Fuzzy subalgebra: Difference between revisions
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In fuzzy logic given a first order language we can interpret it by a ''[[formal fuzzy logic |fuzzy interpretation]]'', i.e. a pair (D,I) such that D is a nonempty set and I, the ''interpretation function'' is a map associating any n-ary functor with an n-ary operation and any constant with an element of D (as in the classical case). Moreover, | {{subpages}} | ||
In [[fuzzy logic]] given a first order language and a valuation structure ''V'', we can interpret it by a ''[[formal fuzzy logic |fuzzy interpretation]]'', i.e. a pair (D,I) such that D is a nonempty set and I, the ''interpretation function'' is a map associating any n-ary functor with an n-ary operation and any constant with an element of D (as in the classical case). Moreover, I associates any n-ary predicate name with a suitable n-ary fuzzy relation in D. The fuzzy subalgebras are particular fuzzy interpretations of a first order language for algebraic structures with a monadic predicate symbol S. The intended meaning of S(x) is that x is an element of the subalgebra. More precisely, a ''fuzzy subalgebra'', is a model of a theory containing, for any ''n''-ary operation name h, the axiom | |||
A1 | A1 <math>\forall</math>x<sub>1</sub>..., <math>\forall</math>x<sub>n</sub>(S(x<sub>1</sub>)ʌ...ʌS(x<sub>n</sub>) → S(h(x<sub>1</sub>,...,x<sub>n</sub>)) | ||
and, for any constant c, the axiom | and, for any constant c, the axiom | ||
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ii) s('''c''') = 1. | ii) s('''c''') = 1. | ||
In a largely studied class of valuation structures the product <math>\odot</math> coincides with the joint operation. 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 <math>\in</math> D : s(x)≥ λ} of s is a subalgebra. | '''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 <math>\in</math> D : s(x)≥ λ} of s is a subalgebra. | ||
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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 | 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 | ||
3) s(x) ≤ s(x<sup>-1</sup>). | 3) s(x) ≤ s(x<sup>-1</sup>). | ||
The notions of fuzzy submonoid and fuzzy subgroup are strictly related with the notions of '' fuzzy order'' and ''fuzzy equivalence'', respectively. | |||
'''Proposition'''. Assume that S is a nonempty set, G a group of transformations in S and (G,s) a fuzzy submonoid (subgroup) of G. Then, by setting | |||
e(x,y) = Sup{s(h) : h is an element in G such that h(x) = y} | 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, | we obtain a fuzzy order (a fuzzy equivalence). Conversely, let e be a fuzzy order (equivalence) in S and define s by setting, for every transformation h of S, | ||
s(h)= Inf{e(x,h(x)): x<math>\in</math>S}. | s(h)= Inf{e(x,h(x)): x<math>\in</math>S}. | ||
Then s defines a fuzzy subgroup of transformation in S. | Then s defines a fuzzy submonoid (fuzzy subgroup) of transformation in S. | ||
== Bibliography == | == Bibliography == | ||
* [[George Klir|Klir, G.]] and Bo Yuan, ''Fuzzy Sets and Fuzzy Logic'' (1995) ISBN 978-0-13-101171-7 | * [[George Klir|Klir, G.]] and Bo Yuan, ''Fuzzy Sets and Fuzzy Logic'' (1995) ISBN 978-0-13-101171-7 | ||
* Zimmermann H., ''Fuzzy Set Theory and its Applications'' (2001), ISBN 978-0-7923-7435-0. | * Zimmermann H., ''Fuzzy Set Theory and its Applications'' (2001), ISBN 978-0-7923-7435-0. | ||
* Demirci M., Recasens J., ''Fuzzy groups, fuzzy functions and fuzzy equivalence relations'', Fuzzy Sets and Systems, 144 (2004), 441-458. | * Demirci M., Recasens J., ''Fuzzy groups, fuzzy functions and fuzzy equivalence relations'', Fuzzy Sets and Systems, 144 (2004), 441-458. | ||
* Di Nola A., Gerla G., ''Lattice valued algebras'', Stochastica, 11 (1987), 137-150. | * Di Nola A., Gerla G., ''Lattice valued algebras'', Stochastica, 11 (1987), 137-150. | ||
* Hájek P., ''Metamathematics of fuzzy logic''. Kluwer 1998. | * Hájek P., ''Metamathematics of fuzzy logic''. Kluwer 1998. | ||
* Klir G. , UTE H. St.Clair and Bo Yuan ''Fuzzy Set Theory Foundations and Applications'',1997. | * Klir G. , UTE H. St.Clair and Bo Yuan ''Fuzzy Set Theory Foundations and Applications'',1997. | ||
