Fuzzy subalgebra

 ( Fuzzy Logic ) 

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.

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Definition Consider a first order language for algebraic structures with a Unary operator symbol S. Then a fuzzy subalgebra is a fuzzy model of a theory containing, for any n-ary operation h, the axioms

$\backslash forall\; x\_1,\; ...,\; \backslash forall\; x\_n\; (S(x\_1)\; \backslash land\; .....\; \backslash land\; S(x\_n)\; \backslash rightarrow\; S(h(x\_1,\; ...,\; x\_n))$

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 $\backslash odot$ 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,...,d_n in D, if h is the interpretation of the n-ary operation symbol h, then

• $s(d\_1)\; \backslash odot...\; \backslash odot\; s(d\_n)\; \backslash leq\; s(\backslash mathbf\{h\}(d\_1,...,d\_n))$

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 $\backslash odot$ 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.

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

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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