Fuzzy Equational Logic (Studies in Fuzziness and Soft by Radim Belohlávek, Vilem Vychodil

By Radim Belohlávek, Vilem Vychodil

The e-book bargains with similarity kin outlined on a collection with features. The capabilities are required to map related components to comparable ones. The ebook provides simple mathematical homes of constructions along with similarity-preserving features and logics for reasoning approximately similarities. The provided textual content is self-contained. The notions and effects are demonstrated through examples that are graphically illustrated. The e-book turns out to be useful for researchers, however it is usually used as a graduate textual content.

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Therefore, if a ⊗ c ≤ b then c ≤ a → b. If c ≤ a → b then, by isotony of ⊗ and by the property of a → b, a ⊗ c ≤ a ⊗ (a → b) ≤ b. e. 48) holds true. 48). If a⊗c ≤ b then clearly c ≤ a → b (a → b is the supremum of such c’s). If c ≤ a → b then a ⊗ c ≤ a ⊗ (a → b) = a ⊗ {d | a ⊗ d ≤ b} = {a ⊗ d | a ⊗ d ≤ b} ≤ b. 48) is true. In what follows, we introduce a particular construction of complete residuated lattices. The construction is called an ordinal sum and will be used later. 23. Suppose I, ≤ is a chain with the least element 0 and the greatest element 1.

Let f be both non-decreasing and left-continuous in x. 52). For a = {aj | j ∈ J}, there are two possibilities. Either a = ({aj | j ∈ J} − {a}) or a > ({aj | j ∈ J} − {a}). If a = ({aj | j ∈ J} − {a}) then for each n ∈ N there exists some aj(n) ∈ {aj | j ∈ J} − {a} such that a − aj(n) < n1 . Clearly, we may safely assume aj(n) ≤ aj(n+1) . Then we have limn→∞ aj(n) = a and aj(n) < a (n ∈ N). A moment’s reflection shows that f( j∈J aj , b) = f ( lim aj(n) , b) = lim f (aj(n) , b) = n→∞ = n→∞ n∈N f (aj(n) , b) ≤ j∈J f (aj , b) by the definition of left-continuity in x.

That is, A ⊆ B means that for each u ∈ U we have A(u ) ≤ B(u ). Common operations with L-sets which generalize the ordinary operations with sets result by componentwise extension of operations on L. That is, any operation o (possibly infinitary) on L induces in a componentwise manner an operation O on LU by putting O(A, . . )(u ) = o(A(u ), . . ) for arbitrary L-sets A, . . and arbitrary u ∈ U . Arguments of O are L-sets in U and the result of O applied to A, . . is an L-set O(A, . . ) in U to which an element u ∈ U belongs to a degree obtained by applying o to A(u ), .

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