%AHaiman, Milan%BJournal Name: Order; Related Information: CHORUS Timestamp: 2023-11-21 19:04:22
%D2023%ISpringer Science + Business Media
%JJournal Name: Order; Related Information: CHORUS Timestamp: 2023-11-21 19:04:22
%K
%MOSTI ID: 10475241
%PMedium: X
%TThe Dimension of Divisibility Orders and Multiset Posets
%XAbstract
The Dushnik–Miller dimension of a posetPis the leastdfor whichPcan be embedded into a product ofdchains. Lewis and Souza isibility order on the interval of integers$$[N/\kappa , N]$$$[N/\kappa ,N]$is bounded above by$$\kappa (\log \kappa )^{1+o(1)}$$$\kappa {(log\kappa )}^{1+o\left(1\right)}$and below by$$\Omega ((\log \kappa /\log \log \kappa )^2)$$$\Omega \left({(log\kappa /loglog\kappa )}^{2}\right)$. We improve the upper bound to$$O((\log \kappa )^3/(\log \log \kappa )^2).$$$O({(log\kappa )}^{3}/{(loglog\kappa )}^{2}).$We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.

%0Journal Article