s_{\alpha }}}=R_n\cap (O\times S_{\alpha })$. Thus $ex{t}_{ORL}(O,S_{\alpha }-\{s_{\alpha }\},R_{n_{S_{\alpha } -\left\{s_{\alpha }\right\}}}\ne ex{t}_{ORL}(O,M,R_n)$  for any $s_{\alpha }\in S_{\alpha }$  is said to be object ranking lattice with reduct of $(O,M,R_n)$.

Definition 13.

Let $(O,M,V_a,R_n)$  be a SE-ISI formal context, if the reduct $S_{\alpha }\subseteq M$  (denoted as a set of attributes) then it is said to be a consistent set of Object Ranking Meet Irreducible (ORMI) $ex{t}_{ORMI}(O,M,R_n)=ex{t}_{ORMI}(O,M,R_{n_{s_{\alpha }}})$. If $S_{\alpha }$  is an ORMI consistent set of $(O,M,R_n)$  then there is no proper subset $T_{\alpha }\subset S_{\alpha }$  such that $T_{\alpha }$  is an ORMI consistent set and $S_{\alpha }$  is a reduct of ORMI consistent set of $I(O,M,R_n)$. Thus, the extent set of object ranking meets irreducible element is defined in $(O,M,R_n)$.

Similarly, the consistent set of Object Ranking Join Irreducible (ORJI) and Object ranking Granular (ORG) element can defined in $\left(O,M,R_n\right)$.

Example 2.

FromTable 2: Consider a context $(O,M,V_a,R_n)$  where $O=\{\mathrm{1,2},\mathrm{3,4}\}$  and $M=\{a,b,c,d,e,f,g\}$  where as $(p_{\alpha },{f_{\alpha }}_i)=1$  if $p_{\alpha }{R}_n{f_{\alpha }}_i,(p_{\alpha },{f_{\alpha }}_i)=? if p_{\alpha }{R}_n{f_{\alpha }}_i$  and $\left(p_{\alpha },{f_{\alpha }}_i\right)=0 if p_{\alpha }\neg R_n{f_{\alpha }}_i\forall p_{\alpha }\in O,{f_{\alpha }}_i\in M$.

Table 2. Information Table.

U	a	b	c	d	e	f	g
1	1	1	0	1	1	0	1
2	?	?	1	0	0	?	?
3	0	0	0	1	0	0	?
4	1	1	1	0	0	1	0

Object Rank: $Rank\left(1\right)=5, Rank\left(2\right)=5, Rank\left(3\right)=2, Rank\left(4\right)=4 \mathrm{Then} O_1=\left\{3\right\},O_2=\left\{4\right\},O_3=\left\{\mathrm{1,2}\right\}$.

Then $L\left[O_0\right]=\left(\mathrm{\varnothing },\left[\mathrm{\varnothing },\mathrm{\varnothing }\right]\right),L\left[O_1\right]=\left(3,\left[d,dg\right]\right),L\left[O_2\right]=\left(4,\left[abcf,abcf\right]\right),L\left[O_3\right]=(1,[abdeg,abdeg\left]\right),(2,[c,abcfg\left]\right) and L\left(O_4\right)=(O,[\underset{\_}{B},\overline{B}\left]\right)$.

Hence, this generates the set $\left(\mathrm{\varnothing },\left[\mathrm{\varnothing },\mathrm{\varnothing }\right]\right),\left(123,\left[\mathrm{\varnothing },g\right]\right),\left(12,\left[\mathrm{\varnothing },abg\right]\right),\left(4,\left[abcf,abcf\right]\right),\left(3,\left[d,dg\right]\right),\left(2,\left[c,abcfg\right]\right),$

$$(1,[abdeg,abdeg\left]\right) and \left(O,\left[\underset{\_}{S_{\alpha }},{\overline{S}}_{\alpha }\right]\right)$$

and Figure 2 illustrates the Hasse diagram.

