By Michael Winter
Goguen different types expand the relational calculus and its specific formalization to the bushy international. ranging from the basic techniques of units, binary relatives and lattices this booklet introduces a number of specific formulations of an summary concept of family members resembling allegories, Dedekind different types and comparable constructions. it's proven that neither idea is adequately wealthy to explain uncomplicated operations on fuzzy relatives. The e-book then introduces Goguen different types and offers a accomplished examine of those buildings together with their illustration idea, and the definability of norm-based operations.
The strength of the idea is tested by way of a finished instance. a definite Goguen type is used to specify and to strengthen a fuzzy controller. in accordance with its summary description in addition to convinced fascinating homes and their formal proofs, a established controller is derived with out compromising the - occasionally - intuitive number of norm-based operations through fuzzy engineers.
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Extra info for Goguen Categories: A Categorical Approach to L-fuzzy Relations
Example text
LATTICES 33 As mentioned above, a special class of filters is used to show that a distributive lattice may be embedded into a power set. 52 A subset F of a lattice L with least element 0 is called a prime filter of L iff for all x, y ∈ L the following properties hold: (1) 0 ∈ F , (2) x ∧ y ∈ F iff x ∈ F and y ∈ F , (3) x ∨ y ∈ F iff x ∈ F or y ∈ F . 49 it is obvious that any prime filter is a filter. That the other implication is not true is shown by the next lemma. 53 Let L be a distributive lattice and x ∈ L.
Definition of ∧ (4) We immediately conclude that ϑ(f )({F }) = f (x) = {F }⊆ψ(x) f (x) = F ∈ψ(x) f (x). x∈F Notice that (2) as well as (3) of the previous lemma implies that ϑ is monotone. The next lemma will show the key property (ext) introduced above. Notice that in the proof of this lemma it is essential that ψ is continuous. 64 If f : L1 → L2 is antitone, then we have τ (ϑ(τ (f ))) = τ (ϑ(f )). Proof. 63 (2), and, hence, τ (ϑ(f )) ≤ τ (ϑ(τ (f ))). ≥: Consider the property (∗) ϑ(τ (f )) ≤ τ (ϑ(f )).
Notice that in the next lemma we identify Rel with the 0–1 crisp relations from L-Rel. 5 Let f : L → Rel[A, B] be an antimorphism, R : A → B an u (αA ◦ f (u)). Then we have Rv ⊆ f (v) for all L-fuzzy relation with R ⊆ u∈L v ∈ L. Proof. Suppose Rv (x, y) = 1. By definition we have v ≤ ( u∈L u (αA ◦ f (u))) (x, y). Now, let M be the set of all w ∈ L such that f (w)(x, y) = 1 holds. Then we have v≤ u (αA ◦ f (u)) (x, y) u∈L u (αA ◦ f (u))(x, y) = definition of ∪ u∈L u (αA (x, w) ∧ f (u)(w, y)) = definition of ◦ u∈L w∈A (u ∧ f (u)(x, y)) = u definition of αA u∈L = M.