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The '''uncovered set''' is defined for a set of [[preferential voting|rank-order]] preferences. Usually, the uncovered set is defined only for situations without pairwise ties. When there are no pairwise ties, then the uncovered set is identical to the set called '''Fishburn winners''':
{{definition|Select the candidate or candidates that are not Fishburn losers. A candidate ''i'' is a Fishburn loser if there is some other candidate ''j'' such that every candidate that pairwise beats ''j'' also pairwise beats ''i'' and there is at least one candidate that pairwise beats ''i'' but does not pairwise beat ''j''.}} It is a nonempty subset of the Smith set. When there are pairwise ties, a likely equivalent definition is: <blockquote>In voting systems, the '''Landau set''' (or '''uncovered set''', or '''Fishburn set''') is the set of candidates ''x'' such that for every other candidate ''z'', there is some candidate ''y'' (possibly the same as ''x'' or ''z'') such that ''y'' is not preferred to ''x'' and ''z'' is not preferred to ''y''. </blockquote> '''IndependenceThe ofuncovered coveredset alternatives'''is saysa thatnonempty ifsubset oneof optionthe (X)[[Smith winsset]]. anThe election,reason andis athat newevery alternativecandidate (Y)in the Smith set is added,preferred Xto willevery wincandidate not in the electionSmith ifset, Ytherefore iseach notcandidate in the [[uncoveredSmith set ]]. Independencecan ofbe coveredconsidered alternativesa impliescandidate [[Smith criterion|Smith]]''x'' and thusbe [[Condorcettheir criterion|Condorcet]].own Ifcandidate ''y''; since a methodcandidate can't be preferred to themselves (''y'' is independentnot ofpreferred coveredto alternatives''x''), thenand since candidates in the methodSmith failsset monotonicitybeing ifpreferred perfectto tiesevery cancandidate alwaysnot bein brokenthe Smith set implies that candidates not in favorthe ofSmith aset choiceare Wnot bypreferred usingto ballotscandidates rankingin Wthe firstSmith set (''z'' is not preferred to ''y''), the uncovered set must be a subset of the Smith set.
'''Independence of covered alternatives''' says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the [[uncovered set]]. Independence of covered alternatives implies [[Smith criterion|Smith]] and thus [[Condorcet criterion|Condorcet]]. If a method is independent of covered alternatives, then the method fails monotonicity if perfect ties can always be broken in favor of a choice W by using ballots ranking W first.
[[Category:Voting theory]]
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