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In the branch of mathematics called
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, a modular lattice is a lattice that satisfies the following self- dual condition, ;Modular law: implies where are arbitrary elements in the lattice,  ≤  is the
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
, and  ∨  and  ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , 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
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
in the sense of
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
. Modular lattices arise naturally in
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the 2nd Isomorphism Theorem. For example, the subspaces of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(and more generally the submodules of a
module over a ring In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
) form a modular lattice. In a not necessarily modular lattice, there may still be elements for which the modular law holds in connection with arbitrary elements and (for ). Such an element is called a modular element. Even more generally, the modular law may hold for any and a fixed pair . Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semimodularity. Modular lattices are sometimes called Dedekind lattices after
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
, who discovered the modular identity in several motivating examples.


Introduction

The modular law can be seen as a restricted associative law that connects the two lattice operations similarly to the way in which the associative law λ(μ''x'') = (λμ)''x'' for vector spaces connects multiplication in the field and scalar multiplication. The restriction is clearly necessary, since it follows from . In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law. It is easy to see that implies in every lattice. Therefore, the modular law can also be stated as ;Modular law (variant): implies . The modular law can be expressed as an equation that is required to hold unconditionally. Since implies and since , replace with in the defining equation of the modular law to obtain: ;Modular identity: . This shows that, using terminology from
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
, the modular lattices form a subvariety of the
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
of lattices. Therefore, all homomorphic images, sublattices and direct products of modular lattices are again modular.


Examples

The lattice of submodules of a
module over a ring In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
is modular. As a special case, the lattice of subgroups of an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
is modular. The lattice of
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
s of a group is modular. But in general the lattice of all subgroups of a group is not modular. For an example, the lattice of subgroups of the dihedral group of order 8 is not modular. The smallest non-modular lattice is the "pentagon" lattice ''N''5 consisting of five elements 0, 1, ''x'', ''a'', ''b'' such that 0 < ''x'' < ''b'' < 1, 0 < ''a'' < 1, and ''a'' is not comparable to ''x'' or to ''b''. For this lattice, :''x'' ∨ (''a'' ∧ ''b'') = ''x'' ∨ 0 = ''x'' < ''b'' = 1 ∧ ''b'' = (''x'' ∨ ''a'') ∧ ''b'' holds, contradicting the modular law. Every non-modular lattice contains a copy of ''N''5 as a sublattice.


Properties

Every distributive lattice is modular. proved that, in every finite modular lattice, the number of join-irreducible elements equals the number of meet-irreducible elements. More generally, for every , the number of elements of the lattice that cover exactly other elements equals the number that are covered by exactly other elements. A useful property to show that a lattice is not modular is as follows: : A lattice is modular if and only if, for any , ::\Big((c\leq a)\text(a\wedge b=c\wedge b)\text(a\vee b=c\vee b)\Big)\Rightarrow(a=c) Sketch of proof: Let G be modular, and let the premise of the implication hold. Then using absorption and modular identity: : ''c'' = (''c''∧''b'') ∨ ''c'' = (''a''∧''b'') ∨ ''c'' = ''a'' ∧ (''b''∨''c'') = ''a'' ∧ (''b''∨''a'') = ''a'' For the other direction, let the implication of the theorem hold in G. Let ''a'',''b'',''c'' be any elements in G, such that ''c'' ≤ ''a''. Let ''x'' = (''a''∧''b'') ∨ ''c'', ''y'' = ''a'' ∧ (''b''∨''c''). From the modular inequality immediately follows that ''x'' ≤ ''y''. If we show that ''x''∧''b'' = ''y''∧''b'', ''x''∨''b'' = ''y''∨''b'', then using the assumption ''x'' = ''y'' must hold. The rest of the proof is routine manipulation with infima, suprema and inequalities.


