Editing Modular lattice (section)

{{distinguish|unimodular lattice}}
[[File:2d modular lattice.svg|thumb|A modular lattice of [[order dimension]] 2. As with all finite 2-dimensional lattices, its [[Hasse diagram]] is an [[st-planar graph|''st''-planar graph]].]]

In the branch of mathematics called [[order theory]], a '''modular lattice''' is a [[lattice (order)|lattice]] that satisfies the following self-[[duality (order theory)|dual]] condition,
;'''Modular law''':{{Math|''a'' ≤ ''b''}} implies {{Math|1=''a'' ∨ (''x'' ∧ ''b'') = (''a'' ∨ ''x'') ∧ ''b''}}

where {{Math|''x'', ''a'', ''b''}} are arbitrary elements in the lattice, &nbsp;≤&nbsp; is the [[partial order]], and &nbsp;∨&nbsp; and &nbsp;∧&nbsp;(called [[join and meet]] respectively) are the operations of the lattice.  This phrasing emphasizes an interpretation in terms of projection onto the sublattice {{Math|[''a'', ''b'']}}, a fact known as the '''diamond isomorphism theorem'''.<ref>{{Cite web|url=https://math.stackexchange.com/q/443947 |title=Why are modular lattices important?|website=Mathematics Stack Exchange|access-date=2018-09-17}}</ref>  An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a [[variety (universal algebra)|variety]] in the sense of [[universal algebra]].

Modular lattices arise naturally in [[algebra]] and in many other areas of mathematics.  In these scenarios, modularity is an abstraction of the [[Second isomorphism theorem|2nd Isomorphism Theorem]].  For example, the subspaces of a [[vector space]] (and more generally the submodules of a [[module over a ring]]) form a modular lattice.

In a not necessarily modular lattice, there may still be elements {{Mvar|b}} for which the modular law holds in connection with arbitrary elements {{Mvar|x}} and {{Mvar|a}} (for {{Math|''a'' ≤ ''b''}}). Such an element is called a '''right modular element'''. Even more generally, the modular law may hold for any {{Mvar|a}} and a fixed pair {{Math|(''x'', ''b'')}}. Such a pair is called a '''modular pair''', and there are various generalizations of modularity related to this notion and to [[semimodular lattice|semimodularity]].

Modular lattices are sometimes called '''Dedekind lattices''' after [[Richard Dedekind]], who discovered the modular identity in [[#History|several motivating examples]].