Ranking for Objects and Attribute Reductions in Intuitionistic Fuzzy Ordered Information Systems

Table 3

(1L)	𝑅𝐴^≀≽(𝑋)⊆𝑋	(Contraction)
(1U)	𝑋⊆𝑅𝐴^≀≽(𝑋)	(Extension)
(2)	𝑅𝐴^≀≽(∼𝑋)=∼𝑅𝐴^≀≽(𝑋)	(Duality)
	𝑅𝐴^≀≽(∼𝑋)=∼𝑅𝐴^≀≽(𝑋)	(Duality)
(3L)	𝑅𝐴^≀≽(𝜙)=𝜙	(Normality)
(3U)	𝑅𝐴^≀≽(𝜙)=𝜙	(Normality)
(4L)	𝑅𝐴^≀≽(𝑈)=𝑈	(Co-normality)
(4U)	𝑅𝐴^≀≽(𝑈)=𝑈	(Co-normality)
(5L)	𝑅𝐴^≀≽(𝑋∩𝑌)=𝑅𝐴^≀≽(𝑋)∩𝑅𝐴^≀≽(𝑌)	(Multiplication)
(5U)	𝑅𝐴^≀≽(𝑋∪𝑌)=𝑅𝐴^≀≽(𝑋)∪𝑅𝐴^≀≽(𝑌)	(Addition)
(5L^′)	𝑅𝐴^≀≽(𝑋∪𝑌)⊃𝑅𝐴^≀≽(𝑋)∪𝑅𝐴^≀≽(𝑌)	(F-Multiplication)
(5U^′)	𝑅𝐴^≀≽(𝑋∩𝑌)⊂𝑅𝐴^≀≽(𝑋)∩𝑅𝐴^≀≽(𝑌)	(F-Addition)
(6L)	𝑋⊆𝑌⇒𝑅𝐴^≀≽(𝑋)⊆𝑅𝐴^≀≽(𝑌)	(Monotone)
(6U)	𝑋⊆𝑌⇒𝑅𝐴^≀≽(𝑋)⊆𝑅𝐴^≀≽(𝑌)	(Monotone)
(7L)	𝑅𝐴^≀≽(𝑅𝐴^≀≽(𝑋))=𝑅𝐴^≀≽(𝑋)	(Idempotency)
(7U)	𝑅𝐴^≀≽(𝑅𝐴^≀≽(𝑋))=𝑅𝐴^≀≽(𝑋)	(Idempotency)