Editing Modular lattice (section)

==Modular pairs and related notions==
[[Image:Centred hexagon lattice D2.svg|thumb|right|The centred hexagon lattice ''S''<sub>7</sub>, also known as ''D''<sub>2</sub>, is M-symmetric but not modular.]]
In any lattice, a '''modular pair''' is a pair (''a,&nbsp;b'') of elements such that for all ''x'' satisfying ''a''&nbsp;∧&nbsp;''b'' ≤ ''x''&nbsp;≤&nbsp;''b'', we have (''x''&nbsp;∨&nbsp;''a'')&nbsp;∧&nbsp;''b''&nbsp;=&nbsp;''x'', i.e. if one half of the diamond isomorphism theorem holds for the pair.<ref>The [[French language|French]] term for modular pair is ''couple modulaire''. A pair (''a,&nbsp;b'') is called a ''paire modulaire'' in French if both (''a,&nbsp;b'') and (''b,&nbsp;a'') are modular pairs.</ref> An element ''b'' of a lattice is called a '''right modular element''' if (''a,&nbsp;b'') is a modular pair for all elements ''a'', and an element ''a'' is called a '''left modular element''' if (''a,&nbsp;b'') is a modular pair for all elements ''b''.<ref>Modular element has been varying defined by different authors to mean right modular ({{harvtxt|Stern|1999|p=74}}), left modular ({{harvtxt|Orlik|Terao|1992|loc=Definition 2.25}}), both left and right modular (or dual right modular) ({{harvtxt|Sagan|1999}}, {{harvtxt|Schmidt|1994|p=43}}), or satisfying a modular rank condition ({{harvtxt|Stanley|2007|loc=Definition 4.12}}).  These notions are equivalent in a semimodular lattice, but not in general.</ref>

A lattice with the property that if (''a,&nbsp;b'') is a modular pair, then (''b,&nbsp;a'') is also a modular pair is called an '''M-symmetric lattice'''.<ref>Some authors, e.g. Fofanova (2001), refer to such lattices as ''semimodular lattices''. Since every M-symmetric lattice is [[semimodular lattice|semimodular]] and the converse holds for lattices of finite length, this can only lead to confusion for infinite lattices.</ref>  Thus, in an M-symmetric lattice, every right modular element is also left modular, and vice-versa. Since a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric. In the lattice ''N''<sub>5</sub> described above, the pair (''b,&nbsp;a'') is modular, but the pair (''a,&nbsp;b'') is not. Therefore, ''N''<sub>5</sub> is not M-symmetric. The centred hexagon lattice ''S''<sub>7</sub> is M-symmetric but not modular. Since ''N''<sub>5</sub> is a sublattice of ''S''<sub>7</sub>, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices.

M-symmetry is not a self-dual notion. A '''dual modular pair''' is a pair which is modular in the [[duality (order theory)|dual]] lattice, and a lattice is called dually M-symmetric or '''M<sup>*</sup>-symmetric''' if its dual is M-symmetric. It can be shown that a finite lattice is modular if and only if it is M-symmetric and M<sup>*</sup>-symmetric. The same equivalence holds for infinite lattices which satisfy the [[ascending chain condition]] (or the descending chain condition).

Several less important notions are also closely related. A lattice is '''cross-symmetric''' if for every modular pair (''a,&nbsp;b'') the pair (''b,&nbsp;a'') is dually modular. Cross-symmetry implies M-symmetry but not M<sup>*</sup>-symmetry. Therefore, cross-symmetry is not equivalent to dual cross-symmetry. A lattice with a least element 0 is '''⊥-symmetric''' if for every modular pair (''a,&nbsp;b'') satisfying ''a''&nbsp;∧&nbsp;''b''&nbsp;=&nbsp;0 the pair (''b,&nbsp;a'') is also modular.