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Modular lattice
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<p>In the branch of mathematics called <a href="/facts/Order_theory/gn6fM3oJ">order theory</a>, a modular lattice is a <a href="/facts/Lattice_(order)/5ak6ufky">lattice</a> that satisfies the following self-<a href="/facts/Duality_(order_theory)/TTTcqYQy">dual</a> condition, </p> Modular law <i>a</i> ≤ <i>b</i> implies <i>a</i> ∨ (<i>x</i> ∧ <i>b</i>) = (<i>a</i> ∨ <i>x</i>) ∧ <i>b</i> <p>where <i>x</i>, <i>a</i>, <i>b</i> are arbitrary elements in the lattice, ≤ is the <a href="/facts/Partial_order/qb4a3FAX">partial order</a>, and ∨ and ∧ (called <a href="/facts/Join_and_meet/cYcLhRIR">join and meet</a> respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice [<i>a</i>, <i>b</i>], a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a <a href="/facts/Variety_(universal_algebra)/vXWYFyDI">variety</a> in the sense of <a href="/facts/Universal_algebra/uMJAL6ER">universal algebra</a>. </p><p>Modular lattices arise naturally in <a href="/facts/Algebra/nMdqczjo">algebra</a> and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the <a href="/facts/Second_isomorphism_theorem/SNpPTkrd">2nd Isomorphism Theorem</a>. For example, the subspaces of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> (and more generally the submodules of a <a href="/facts/Module_over_a_ring/jP0LIhFL">module over a ring</a>) form a modular lattice. </p><p>In a not necessarily modular lattice, there may still be elements b for which the modular law holds in connection with arbitrary elements x and a (for <i>a</i> ≤ <i>b</i>). Such an element is called a right modular element. Even more generally, the modular law may hold for any a and a fixed pair (<i>x</i>, <i>b</i>). Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to <a href="/facts/Semimodular_lattice/P4abbawa">semimodularity</a>. </p><p>Modular lattices are sometimes called Dedekind lattices after <a href="/facts/Richard_Dedekind/CLkWDhY3">Richard Dedekind</a>, who discovered the modular identity in several motivating examples. </p>