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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

∀ x 1 , . . . , ∀ x n ( S ( x 1 ) ∧ . . . . . ∧ S ( x n ) → S ( h ( x 1 , . . . , x n ) ) {\displaystyle \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 ⊙ {\displaystyle \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 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

• s ( d 1 ) ⊙ . . . ⊙ s ( d n ) ≤ s ( h ( d 1 , . . . , d n ) ) {\displaystyle 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 ⊙ {\displaystyle \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 ( u ) = 1 {\displaystyle s(\mathbf {u} )=1}
2. s ( x ) ⊙ s ( y ) ≤ s ( x ⋅ y ) {\displaystyle 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.

Bibliography

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