Fuzzy subalgebra

In fuzzy logic given a first order language and a valuation structure V, we can interpret it by a fuzzy interpretation, i.e. a pair (D,I) such that D is a nonempty set and I, the interpretation function is a map associating any n-ary functor with an n-ary operation and any constant with an element of D (as in the classical case). Moreover, I associates any n-ary predicate name with a suitable n-ary fuzzy relation in D. The fuzzy subalgebras are particular fuzzy interpretations of a first order language for algebraic structures with a monadic predicate symbol S. The intended meaning of S(x) is that x is an element of the subalgebra. More precisely, a fuzzy subalgebra, is a model of a theory containing, for any n-ary operation name h, the axiom

A1 ∀x₁..., ∀x_n(S(x₁)∧.....∧ S(x_n) → S(h(x₁,...,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 d₁,...,d_n in D, if h is the interpretation of the n-ary operation symbol h, then

i) s(d₁)$\odot ...\odot$s(d_n)≤ s(h(d₁,...,d_n))

Moreover, if c is the interpretation of a constant c

ii) s(c) = 1.

In a largely studied class of valuation structures the product $\odot$ coincides with the joint operation. 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(x-1).

The notions of fuzzy submonoid and fuzzy subgroup are strictly related with the notions of fuzzy order and fuzzy equivalence, respectively.

Proposition. Assume that S is a nonempty set, G a group of transformations in S and (G,s) a fuzzy submonoid (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 order (a fuzzy equivalence). Conversely, let e be a fuzzy order (equivalence) in S and define s by setting, for every transformation h of S,

 s(h)= Inf{e(x,h(x)): x$\in$S}. 

Then s defines a fuzzy submonoid (fuzzy subgroup) of transformation in S.

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