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Lattice (order)
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== Examples of non-lattices == {| style="float:right" | [[File:Pow3nonlattice.svg|thumb|x150px|'''Pic. 8:''' Non-lattice poset: <math>a</math> and <math>b</math> have common lower bounds <math>0, d, g, h,</math> and <math>i,</math> but none of them is the [[greatest lower bound]].]] |} {| style="float:right" | [[File:NoLatticeDiagram.svg|thumb|x150px|'''Pic. 7:''' Non-lattice poset: <math>b</math> and <math>c</math> have common upper bounds <math>d, e,</math> and <math>f,</math> but none of them is the [[least upper bound]].]] |} {| style="float:right" | [[File:KeinVerband.svg|thumb|x150px|'''Pic. 6:''' Non-lattice poset: <math>c</math> and <math>d</math> have no common upper bound.]] |} Most partially ordered sets are not lattices, including the following. * A discrete poset, meaning a poset such that <math>x \leq y</math> implies <math>x = y,</math> is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice. * Although the set <math>\{1, 2, 3, 6\}</math> partially ordered by divisibility is a lattice, the set <math>\{1, 2, 3\}</math> so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in <math>\{2, 3, 6\}.</math> * The set <math>\{1, 2, 3, 12, 18, 36\}</math> partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12, 18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other). <!---stop floating mode before next section--->{{clear}}
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