Consider a reduct ${S_{\alpha }}_1=\{a,d,f,g\}$ whose corresponding concept lattice are $L(O,M,{R_n}_{S_{\alpha }})$. Then $L\left[O_0\right]=(\mathrm{\varnothing },[\mathrm{\varnothing },\mathrm{\varnothing }\left]\right),L\left[O_1\right]=(3,[d,dg\left]\right),(4,[af,af\left]\right),L\left[O_2\right]=(1,[adg,adg\left]\right),(2,[\mathrm{\varnothing },afg\left]\right)\mathrm{and}L\left(O_3\right)=(O,[\underset{\_}{B},\overline{B}\left]\right)$.

Figure 2. Illustrates the Hasse diagram From Table 2.

Figure 2. Illustrates the Hasse diagram From Table 2.

Hence this generates the set $\left(\mathrm{\varnothing },\left[\mathrm{\varnothing },\mathrm{\varnothing }\right]\right),\left(123,\left[\mathrm{\varnothing },g\right]\right),\left(12,\left[\mathrm{\varnothing },ag\right]\right),\left(4,\left[af,af\right]\right),\left(3,\left[d,dg\right]\right),\left(2,\left[\varnothing ,afg\right]\right),\left(1,\left[adg,adg\right]\right) and (O,[\underset{\_}{{S_{\alpha }}_1},\overline{{S_{\alpha }}_1}\left]\right)$ and Figure 3 expressions the hasse diagram.

Figure 3. Reduct ${S_{\alpha }}_1$.

Consider a reduct ${S_{\alpha }}_2=\{b,c,d,g\}$ whose corresponding concept lattice is $L(O,M,{R_n}_{S_{\alpha }})$. Then $L\left[O_0\right]=(\mathrm{\varnothing },[\mathrm{\varnothing },\mathrm{\varnothing }\left]\right),L\left[O_1\right]=(3,[d,dg\left]\right),(4,[bc,bc\left]\right),L\left[O_2\right]=(1,[bdg,bdg\left]\right),(2,[c,bg\left]\right) \mathrm{and} L\left(O_3\right)=(O,[\underset{\_}{B},\overline{B}\left]\right)$.

Hence, this generates the set $\left(\mathrm{\varnothing },\left[\mathrm{\varnothing },\mathrm{\varnothing }\right]\right),\left(123,\left[\mathrm{\varnothing },g\right]\right),\left(12,\left[\mathrm{\varnothing },bg\right]\right),\left(4,\left[bc,bc\right]\right),\left(3,\left[d,dg\right]\right),$

$$\left(2,\left[c,bcg\right]\right),\left(1,\left[bdg,bdg\right]\right) and (O,[\underset{\_}{{S_{\alpha }}_2},\overline{{S_{\alpha }}_2}\left]\right)$$

and Figure 4 expressions the hasse diagram.

Figure 4. Reduct ${S_{\alpha }}_2$.

Consider a reduct ${S_{\alpha }}_3=\{d,f,g\}$ whose corresponding concept lattice is $L\left(O,M,{R_n}_{S_{\alpha }}\right). \mathrm{Then} L\left[O_0\right]=(\mathrm{\varnothing },[\mathrm{\varnothing },\mathrm{\varnothing }\left]\right),L\left[O_1\right]=(4,[f,f\left]\right),L\left[O_2\right]=(1,[dg,dg\left]\right),(2,[\mathrm{\varnothing },fg\left]\right),(3,[d,dg\left]\right), \mathrm{and} L\left(O_3\right)=(O,[\underset{\_}{B},\overline{B}\left]\right)$.

Hence, this generates the set $\left(\mathrm{\varnothing },\left[\mathrm{\varnothing },\mathrm{\varnothing }\right]\right),\left(123,\left[\mathrm{\varnothing },g\right]\right),\left(4,\left[f,f\right]\right),\left(3,\left[d,dg\right]\right),\left(2,\left[\mathrm{\varnothing },fg\right]\right),$

$$(1,[dg,dg\left]\right) and (O,[\underset{\_}{{S_{\alpha }}_3},\overline{{S_{\alpha }}_3}\left]\right)$$

and Figure 5 expressions the hasse diagram.

