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Title:Orthomodular Lattices and Cut Elimination
Author(s):Marble, Robert Patrick
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
Abstract:The thesis is a start toward a positive solution of the word problem for freely generated orthomodular lattices. It was a proof theoretical 'partial' cut elimination procedure.
A Gentzen type sequent calculus (OMO) is defined which can be seen to characterize a free orthomodular lattice L in the sense that two words u and v (on the generating set of L) are equal in L if and only if u (--->) v and v (--->) u are sequents which are derivable in the calculus OMO. Several types of applications of the cut rules of OMO are singled out and called benign cuts. They have the property that their application does not lead to the complete elimination of the atomic components of their cut formulas from their proof branches. Their use, then, does not hinder the recovery from an endsequent of information about all formulas used in a proof of that sequent, during an algorithmic process of deciding about the derivability of that sequent.
The inference rule which is used to manifest the orthomodularity of models of OMO is the rule OM:
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
It is shown that an OMO proof of the form
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
where P and Q are cut-free and contain no applications of rule OM, can be reconstructed to prove the same endsequent without the use of any cuts which are not benign.
Issue Date:1981
Type:Text
Description:89 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981.
URI:http://hdl.handle.net/2142/71196
Other Identifier(s):(UMI)AAI8203523
Date Available in IDEALS:2014-12-16
Date Deposited:1981


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