* Gerla G., | * Gerla G., Fuzzy submonoids, fuzzy preorders and quasi-metrics, Fuzzy Sets and Systems 157 (2006) 2356-2370. | ||
*Mordeson J., Kiran R. Bhutani and Azriel Rosenfeld. ''Fuzzy Group Theory'', Springer Series: Studies in Fuzziness and Soft Computing, Vol. 182, 2005. | *Mordeson J., Kiran R. Bhutani and Azriel Rosenfeld. ''Fuzzy Group Theory'', Springer Series: Studies in Fuzziness and Soft Computing, Vol. 182, 2005. | ||
* Rosenfeld A., ''Fuzzy groups'', J. Math. Anal. Appl., 35 (1971), 512-517. | * Rosenfeld A., ''Fuzzy groups'', J. Math. Anal. Appl., 35 (1971), 512-517. | ||
* Zadeh L.A., ''Fuzzy Sets'', ‘’Information and Control’’, 8 (1965) 338353. | * Zadeh L.A., ''Fuzzy Sets'', ‘’Information and Control’’, 8 (1965) 338353. | ||
* Zadeh L.A., ''Similarity relations and fuzzy ordering'', Inform. Sci. 3 (1971) 177–200. | * Zadeh L.A., ''Similarity relations and fuzzy ordering'', Inform. Sci. 3 (1971) 177–200.[[Category:Suggestion Bot Tag]] | ||
[[Category: |
Latest revision as of 17:00, 19 August 2024
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In fuzzy logic given a first order language and a valuation structure V, we can interpret it by a fuzzy interpretation, i.e. a pair (D,I) such that D is a nonempty set and I, the interpretation function is a map associating any n-ary functor with an n-ary operation and any constant with an element of D (as in the classical case). Moreover, I associates any n-ary predicate name with a suitable n-ary fuzzy relation in D. The fuzzy subalgebras are particular fuzzy interpretations of a first order language for algebraic structures with a monadic predicate symbol S. The intended meaning of S(x) is that x is an element of the subalgebra. More precisely, a fuzzy subalgebra, is a model of a theory containing, for any n-ary operation name h, the axiom
A1 x1..., xn(S(x1)ʌ...ʌS(xn) → S(h(x1,...,xn))
and, for any constant c, the axiom
A2 S(c).
A1 expresses the closure of S with respect to the operation h, A2 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 it is easy to see that 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
i) s(d1)s(dn)≤ s(h(d1,...,dn))
Moreover, if c is the interpretation of a constant c
ii) s(c) = 1.
In a largely studied class of valuation structures the product coincides with the joint operation. 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.
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
2) s(x)s(y) ≤ s(x•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
3) s(x) ≤ s(x-1).
The notions of fuzzy submonoid and fuzzy subgroup are strictly related with the notions of fuzzy order and fuzzy equivalence, respectively.
Proposition. Assume that S is a nonempty set, G a group of transformations in S and (G,s) a fuzzy submonoid (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 order (a fuzzy equivalence). Conversely, let e be a fuzzy order (equivalence) in S and define s by setting, for every transformation h of S,
s(h)= Inf{e(x,h(x)): xS}.
Then s defines a fuzzy submonoid (fuzzy subgroup) of transformation in S.
Bibliography
- Klir, G. and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 978-0-13-101171-7
- Zimmermann H., Fuzzy Set Theory and its Applications (2001), ISBN 978-0-7923-7435-0.
- Demirci M., Recasens J., Fuzzy groups, fuzzy functions and fuzzy equivalence relations, Fuzzy Sets and Systems, 144 (2004), 441-458.
- Di Nola A., Gerla G., Lattice valued algebras, Stochastica, 11 (1987), 137-150.
- Hájek P., Metamathematics of fuzzy logic. Kluwer 1998.
- Klir G. , UTE H. St.Clair and Bo Yuan Fuzzy Set Theory Foundations and Applications,1997.
- Gerla G., Fuzzy submonoids, fuzzy preorders and quasi-metrics, Fuzzy Sets and Systems 157 (2006) 2356-2370.
- Mordeson J., Kiran R. Bhutani and Azriel Rosenfeld. Fuzzy Group Theory, Springer Series: Studies in Fuzziness and Soft Computing, Vol. 182, 2005.
- Rosenfeld A., Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
- Zadeh L.A., Fuzzy Sets, ‘’Information and Control’’, 8 (1965) 338353.
- Zadeh L.A., Similarity relations and fuzzy ordering, Inform. Sci. 3 (1971) 177–200.