Figure 5. Reduct ${S_{\alpha }}_3$.

From Table 2 and Figure 4, the reduction ${S{}_{\alpha }}_1=\{a,d,f,g\}$ is an $ORL$ SE-ISI concept lattice which has satisfied the isomorphic to the concept lattice. Hence, from Figure 5 the reduction ${S{}_{\alpha }}_2=\{b,c,d,g\}$ is a $ORMI$ SE-ISI concept lattice has which maintains the extents of meet irreduciable lattice and from Figure 6 the reduction ${S{}_{\alpha }}_3=\{d,f,g\}$ is an $ORG$ SE-ISI concept lattice which has conserved the extents of join irreducible and granular lattice.

Figure 6. From Table $S_{\alpha }$.

Hence, from the above study we can extend the theorems of a consistent set of object ranking lattice and its equivalent relation to meet (join) irreducible set.

Theorem 3.

Let $(O,M,V_a,R_n)$  be a SE-ISI formal context then the consistency set of object ranking lattice is equal to the consistency set of object ranking meet irreducible set.

Proof. 

Let $S_{\alpha }\in ORL$  then by the Definition 12 $ex{t}_{ORL}(O,M,R_n)=ex{t}_{ORL}(O,M,{R_n}_{s_{\alpha }})$, hence, for any $P_{\alpha }\in ex{t}_{ORMI}(O,M,R_n)$, by Definition 13, $P_{\alpha }\ne Q_{\alpha }\vee R_{\alpha }$, where $Q_{\alpha }\in (p_{\alpha },g_{\alpha },1)$  and $R\in (p_{\alpha },b_{\alpha },?)$  which implies $P_{\alpha }\ne Q_{\alpha }$  and $P_{\alpha }\ne R_{\alpha }$, since $ex{t}_{ORL}(O,M,R_n)=ex{t}_{ORML}(O,M,R_{n_{s_{\alpha }}})$  obtained that $P_{\alpha }\in ex{t}_{ORL}(O,M,R_{n_{s_{\alpha }}})\Rightarrow Q_{\alpha },R_{\alpha }\in ex{t}_{ORL}(O,M,R_{n_s})$. Thus $ex{t}_{ORMI}(O,M,R_n)=ex{t}_{ORMI}(O,M,R_{n_{s_{\alpha }}})$. Similarly, it can prove the remining reductions. Therefore, $ex{t}_{ORMI}(O,M,R_n)=ex{t}_{ORMI}(O,M,{R_n}_s)$  which implies $S_{\alpha }\in ORMI$. Thus $ORL\subseteq ORMI$.

Suppose $S_{\alpha }\in ORMI$, since $S_{\alpha }\subseteq M$  then $ex{t}_{ORL}(O,M,R_{n_{s_{\alpha }}})=ex{t}_{ORL}(O,M,R_n)$, For any $P_{\alpha }\in ex{t}_{ORL}(O,M,R_n)$  then there exist $P_{{\alpha }_i}\in ex{t}_{ORL}(O,M,R_n) \forall i\in I$ such that $P_{\alpha }={\displaystyle \underset{i}{\cap }{P}_{{\alpha }_i}}$. By Definition 7, Thus $ex{t}_{ORMI}(O,M,R_n)\subseteq ex{t}_{ORMI}(O,M,R_{n_{s_{\alpha }}})$. Since $P_{\alpha }={\displaystyle \underset{i}{\cap }{P}_{{\alpha }_i}}$  then $ex{t}_{ORL}(O,M,R_n)\subseteq ex{t}_{ORL}(O,M,R_{n_{s_{\alpha }}})$  which implies $S_{\alpha }\in ORL$. Hence $ORML\subseteq ORL$. Therefore, $ORL=ORMI$. □

Corollary 1.

