Fuzzy subalgebra

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.

Definition

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 h, the axioms

$\forall x_1, ..., \forall x_n (S(x_1) \land ..... \land S(x_n) \rightarrow S(h(x_1, ..., x_n))$

and, for any constant c, S(c).

The first axiom expresses the closure of S with respect to the operation h, and the second expresses the fact that c is an element in S. As an example, assume that the valuation structure is defined in ,1and denote by $\odot$ the operation in ,1used to interpret the conjunction. Then a fuzzy subalgebra of an algebraic structure whose domain is D is defined by a fuzzy subset of D such that, for every d₁,...,d_n in D, if h is the interpretation of the n-ary operation symbol h, then

* $s(d_1) \odot... \odot s(d_n) \leq s(\mathbf(d_1,...,d_n))$

Moreover, if c is the interpretation of a constant c such that 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 ,1 the closed cut of s is a subalgebra.

Fuzzy subgroups and submonoids

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

# $s(\mathbf)=1$
# $s(x) \odot s(y) \leq s(x \cdot y)$

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
* 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

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.

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

Bibliography

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Fuzzy logic