By Radim Belohlávek, Vilem Vychodil

The publication offers with similarity kinfolk outlined on a suite with capabilities. The capabilities are required to map related components to related ones. The booklet offers simple mathematical houses of constructions along with similarity-preserving services and logics for reasoning approximately similarities. The offered textual content is self-contained. The notions and effects are confirmed via examples that are graphically illustrated. The ebook turns out to be useful for researchers, however it is additionally used as a graduate text.

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**Sample text**

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.

Similarity of objects is often related to a collection of attributes of these objects. Two objects are considered similar if they are similar according to having some attributes. The following assertion says that fuzzy equivalences are just fuzzy relations which result by the criterion of having the same attributes. 76 (Leibniz equivalence). 80) is an L-equivalence in U . For each L-equivalence θ in U , there is some S ⊆ LU such that θ = θS . Moreover, θS is an L-equality iff for every distinct u and v there is some A ∈ S such that A(u ) = A(v ).

Indeed, for np ≤ nq , cp ⊗ (a → b)np ⊗ cq ⊗ anq ≤ cp ⊗ (a → b)np ⊗ cq ⊗ anp ≤ ≤ cp ⊗ cq ⊗ anp ⊗ (a → b)np ≤ cp ⊗ cq ⊗ bnp ≤ cp ⊗ bnp ≤ k i=1 ci ⊗ bni , and analogously for np > nq . 50 (implicational truth stressers). Each implicational truth stresser on a complete residuated lattice satisﬁes ∗ (a ⊗ b) = a∗ ⊗ b∗ . 58) ∗ Proof. 57), it is enough to see (a ⊗ b) ≤ a∗ ⊗b∗ . 18) and monotony of ∗ yield (a ⊗ b) = (a ⊗ b) ⊗ ∗ (a ⊗ b) ≤ a∗ ⊗ b∗ . 51. 58) one can ask under what conditions L∗ satisﬁes ∗ (a → b) = a∗ → b∗ .