Let $(O,M,V_a,R_n)$  be a SE-ISI formal context then the consistency set of an object ranking lattice equal to the consistency set of object ranking join irreducible set.

Theorem 4.

Let $(O,M,V_a,R_n)$  be a SE-ISI formal context and $P_{\alpha }\subseteq O, B_{\alpha }\subseteq M$  then $\underset{¯}{B_{\alpha }}, \overline{B_{\alpha }}\subseteq M$  then $(P_{\alpha },[\underset{¯}{B_{\alpha }},\overline{B_{\alpha }}])$  is a join irreducible element $\left(ORJI\right)$  which implies $(P_{\alpha },[\underset{¯}{B_{\alpha }},\overline{B_{\alpha }}])$  is a object ranking lattice $\left(ORL\right)$.

Proof. 

Let $(P_{\alpha },[\underset{¯}{B_{\alpha }},\overline{B_{\alpha }}]) \in ex{t}_{ORL}(O,M,R_n)$  then ${\displaystyle \underset{p_{{\alpha }_i}\in P_{\alpha }}{\cup }(p_{{\alpha }_i}{}^{\oplus \odot },}{p}_{{\alpha }_i}{}^{\oplus })$.

By Properties 1 and Definition 2,

$\begin{array}{l}({\displaystyle \underset{p_{{\alpha }_i}\in P_{\alpha }}{\cup }{p}_{{\alpha }_i}{}^{\oplus \odot }{)}^{\oplus \odot }={({\displaystyle \underset{p_{{\alpha }_i}\in P_{\alpha }}{\cap }{p}_{{\alpha }_i}{}^{\oplus \odot }})}^{\oplus \odot }=}{\displaystyle \underset{p_{{\alpha }_i}\in P_{\alpha }}{\cap }{p}_{{\alpha }_i}}=B_{\alpha }\\ which implies {\displaystyle \underset{p_{{\alpha }_i}\in P_{\alpha }}{\cap }{p}_{{\alpha }_i}}=\underset{¯}{B_{\alpha }}\subseteq B_{\alpha }\subseteq \overline{B_{\alpha }}\\ Thus (P_{\alpha },[\underset{¯}{B_{\alpha }},\overline{B_{\alpha }}])={\displaystyle \underset{p_{{\alpha }_i}\in P_{\alpha }}{\cup }(}{p}_{{\alpha }_i}{}^{\oplus \odot },p_{{\alpha }_i}{}^{\oplus })\\ Therefore (P_{\alpha },(p_{{\alpha }_i}{}^{\oplus \odot },p_{{\alpha }_i}{}^{\oplus }))=(Q_{\alpha },(p_{{\alpha }_j}{}^{\oplus \odot },p_{{\alpha }_j}{}^{\oplus }))\vee (R_{\alpha },(p_{{\alpha }_k}{}^{\oplus \odot },p_{{\alpha }_k}{}^{\oplus }))\\ By properties 1 and Definition 2 ,\\ =((Q_{\alpha }{\displaystyle \cup R_{\alpha }{)}^{\oplus \odot }},(p_{{\alpha }_i}{}^{\oplus \odot },p_{{\alpha }_j}{}^{\oplus }){\displaystyle \cap (p_{{\alpha }_k}{}^{\oplus \odot },p_{{\alpha }_k}{}^{\oplus })}) \forall p_{{\alpha }_i},p_{{\alpha }_j},p_{{\alpha }_k}\in P_{\alpha }\\ = ((Q_{\alpha }{\displaystyle \cup R_{\alpha }{)}^{\oplus \odot }},(p_{{\alpha }_i}{}^{\oplus \odot }{\displaystyle \cap p_{{\alpha }_k}{}^{\oplus \odot }},p_{{\alpha }_j}{}^{\oplus }{\displaystyle \cap p_{{\alpha }_k}{}^{\oplus }}))\end{array}$

Here $(Q_{\alpha },(p_{{\alpha }_j}{}^{\oplus \odot },p_{{\alpha }_j}{}^{\oplus })) and (R_{\alpha },(p_{{\alpha }_k}{}^{\oplus \odot },p_{{\alpha }_k}{}^{\oplus }))$  are the element of join irreducible in object ranking lattice. Hence, By the parent and child concept, it is easy to define the join irreducible element of $ORL(O,M,R_n)$.

