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A Monotonicity Property of Partial Orders
Author(s) -
Kleitman D. J.,
Shearer J. B.
Publication year - 1981
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm198165181
Subject(s) - monotonic function , mathematics , combinatorics , property (philosophy) , partially ordered set , set (abstract data type) , order (exchange) , discrete mathematics , mathematical analysis , philosophy , epistemology , computer science , programming language , finance , economics
A proof using the FKG inequalities of the following result is obtained. Let P be a partially ordered set on a 1 ⩽ a 2 ⩽ ⋯ ⩽ a m and b 1 ⩽ b 2 ⩽ ⋯ ⩽ b n . Let P ( x ) be the proportion of linear extentions of P for which x holds. If x and y are disjunctions of conjunctions of additional inequalities of the form a i ⩾ b j , then P ( x and y ) ⩾ P ( x ) P ( y ). An example is provided that shows the result can be false if we don't assume the { a i } and { b j } are linearly ordered in P .