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dc.contributor.advisorEmamy-K., M. Reza
dc.contributor.authorMeléndez Ríos, Gustavo
dc.date.accessioned2022-07-14T21:59:21Z
dc.date.available2022-07-14T21:59:21Z
dc.date.issued2022-04-26
dc.identifier.urihttps://hdl.handle.net/11721/2864
dc.description.abstractThis thesis proposes two new classes of lattices: <em>ST</em>-distributive and <em>ST</em>-modular lattices. The idea is to define relative distributive and modular properties that are satisfied by some elements over two subsets <em>S</em> and <em>T</em> of a given lattice <em>L</em>. These new classes include the usual distributive and modular lattices. Our main results are (1) establishing some basic properties, (2) completely characterizing the maximal <em>S</em> and <em>T</em> (with some restrictions) to form <em>ST</em>-distributive lattices in the lattice families <strong>M</strong><sub>n</sub> and <strong>M</strong><sub>n,n</sub>, and (3) presenting an application of <em>ST</em>-modular lattices to convex sets. This application has been the first example of <em>ST</em>-modularity and the original motivation of our work [7]. All this extends our work in [10]. In addition, we discuss two other problems with new results. First, we present two shortcuts to the <strong>M</strong><sub>3</sub>-<strong>N</strong><sub>5</sub> Theorem proof in [5, 4] and design two methods to compare the lengths of the three proofs [11]. Second, we introduce a poset on the cut-complexes of the 4-cube and show that it is a distributive lattice [9].en_US
dc.language.isoen_USen_US
dc.rightsAttribution-NonCommercial-NoDerivs 3.0 United States*
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/us/*
dc.subjectCut-complexesen_US
dc.subjectBirkhoff's Representation Theoremen_US
dc.subjectHasse diagramsen_US
dc.subjectPosetsen_US
dc.subject.lcshConvex polytopesen_US
dc.subject.lcshLattice theoryen_US
dc.subject.lcshLattices, Distributiveen_US
dc.subject.lcshModular latticesen_US
dc.titleST-distributive and ST-modular latticesen_US
dc.typeThesisen_US
dc.rights.holder© 2022 Gustavo A. Meléndez Ríosen_US
dc.contributor.committeeMedina Rivera, Luis A.
dc.contributor.committeeRubio Canabal, Ivelisse
dc.contributor.campusUniversity of Puerto Rico, Río Piedras Campusen_US
dc.description.graduationSemesterSpring (2nd Semester)en_US
dc.description.graduationYear2022en_US
thesis.degree.disciplineMathsen_US
thesis.degree.levelM.S.en_US


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Attribution-NonCommercial-NoDerivs 3.0 United States
Except where otherwise noted, this item's license is described as Attribution-NonCommercial-NoDerivs 3.0 United States