REDUCING FUZZY ALGEBRA TO CLASSICAL ALGEBRA

Arthur Weinberger ()
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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, 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 setX, there is a setYsuch that the lattice of fuzzy subsets ofXis isomorphic to a sublattice of the classical subsets ofY. In fact it is further proved that ifXis infinite, then we can chooseY = Xand get the surprising result that the lattice of fuzzy subsets ofXis isomorphic to a sublattice of the classical subsets ofXitself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval𝕀 = [0,1]embeds into the lattice of classical subsets of𝕀.The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebraAembeds into the lattice of classical subalgebras of a closely related algebraA′. 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
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DOI: 10.1142/S1793005705000020

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