Therefore, $P_{\alpha }\subseteq M,$  the joint irreducible element where extent to the set using the object ranking lattice. □

5. Observational Study

This section, we analyze the real-life application of attribute reduction with an example. The Table 3 represents the database consists of 6 patients in an ICU ward.

Table 3. Information Table.

Patients	BP	ECG	Pluse	Condition
1	Low	Good	Medium	Excellent
2	High	Good	High	Good
3	Low	Good	High	Good
4	Low	Better	Medium	Good
5	High	Good	High	?
6	High	Better	Low	Good

Work Rule for Object Ranking

Step 1: Consider incomplete information Table 3.

Step 2: Examine the indiscernibility relation.

Example 3.

The Table 4 represent, BP data as $\{Low,High\}$  denoted as $\{l_1,l_2\}$ respectively. If the temperature $low$ is related to $l_1$ then the value refers as $1$ else the value of $l_1$ refer as $0$. similarly, ECG data as  $\{Good,Better\}$ as $\{m_1,m_2\}$ respectively, pulse data as $\{High,Medium,Low\}$ as $\{n_1,n_2,n_3\}$ respectively, condition data as $\{Excellent,Good,?\}$ as $\{o_1,o_2,o_3\}$ respectively.

Step 3: Finding the rank of each object and also find the generating set. Eg: rank of object 1:4, object 2:4, object 3:4, object 4:4, object 5:6, object 6:4. Also find the set $O_1=\{\mathrm{1,2},\mathrm{3,4},6\}$  and $O_2=\left\{5\right\}$.

Step 4: From the generating set construct the hasse diagram. Eg: $\{L_0,L_1....L_{21}\}$.

Step 5: Determine the reduction set $\{{S_{\alpha }}_1,{S_{\alpha }}_2,{S_{\alpha }}_3\}$ and construct the hasse diagram.

Step 6: From the construction, $ORL,ORMI,ORJI,ORG$ are analyzed.

In this example, we have analyzed on 6 objects using the formal concepts of SE-ISI context and its structure. Furthermore, a large number of objects can be investigated by using an algorithm. This process can be evaluated for ‘n’ number of object using the object ranking efficiencies toward the consistency set to determine with meet and join irreducible element.

Table 4. Indiscernibility Table.

	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{m}}_1$	${\mathit{m}}_2$	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	${\mathit{o}}_1$	${\mathit{o}}_2$	${\mathit{o}}_3$
1	0	1	1	0	0	1	0	0	0	1
2	1	0	1	0	1	0	0	0	1	0
3	0	1	1	0	1	0	0	0	1	0
4	0	1	0	1	0	1	0	0	1	0
5	1	0	1	0	1	0	0	?	?	?
6	1	0	0	1	0	0	1	0	1	0

Table 4. Indiscernibility Table.

	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{m}}_1$	${\mathit{m}}_2$	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	${\mathit{o}}_1$	${\mathit{o}}_2$	${\mathit{o}}_3$
1	0	1	1	0	0	1	0	0	0	1
2	1	0	1	0	1	0	0	0	1	0
3	0	1	1	0	1	0	0	0	1	0
4	0	1	0	1	0	1	0	0	1	0
5	1	0	1	0	1	0	0	?	?	?
6	1	0	0	1	0	0	1	0	1	0

Example 4.

