{"history":[],"article":{"id":"52d9f122-5c2c-4c2f-9ee5-84005788fc1a","title":"Modular lattice","slug":"modular-lattice","content":":::summary\n:::\n\n# \nIn the branch of mathematics called [order theory](https://bliptext.com/articles/order-theory), a **modular [lattice](https://bliptext.com/articles/lattice-order)** is a [lattice](https://bliptext.com/articles/lattice-order) that satisfies the following self-[dual](https://bliptext.com/articles/duality-order-theory) condition,\na ≤ b\nimplies\na ∨ (x ∧ b) = (a ∨ x) ∧ b\nwhere\nx, a, b\nare arbitrary elements in the lattice, ≤ is the [partial order](https://bliptext.com/articles/partial-order), and ∨ and ∧ (called [join and meet](https://bliptext.com/articles/join-and-meet) respectively) are the operations of the lattice.\nThis phrasing emphasizes an interpretation in terms of projection onto the sublattice\n[a, b]\n, a fact known as the **diamond isomorphism theorem**.\nAn alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a [variety](https://bliptext.com/articles/variety-universal-algebra) in the sense of [universal algebra](https://bliptext.com/articles/universal-algebra).\nModular lattices arise naturally in [algebra](https://bliptext.com/articles/algebra) and in many other areas of mathematics.\nIn these scenarios, modularity is an abstraction of the [2nd Isomorphism Theorem](https://bliptext.com/articles/second-isomorphism-theorem).\nFor example, the subspaces of a [vector space](https://bliptext.com/articles/vector-space) (and more generally the submodules of a [module over a ring](https://bliptext.com/articles/module-over-a-ring)) form a modular lattice.\nIn 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\na ≤ b\n). Such an element is called a **right modular element**. Even more generally, the modular law may hold for any a and a fixed pair\n(x, b)\n. Such a pair is called a **modular pair**, and there are various generalizations of modularity related to this notion and to [semimodularity](https://bliptext.com/articles/semimodular-lattice).\nModular lattices are sometimes called **Dedekind lattices** after [Richard Dedekind](https://bliptext.com/articles/richard-dedekind), who discovered the modular identity in [several motivating examples](https://bliptext.com/articles/).\n# [A modular lattice of order dimension 2. As with all finite 2-dimensional lattices, its Hasse diagram is an st-planar graph. | upload.wikimedia.org/wikipedia/commons/7/79/2d///modular///lattice.svg]\n\n# Introduction\nThe modular law can be seen as a restricted [associative law](https://bliptext.com/articles/associativity) that connects the two lattice operations similarly to the way in which the [associative law](https://bliptext.com/articles/associativity) λ(μx) = (λμ)x for vector spaces connects multiplication in the field and scalar multiplication.\nThe restriction\na ≤ b\nis clearly necessary, since it follows from\na ∨ (x ∧ b) = (a ∨ x) ∧ b\n. In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law.\nIt is easy to see that\na ≤ b\nimplies\na ∨ (x ∧ b) ≤ (a ∨ x) ∧ b\nin every lattice.\nTherefore, the modular law can also be stated as\na ≤ b\nimplies\n(a ∨ x) ∧ b ≤ a ∨ (x ∧ b)\n.\nThe modular law can be expressed as an equation that is required to hold unconditionally.\nSince\na ≤ b\nimplies\nand since\na ∧ b ≤ b\n, replace\na\nwith\na ∧ b\nin the defining equation of the modular law to obtain:\n(a ∧ b) ∨ (x ∧ b) = ((a ∧ b) ∨ x) ∧ b\n. This shows that, using terminology from [universal algebra](https://bliptext.com/articles/universal-algebra), the modular lattices form a sub[variety](https://bliptext.com/articles/variety-universal-algebra) of the [variety](https://bliptext.com/articles/variety-universal-algebra) of lattices.\nTherefore, all homomorphic images, [sublattices](https://bliptext.com/articles/sublattice) and direct products of modular lattices are again modular.\n\n# Examples\nThe lattice of submodules of a [module over a ring](https://bliptext.com/articles/module-over-a-ring) is modular.\nAs a special case, the lattice of subgroups of an [abelian group](https://bliptext.com/articles/abelian-group) is modular.\nThe lattice of [normal subgroups](https://bliptext.com/articles/normal-sub[group](https://bliptext.com/articles/group-mathematics)) of a [group](https://bliptext.com/articles/group-mathematics) is modular.\nBut in general the [lattice of all subgroups](https://bliptext.com/articles/lattice-of-subgroups) of a group is not modular.\nFor an example, the lattice of subgroups of the dihedral group of order 8 is not modular.