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.
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
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
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 s : D → [0,1] of D such that, for every d1,...,dn in D, if h is the interpretation of the n-ary operation symbol h, then 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
In such a case it is immediate to prove the following 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.
In such a case a fuzzy subset s of a monoid (M,•,u) is a fuzzy submonoid if and only if where u is the neutral element in A.
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 we obtain a fuzzy equivalence.