Weak Monoidal DB Topos - Big Data Integration Theory - page 486

connection with

= a R ≤

= a ∈ W |∀ b b ∈ S implies (a,b) ∈ R − 1
≤

λ(S) ∈ W |∀ b ∈ S.(b,a) ∈

= a b (a,b) / S

R − 1
≤

|∀ ∈ implies b/ ∈

= a

b (a,b)

K

R − 1

|∀ ∈ ∈

does not imply b φ

≤

= a b (a,b) M |= b φ

R − 1 |∀ ∈ does not imply

≤

={ a | M |= a R − 1

≤ ¬ c φ

}= R − 1

≤ ¬ c φ

K .

Thus, the modal additive algebraic negation λ is represented by a composition
R − 1

≤ ¬ c of the universal modal operator

R − 1
≤

and classic (standard) logical nega-

tion ¬ c . Analogously,

= a R ≤

ρ(S) ∈ W |∀ b ∈ S.(a,b) ∈

= a b b R ≤

∈ W |∀ ∈ S implies (a,b) ∈

= a b (a,b) / S |∀ ∈ R ≤ implies b/ ∈

= a |∀ b (a,b) ∈ R ≤ does not imply b ∈ φ K

= a b (a,b) M |= b φ

|∀ ∈ R ≤ does not imply

={ a | M |= a R ≤ ¬ c φ }

= R ≤ ¬ c φ K .

Thus, the modal multiplicative algebraic negation ρ is represented by a composition
R ≤ ¬ c of the universal modal operator R ≤ and classic (standard) logical negation

¬ c .
Consequently, the closure operator Γ = ρλ is represented by the composition

( R ≤ ¬ c )( R − 1

≤ ¬ c ).

That is, from the fact that ¬ c R − 1

≤ ¬ c is equal to the existential modal operator

R − 1
≤

,

R ≤ R − 1
≤

we obtain that Γ is represented by modal composition .

In fact, it is easy to verify that the Kripke semantics of the closure operator Γ
corresponds to

b ∀ b ,

M |= a R ≤ R − 1
≤

∀ M |= c φ implies c ≤ ≤

φ iff c( b) implies a