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A1 ∀x1..., ∀xn(S(x1)∧.....∧ S(xn) → S(h(x1,...,xn)) and, for any constant c, the axiom A2 S(c). A1 expresses the closure of S with respect to the operation h, A2 expresses the fact that c is an element in S. As an example, assume that the valuation structure is defined in and denote by the operation in [0,1 used to interpret the conjunction. Then it is easy to see that a fuzzy subalgebra of an algebraic structure whose domain is D is defined by a fuzzy subset s : D → [0,1] of D such that, for every d1,...,dn in D, if h is the interpretation of the n-ary operation symbol h, then i) s(d1)s(dn)≤ s(h(d1,...,dn)) Moreover, if c is the interpretation of a constant c ii) s(c) = 1. A largely studied class of fuzzy subalgebras is the one in which the operation coincides with the minimum. In such a case it is immediate to prove the following proposition. Proposition. A fuzzy subset s of an algebraic structure defines a fuzzy subalgebra if and only if for every λ in {Link without Title} , the closed cut {x D : s(x)≥ λ} of s is a subalgebra. The ''fuzzy subgroups'' and the ''fuzzy submonoids'' are particularly interesting classes of fuzzy subalgebras. In such a case a fuzzy subset ''s'' of a monoid (M,•,u) is a fuzzy submonoid if and only if 1) s(u) =1 2) s(x)s(y) ≤ s(x•y) where u is the neutral element in A. Given a group G, a ''fuzzy subgroup'' of G is a fuzzy submonoid s of G such that 3) s(x) ≤ s(x-1). It is possible to prove that the notion of fuzzy subgroup is strictly related with the notions of ''fuzzy equivalence''. In fact, assume that S is a set, G a group of transformations in S and (G,s) a fuzzy subgroup of G. Then, by setting e(x,y) = Sup{s(h) : h is an element in G such that h(x) = y} we obtain a fuzzy equivalence. Conversely, let e be a fuzzy equivalence in S and, for every transformation h of S, set s(h)= Inf{e(x,h(x)): xS}. Then s defines a fuzzy subgroup of transformation in S. In a similar way we can relate the fuzzy submonoids with the fuzzy orders. BIBLIOGRAPHY
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