From the above Table 3 we generate Table 4 by an indiscernibility relation, using SE-ISI formal context. Also, we analyze an incomplete information context based on the reduction of attribute. From Table 5, the SE-ISI context was implement for defining the structure as Figure 6. Hence, we investigate $ORL,ORMI,ORJI,and ORG$ using the structural features.

Table 5. $S$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_2\left]\right)$
$L_1$	$(1235,[m_1,m_1\left]\right)$	$L_{12}$	$(13,[l_2{m}_1,l_2{m}_1\left]\right)$
$L_2$	$(235,[m_1{n}_1,m_1{n}_1{o}_2\left]\right)$	$L_{13}$	$(25,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[m_1,m_1{o}_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,\left[l_1{m}_2{n}_3{o}_2,l_1{m}_2{n}_3{o}_2\right])$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{n}_2{o}_2,l_2{m}_2{n}_2{o}_2\left]\right)$
$L_6$	$(256,[l_1,l_1\left]\right)$	$L_{17}$	$(3,[l_2{m}_1{n}_1{o}_2,l_2{m}_1{n}_1{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{m}_1{n}_1{o}_2,l_1{m}_1{n}_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{m}_1{n}_2{o}_3,l_2{m}_1{n}_2{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2{n}_2,l_2{n}_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{\alpha }},\overline{S_{\alpha }}\left]\right)$

Table 5. $S$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_2\left]\right)$
$L_1$	$(1235,[m_1,m_1\left]\right)$	$L_{12}$	$(13,[l_2{m}_1,l_2{m}_1\left]\right)$
$L_2$	$(235,[m_1{n}_1,m_1{n}_1{o}_2\left]\right)$	$L_{13}$	$(25,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[m_1,m_1{o}_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,\left[l_1{m}_2{n}_3{o}_2,l_1{m}_2{n}_3{o}_2\right])$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{n}_2{o}_2,l_2{m}_2{n}_2{o}_2\left]\right)$
$L_6$	$(256,[l_1,l_1\left]\right)$	$L_{17}$	$(3,[l_2{m}_1{n}_1{o}_2,l_2{m}_1{n}_1{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{m}_1{n}_1{o}_2,l_1{m}_1{n}_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{m}_1{n}_2{o}_3,l_2{m}_1{n}_2{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2{n}_2,l_2{n}_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{\alpha }},\overline{S_{\alpha }}\left]\right)$

Example 5.

Consider the reduct$S_1=\left\{l_1,l_2,m_1,m_2,n_2,o_2,o_3\right\}$ we obtain Table 6 and Figure 7, $S_2=\left\{l_1,l_2,m_2,n_2,o_2,o_3\right\}$ we obtain Table 7 and Figure 8, $S_3=\left\{l_1,l_2,m_1,m_2,o_2,o_3\right\}$ we obtain Table 8 and Figure 9. Hence, the study how its preserves the construction about the incomplete context.

Table 6. $S_1$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_1$	$(1235,[m_1,m_1\left]\right)$	$L_{12}$	$(13,[l_2{m}_1,l_2{m}_1\left]\right)$
$L_2$	$(235,[m_1,m_1{o}_2\left]\right)$	$L_{13}$	$(25,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[m_1,m_1{o}_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,[l_1{m}_2{o}_2,l_1{m}_2{o}_2\left]\right)$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{n}_2{o}_2,l_2{m}_2{n}_2{o}_2\left]\right)$
$L_6$	$(46,[m_2{o}_2,m_2{o}_2\left]\right)$	$L_{17}$	$(3,[l_2{m}_1{o}_2,l_2{m}_1{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{m}_1{o}_2,l_1{m}_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{m}_1{n}_2{o}_3,l_2{m}_1{n}_2{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1{m}_1,l_1{m}_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2{n}_2,l_2{n}_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{{\alpha }_1}},\overline{S_{{\alpha }_1}}\left]\right)$