\nThe smallest non-modular lattice is the \"pentagon\" lattice N5 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.\nFor this lattice,\nholds, contradicting the modular law.\nEvery non-modular lattice contains a copy of N5 as a sublattice.\n# [N5, the smallest non-modular lattice: x∨(a∧b) = x∨0 = x ≠ b = 1∧b =(x∨a)∧b. | upload.wikimedia.org/wikipedia/commons/5/5b/Smallest///nonmodular///lattice///2.svg]\n\n# Properties\nEvery [distributive lattice](https://bliptext.com/articles/distributive-lattice) is modular.\nproved that, in every finite modular lattice, the number of join-irreducible elements equals the number of meet-irreducible elements.\nMore generally, for every k, the number of elements of the lattice that cover exactly k other elements equals the number that are covered by exactly k other elements.\nA useful property to show that a lattice is not modular is as follows:\na, b, c ∈ G\n,\nSketch of proof: Let G be modular, and let the premise of the implication hold.\nThen using absorption and modular identity:\nFor 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.\nLet x = (a∧b) ∨ c, y = a ∧ (b∨c).\nFrom the modular inequality immediately follows that x ≤ y.\nIf we show that x∧b = y∧b, x∨b = y∨b, then using the assumption x = y must hold.\nThe rest of the proof is routine manipulation with infima, suprema and inequalities.\n\n# Diamond isomorphism theorem\nFor any two elements a,b of a modular lattice, one can consider the intervals [a ∧ b, b] and [a, a ∨ b].\nThey are connected by order-preserving maps\nthat are defined by φ(x) = x ∨ a and ψ(y) = y ∧ b.\nThe 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.\nThe example shows that this inequality can be strict in general.\nIn a modular lattice, however, equality holds.\nSince 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.\nThis result is sometimes called the **diamond isomorphism theorem** for modular lattices.\nA lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.\nThe diamond [isomorphism theorem](https://bliptext.com/articles/isomorphism-theorem) for modular lattices is analogous to the second [isomorphism theorem](https://bliptext.com/articles/isomorphism-theorem) in algebra, and it is a generalization of the [lattice theorem](https://bliptext.com/articles/lattice-theorem).\n\n# Modular pairs and related notions\nIn 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.\nAn element b of a lattice is called a **right modular element** if (a, b) is a modular pair for all elements a, and an element a is called a **left modular element** if (a, b) is a modular pair for all elements b.\nA 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**.\nThus, in an M-symmetric lattice, every right modular element is also left modular, and vice-versa.\nSince a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric.\nIn the lattice N5 described above, the pair (b, a) is modular, but the pair (a, b) is not.\nTherefore, N5 is not M-symmetric.\nThe centred hexagon lattice S7 is M-symmetric but not modular.\nSince N5 is a sublattice of S7, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices.\nM-symmetry is not a self-dual notion.\nA **[dual](https://bliptext.com/articles/duality-order-theory) modular pair** is a pair which is modular in the [dual](https://bliptext.com/articles/duality-order-theory) lattice, and a lattice is called [dual](https://bliptext.com/articles/duality-order-theory)ly M-symmetric or **M*-symmetric** if its [dual](https://bliptext.com/articles/duality-order-theory) is M-symmetric.\nIt can be shown that a finite lattice is modular if and only if it is M-symmetric and M*-symmetric.\nThe same equivalence holds for infinite lattices which satisfy the [ascending chain condition](https://bliptext.com/articles/ascending-chain-condition) (or the descending chain condition).\nSeveral less important notions are also closely related.\nA lattice is **cross-symmetric** if for every modular pair (a, b) the pair (b, a) is dually modular.\nCross-symmetry implies M-symmetry but not M*-symmetry.\nTherefore, cross-symmetry is not equivalent to dual cross-symmetry.\nA 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.\n# [The centred hexagon lattice S7, also known as D2, is M-symmetric but not modular. | upload.wikimedia.org/wikipedia/commons/1/1a/Centred///hexagon///lattice///D2.svg]\n\n# History\nThe definition of modularity is due to [Richard Dedekind](https://bliptext.com/articles/richard-dedekind), who published most of the relevant papers after his retirement.