Diamond isomorphism theorem

For any two elements ''a'',''b'' of a modular lattice, one can consider the intervals 'a'' ∧ ''b'', ''b''and 'a'', ''a'' ∨ ''b'' They are connected by order-preserving maps ::φ: 'a'' ∧ ''b'', ''b'''a'', ''a'' ∨ ''b''and ::ψ: 'a'', ''a'' ∨ ''b'''a'' ∧ ''b'', ''b''that are defined by φ(''x'') = ''x'' ∨ ''a'' and ψ(''y'') = ''y'' ∧ ''b''. Image:Modular_pair.svg, In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms. Image:Not a modular pair.svg, Failure of the diamond isomorphism theorem in a non-modular lattice. The composition ψφ is an order-preserving map from the interval 'a'' ∧ ''b'', ''b''to itself which also satisfies the inequality ψ(φ(''x'')) = (''x'' ∨ ''a'') ∧ ''b'' ≥ ''x''. The example shows that this inequality can be strict in general. In a modular lattice, however, equality holds. Since the dual of a modular lattice is again modular, φψ is also the identity on 'a'', ''a'' ∨ ''b'' and therefore the two maps φ and ψ are isomorphisms between these two intervals. This result is sometimes called the diamond isomorphism theorem for modular lattices. A lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements. The diamond isomorphism theorem for modular lattices is analogous to the second
isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exis ...
in algebra, and it is a generalization of the lattice theorem.


Modular pairs and related notions

In any lattice, a modular pair is a pair (''a, b'') of elements such that for all ''x'' satisfying ''a'' ∧ ''b'' ≤ ''x'' ≤ ''b'', we have (''x'' ∨ ''a'') ∧ ''b'' = ''x'', i.e. if one half of the diamond isomorphism theorem holds for the pair. An element ''b'' of a lattice is called a (right) modular element if (''a, b'') is a modular pair for all elements ''a''. A lattice with the property that if (''a, b'') is a modular pair, then (''b, a'') is also a modular pair is called an M-symmetric lattice. 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''5 described above, the pair (''b, a'') is modular, but the pair (''a, b'') is not. Therefore, ''N''5 is not M-symmetric. The centred hexagon lattice ''S''7 is M-symmetric but not modular. Since ''N''5 is a sublattice of ''S''7, 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 dual lattice, and a lattice is called dually M-symmetric or M*-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*-symmetric. The same equivalence holds for infinite lattices which satisfy the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These c ...
(or the descending chain condition). Several less important notions are also closely related. A lattice is cross-symmetric if for every modular pair (''a, b'') the pair (''b, a'') is dually modular. Cross-symmetry implies M-symmetry but not M*-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, b'') satisfying ''a'' ∧ ''b'' = 0 the pair (''b, a'') is also modular.


History

The definition of modularity is due to
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
, who published most of the relevant papers after his retirement. In a paper published in 1894 he studied lattices, which he called ''dual groups'' (german: Dualgruppen) as part of his "algebra of
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
" and observed that ideals satisfy what we now call the modular law. He also observed that for lattices in general, the modular law is equivalent to its dual. In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility. In a digression he introduced and studied lattices formally in a general context. He observed that the lattice of submodules of a module satisfies the modular identity. He called such lattices ''dual groups of module type'' (). He also proved that the modular identity and its dual are equivalent. In the same paper, Dedekind also investigated the following stronger form of the modular identity, which is also self-dual: : (''x'' ∧ ''b'') ∨ (''a'' ∧ ''b'') = 'x'' ∨ ''a''∧ ''b''. He called lattices that satisfy this identity ''dual groups of ideal type'' (). In modern literature, they are more commonly referred to as distributive lattices. He gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type. A paper published by Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements (see picture).


See also

* Modular graph, a class of graphs that includes the Hasse diagrams of modular lattices * Young–Fibonacci lattice, an infinite modular lattice defined on strings of the digits 1 and 2 *
Orthomodular lattice In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b''&nb ...
*
Iwasawa group __NOTOC__ In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group ''G'' is called an Iwasawa group when every subgroup of ''G'' is permutable in ''G'' . proved ...


Notes


References

* * * * * *


External links

* * {{OEIS el, A006981, Number of unlabeled modular lattices with ''n'' elements
Free Modular Lattice Generator
An open-source browser-based web application that can generate and visualize some free modular lattices. Lattice theory