Table 6. $S_1$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_1$	$(1235,[m_1,m_1\left]\right)$	$L_{12}$	$(13,[l_2{m}_1,l_2{m}_1\left]\right)$
$L_2$	$(235,[m_1,m_1{o}_2\left]\right)$	$L_{13}$	$(25,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[m_1,m_1{o}_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,[l_1{m}_2{o}_2,l_1{m}_2{o}_2\left]\right)$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{n}_2{o}_2,l_2{m}_2{n}_2{o}_2\left]\right)$
$L_6$	$(46,[m_2{o}_2,m_2{o}_2\left]\right)$	$L_{17}$	$(3,[l_2{m}_1{o}_2,l_2{m}_1{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{m}_1{o}_2,l_1{m}_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{m}_1{n}_2{o}_3,l_2{m}_1{n}_2{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1{m}_1,l_1{m}_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2{n}_2,l_2{n}_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{{\alpha }_1}},\overline{S_{{\alpha }_1}}\left]\right)$

Hence, from the Figure 7 it concludes that the SE-ISI concept lattice is isomorphic and also classifies the patients by a defined structural form from the concepts of SE-ISI context lattice that was unchanged. Hence, we can come to the conclusion that the number of attributes is purely based on $m_1$ and $O_2$.

Figure 7. From Table ${S_{\alpha }}_1$.

Figure 7. From Table ${S_{\alpha }}_1$.

Table 7. $S_2$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_2\left]\right)$
		$L_{12}$	$(13,[l_2,l_2\left]\right)$
$L_2$	$(235,[\varnothing ,o_2\left]\right)$	$L_{13}$	$(25,[l_1,l_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[\varnothing ,o_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,[l_1{m}_2{o}_2,l_1{m}_2{o}_2\left]\right)$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{n}_2{o}_2,l_2{m}_2{n}_2{o}_2\left]\right)$
$L_6$	$(46,[m_2{o}_2,m_2{o}_2\left]\right)$	$L_{17}$	$(3,[l_2{o}_2,l_2{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{o}_2,l_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{n}_2{o}_3,l_2{n}_2{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1,l_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2{n}_2,l_2{n}_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{{\alpha }_2}},\overline{S_{{\alpha }_2}}\left]\right)$

Table 7. $S_2$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1{n}_1,l_1{m}_1{n}_1{o}_2\left]\right)$
		$L_{12}$	$(13,[l_2,l_2\left]\right)$
$L_2$	$(235,[\varnothing ,o_2\left]\right)$	$L_{13}$	$(25,[l_1,l_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[\varnothing ,o_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,[l_1{m}_2{o}_2,l_1{m}_2{o}_2\left]\right)$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{n}_2{o}_2,l_2{m}_2{n}_2{o}_2\left]\right)$
$L_6$	$(46,[m_2{o}_2,m_2{o}_2\left]\right)$	$L_{17}$	$(3,[l_2{o}_2,l_2{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{o}_2,l_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{n}_2{o}_3,l_2{n}_2{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1,l_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2{n}_2,l_2{n}_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{{\alpha }_2}},\overline{S_{{\alpha }_2}}\left]\right)$

Hence from Figure 8, ORJI reduction cannot have the defined structure form of concept of SE-ISI. Therefore, from Figure 8, $m_1$ has been eliminated after joining irreducible reduction. Thus, from the basic construction of the join irreducible elements, ORJI reduction is essential in SE-ISI lattice.

Figure 8. From Table ${S_{\alpha }}_2$.

Figure 8. From Table ${S_{\alpha }}_2$.

