PDF - Computer Science - Brock University

More documents

Recommendations

Info

with β,γ ≤ α. Wehave (N; N ⌣ )(β,δ) = ⊔ N(β,γ); N ⌣ (γ,δ) γ≤α = ⊔ N(β,γ); N(δ, γ) ⌣ = = = γ≤α ⊔ N(β,γ); N(δ, γ) ⌣ max(β,δ)≤γ≤α ⊔ max(β,δ)≤γ≤α ⊔ max(β,δ)≤γ≤α ⎛ = R β ; ⎝ ⊔ max(β,δ)≤γ≤α R β ; M(β,γ); (R δ ; M(δ, γ)) ⌣ R β ; M(β,γ); M(γ,δ); R ⌣ δ ⎞ M(β,γ); M(γ,δ) ⎠ ; R ⌣ δ Def. of N M symmetric If β ≠ δ, then we obtain ⎛ R β ; ⎝ ⊔ ⎞ M(β,γ); M(γ,δ) ⎠ ; R ⌣ δ max(β,δ)≤γ≤α ⎛ ⎞ ⊑ R β ; ⎝ ⊔ M(β,γ); M(γ,δ) ⎠ ; R ⌣ δ γ≤α = R β ;(M; M)(β,δ); R ⌣ δ = R β ; M(β,δ); R ⌣ δ M idempotent = ⊥ Aβ A δ , where the last equality follows from Lemma 6(1) since we have either β
i.e., we have just shown that N; N ⌣ = I g . In order to verify that N ⌣ ; N = M consider (N ⌣ ; N)(β,δ) = ⊔ N ⌣ (β,γ); N(γ,δ) γ≤α = ⊔ N(γ,β) ⌣ ; N(γ,δ) = = = γ≤α ⊔ γ≤min(β,γ) ⊔ γ≤min(β,γ) ⊔ γ≤min(β,γ) We immediately obtain from Lemma 4(1) N(γ,β) ⌣ ; N(γ,δ) M(β,γ); R ⌣ γ ; R γ ; M(γ,δ) M(β,γ); Ξ γ ; M(γ,δ). Def. N Def. N M(β,γ); Ξ γ ; M(γ,δ) ⊑ M(β,γ); M(γ,γ); M(γ,δ) ⊑ M(β,δ), and, hence, (N ⌣ ; N)(β,δ) ⊑ M(β,δ). For the converse inclusion assume β ≤ δ. Then we have M(β,δ) = M(β,β); M(β,δ) Lemma 4(2) ⎛ ⎞ = ⎝ ⊔ M(β,γ); M(γ,β) ⎠ ; M(β,δ) Lemma 4(3) and (2) for γ = β γ≤β ⎛ ⎞ = ⎝ ⊔ M(β,γ); Ξ γ ; M(γ,β) ⎠ ; M(β,δ) Lemma 6(3) γ≤β = ⊔ M(β,γ); Ξ γ ; M(γ,β); M(β,δ) γ≤β ⊑ ⊔ M(β,γ); Ξ γ ; M(γ,δ). Lemma 4(3) γ≤β The case δ ≤ β is shown analogously. This completes the proof. ⊓⊔ We want to illustrate the previous theorem by an example. In this example we use B-fuzzy relations where B is a Boolean algebra. This is a special case of so-called L-fuzzy relations where L is a Heyting algebra. For further details on these kind of fuzzy relations we refer to [24]. As already mentioned above every Boolean algebra is also a relation algebra where composition is given by the meet operation and converse is the identity. Let B abc be the Boolean algebra with the three atoms a, b, c. We will denote arbitrary elements of B abc by the sequence of atoms below that element, e.g., ab or bc or abc, or 0 for the least
Page 1 and 2: Brock University Department of Comp
Page 3 and 4: standard model one will always obta
Page 5 and 6: 2. F (R) =F (R), 3. F (R ⌣ )=F (R
Page 7 and 8: 3 Matrix Algebras Given a heterogen
Page 9 and 10: The following theorem was shown in
Page 11 and 12: = ⊔ M(β,γ); M(γ,β); M(β,γ);
Page 13: = M(β,δ ′ + 1); M(δ ′ +1,β)
Page 17 and 18: algebra approach. Decision diagrams
Page 19 and 20: the essential part of R in both gra
Page 21: 5. Berghammer, R., Rusinowska, A.,

PDF - Computer Science - Brock University

Create successful ePaper yourself

Delete template?

Save as template?