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


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

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 [0,1], the closed cut {x ∈ D : s(x)≥ λ} 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


 * 1) $$s(\mathbf{u})=1$$
 * 2) $$s(x) \odot s(y) \leq s(x \cdot y)$$

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{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∈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.