Fuzzy subalgebras theory is a chapter of fuzzy set theory

Fuzzy set

Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to...

. 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
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 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)s(d_n)≤ s(h(d₁,...,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 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 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.