## A study of an independent system arising in group choice via the weak Bruhat order |

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### Contents

Lower Bounds for Consistent Sets | 13 |

IH The Weak Bruhat Order of SE Consistent Sets | 43 |

Algorithms for Consistent Sets | 68 |

### Common terms and phrases

Abello adjacent transpositions algorithm Arrow's Impossibility Theorem Arrow's Theorem Catalan number Chapter class of consistent connected consistent set constructions contain a latin contain a maximal contain triples contradiction corresponding Coxeter Coxeter group cyclic denote elements entries EXP(I EXP(p EXPtp Fact Figure following corollary give Hamiltonian path Hasse diagram i+ji implies induction latin square lattice LC(p lower bound majority rule matroid maximal chain maximal connected consistent maximal consistent set maximal consistent subset maximal pseudochain maximal unimodal maximum minimum spanning tree Notation Notice obtained operations ordered pair p,pR pair t,p permutation preceding theorem Proof prove Quadrilateral rule saturated chain semimodular lattice sets of cardinality sets which contain simple majority SPATH(q,p subset of Sj symbols symmetric group T(EXP(p tion TRAN(SE transitive relations TSM domains unimodal sets unimodal with respect upper bound weak Bruhat order weak order