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Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure. Indeed, consider a first order language for algebraic structures with a monadic predicate symbol S. Then a fuzzy subalgebra, is a fuzzy model of a theory containing, for any n-ary operation name h, the axiom
## Bibliography

A1 ∀x_{1}..., ∀x_{n}(S(x_{1})∧.....∧ S(x_{n}) → S(h(x_{1},...,x_{n}))

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 [0,1] and denote by $odot$ 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 d_{1},...,d_{n} in D, if h is the interpretation of the n-ary operation symbol h, then

i) s(d_{1})$odot...\; odot$s(d_{n})≤ s(h(d_{1},...,d_{n}))

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 $odot$ 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 [0,1], the closed cut {x $in$ 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)$odot$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(xIt 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^{-1}).

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)): x$in$S}.

Then s defines a fuzzy subgroup of transformation in S. In a similar way we can relate the fuzzy submonoids with the fuzzy orders.

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Last updated on Sunday August 19, 2007 at 14:45:02 PDT (GMT -0700)

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