EconPapers    
Economics at your fingertips  
 

REDUCING FUZZY ALGEBRA TO CLASSICAL ALGEBRA

Arthur Weinberger ()
Additional contact information
Arthur Weinberger: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902-6000, USA

New Mathematics and Natural Computation (NMNC), 2005, vol. 01, issue 01, pages 27-64

Abstract: This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras.The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets.The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀.The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.

Keywords: Fuzzy sets; fuzzy algebra; lattices; metatheory; universal algebra (search for similar items in EconPapers)
Date: 2005
References: Add references at CitEc
Citations Track citations by RSS feed

Downloads: (external link)
http://www.worldscinet.com/cgi-bin/details.cgi?type=pdf&id=pii:S1793005705000020 (application/pdf)
http://www.worldscinet.com/cgi-bin/details.cgi?typ ... ii:S1793005705000020 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: http://EconPapers.repec.org/RePEc:wsi:nmncxx:v:01:y:2005:i:01:p:27-64

Ordering information: This journal article can be ordered from

Access Statistics for this article

New Mathematics and Natural Computation (NMNC) is edited by Paul P Wang

More articles in New Mathematics and Natural Computation (NMNC) from World Scientific Publishing Co. Pte. Ltd.
Series data maintained by Tai Tone Lim ().

 
Page updated 2014-01-04
Handle: RePEc:wsi:nmncxx:v:01:y:2005:i:01:p:27-64