Table 8. $S_3$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_1$	$(1235,[m_1,m_1\left]\right)$	$L_{12}$	$(13,[l_2{m}_1,l_2{m}_1\left]\right)$
$L_2$	$(235,[m_1,m_1{o}_2\left]\right)$	$L_{13}$	$(25,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[m_1,m_1{o}_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,[l_1{m}_2{o}_2,l_1{m}_2{o}_2\left]\right)$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{o}_2,l_2{m}_2{o}_2\left]\right)$
$L_6$	$(46,[m\_2o\_2,m\_2o\_2\left]\right)$	$L_{17}$	$(3,[l_2{m}_1{o}_2,l_2{m}_1{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{m}_1{o}_2,l_1{m}_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{m}_1{o}_3,l_2{m}_1{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1{m}_1,l_1{m}_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2,l_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{{\alpha }_3}},\overline{S_{{\alpha }_3}}\left]\right)$

Table 8. $S_3$.

Lattice	Concepts	Lattice	Concept
$L_0$	$(\varnothing ,[\varnothing ,\varnothing \left]\right)$	$L_{11}$	$(35,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_1$	$(1235,[m_1,m_1\left]\right)$	$L_{12}$	$(13,[l_2{m}_1,l_2{m}_1\left]\right)$
$L_2$	$(235,[m_1,m_1{o}_2\left]\right)$	$L_{13}$	$(25,[l_1{m}_1,l_1{m}_1{o}_2\left]\right)$
$L_3$	$(134,[l_2,l_2\left]\right)$	$L_{14}$	$(15,[m_1,m_1{o}_3\left]\right)$
$L_4$	$(2346,[o_2,o_2\left]\right)$	$L_{15}$	$(6,[l_1{m}_2{o}_2,l_1{m}_2{o}_2\left]\right)$
$L_5$	$(256,[l_1,l_1\left]\right)$	$L_{16}$	$(4,[l_2{m}_2{o}_2,l_2{m}_2{o}_2\left]\right)$
$L_6$	$(46,[m\_2o\_2,m\_2o\_2\left]\right)$	$L_{17}$	$(3,[l_2{m}_1{o}_2,l_2{m}_1{o}_2\left]\right)$
$L_7$	$(65,[l_1,l_1{o}_2\left]\right)$	$L_{18}$	$(2,[l_1{m}_1{o}_2,l_1{m}_1{o}_2\left]\right)$
$L_8$	$(34,[l_2{o}_2,l_2{o}_2\left]\right)$	$L_{19}$	$(1,[l_2{m}_1{o}_3,l_2{m}_1{o}_3\left]\right)$
$L_9$	$(26,[l_1{o}_2,l_1{o}_2\left]\right)$	$L_{20}$	$(5,[l_1{m}_1,l_1{m}_1{o}_2{o}_3\left]\right)$
$L_{10}$	$(14,[l_2,l_2\left]\right)$	$L_{21}$	$(O,[\underset{\_}{S_{{\alpha }_3}},\overline{S_{{\alpha }_3}}\left]\right)$

From Figure 9 shows that ORG reduct cannot be constructed for the SE-ISI context. Thus, SE-ISI context was unchanged. Hence, we determined the ORG reduction lattice with the loss of the information context.

Figure 9. From Table $S_{{\alpha }_3}$.

Figure 9. From Table $S_{{\alpha }_3}$.

From the above real-time experimental study concludes that the various reduction of attributes with an incomplete information context have been analyzed without any loss of knowledge. The construction of the object ranking lattice has the same elements when compared with $ORMI,ORJI,\mathrm{and} ORG$ reduction. Hence, the attribute BP and ECG, which are essential for the diagonalization of the patient condition to shift to the general ward. In addition, it can be verified with pulse.

6. Conclusions

Formal concept analysis is a powerful tool for data mining and knowledge discovery. In this paper, the reduction of the attributes is investigated alongside the methodology of object ranking concept to determine from the SE-ISI concept. The four features of the reduction of attributes and their relationship were analyzed with the existence of a consistency set. The basic knowledge of the formal concept has been analyzed by defining the three-way concept with an enhancement of object ranking concepts. The existing attribute reduction is obtained from the incomplete context. The consistency set determined with join (meet) irreducible set towards the object ranking concepts. A real-time example is described in this paper for various reductions of attributes and a structural study on four features of the consistency set.