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
∀ 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
s ( u ) = 1 {\displaystyle s(\mathbf {u} )=1}
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.
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