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The Marica‐Schönheim Inequality in Lattices
Author(s) -
Lengvárszky Zsolt
Publication year - 1996
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/28.5.449
Subject(s) - mathematics , distributive property , partition (number theory) , combinatorics , inequality , lattice (music) , join (topology) , modular design , distributive lattice , discrete mathematics , pure mathematics , mathematical analysis , physics , computer science , acoustics , operating system
The Marica‐Schönheim Inequality says that if A is a finite family of sets, then | A −|⩾| A | where A − A =[ A 1 ∖ A 2 : A 1 , A 2 ∈ A ]. For a finite lattice L and A ⊆ L , we define a − b =∨( J a ∖ J b ) where J a =[ j ∈ L : j ⩽ a and j is join‐irreducible], and if A ⊆ L then we let A − A =[ a 1 − a 2 : a 1 , a 2 ∈ A ]. Then the analogue of the Marica‐Schöonheim Inequality is | A − A ⩾| A | for all A ⊆ L . We prove that this is true if L is distributive or complemented and modular or L is a partition lattice.