\nIn a paper published in 1894 he studied lattices, which he called dual groups as part of his \"algebra of [modules](https://bliptext.com/articles/module-mathematics)\" and observed that ideals satisfy what we now call the modular law.\nHe also observed that for lattices in general, the modular law is equivalent to its dual.\nIn 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.\nIn a digression he introduced and studied lattices formally in a general context.\nHe observed that the lattice of submodules of a module satisfies the modular identity.\nHe called such lattices dual groups of module type (Dualgruppen vom Modultypus).\nHe also proved that the modular identity and its dual are equivalent.\nIn the same paper, Dedekind also investigated the following stronger form of the modular identity, which is also self-dual:\nHe called lattices that satisfy this identity dual groups of ideal type (Dualgruppen vom Idealtypus).\nIn modern literature, they are more commonly referred to as [distributive lattices](https://bliptext.com/articles/distributive-lattice).\nHe gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type.\nA 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).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:13:14.704Z","updated_at":"2025-01-25T21:13:14.704Z","search_vector":"'lattic':2 'modular':1","search_content":" \n \n\n# \nIn the branch of mathematics called [order theory](https://bliptext.com/articles/order-theory), a **modular [lattice](https://bliptext.com/articles/lattice-order)** is a [lattice](https://bliptext.com/articles/lattice-order) that satisfies the following self-[dual](https://bliptext.com/articles/duality-order-theory) condition,\na ≤ b\nimplies\na ∨ (x ∧ b) = (a ∨ x) ∧ b\nwhere\nx, a, b\nare arbitrary elements in the lattice, ≤ is the [partial order](https://bliptext.com/articles/partial-order), and ∨ and ∧ (called [join and meet](https://bliptext.com/articles/join-and-meet) respectively) are the operations of the lattice.\nThis phrasing emphasizes an interpretation in terms of projection onto the sublattice\n[a, b]\n, a fact known as the **diamond isomorphism theorem**.\nAn alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a [variety](https://bliptext.com/articles/variety-universal-algebra) in the sense of [universal algebra](https://bliptext.com/articles/universal-algebra).\nModular lattices arise naturally in [algebra](https://bliptext.com/articles/algebra) and in many other areas of mathematics.\nIn these scenarios, modularity is an abstraction of the [2nd Isomorphism Theorem](https://bliptext.com/articles/second-isomorphism-theorem).\nFor example, the subspaces of a [vector space](https://bliptext.com/articles/vector-space) (and more generally the submodules of a [module over a ring](https://bliptext.com/articles/module-over-a-ring)) form a modular lattice.\nIn 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\na ≤ b\n). Such an element is called a **right modular element**. Even more generally, the modular law may hold for any a and a fixed pair\n(x, b)\n. Such a pair is called a **modular pair**, and there are various generalizations of modularity related to this notion and to [semimodularity](https://bliptext.com/articles/semimodular-lattice).\nModular lattices are sometimes called **Dedekind lattices** after [Richard Dedekind](https://bliptext.com/articles/richard-dedekind), who discovered the modular identity in [several motivating examples](https://bliptext.com/articles/).\n# [A modular lattice of order dimension 2. As with all finite 2-dimensional lattices, its Hasse diagram is an st-planar graph. | upload.wikimedia.org/wikipedia/commons/7/79/2d///modular///lattice.svg]\n\n# Introduction\nThe modular law can be seen as a restricted [associative law](https://bliptext.com/articles/associativity) that connects the two lattice operations similarly to the way in which the [associative law](https://bliptext.com/articles/associativity) λ(μx) = (λμ)x for vector spaces connects multiplication in the field and scalar multiplication.\nThe restriction\na ≤ b\nis clearly necessary, since it follows from\na ∨ (x ∧ b) = (a ∨ x) ∧ b\n. In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law.\nIt is easy to see that\na ≤ b\nimplies\na ∨ (x ∧ b) ≤ (a ∨ x) ∧ b\nin every lattice.\nTherefore, the modular law can also be stated as\na ≤ b\nimplies\n(a ∨ x) ∧ b ≤ a ∨ (x ∧ b)\n.\nThe modular law can be expressed as an equation that is required to hold unconditionally.\nSince\na ≤ b\nimplies\nand since\na ∧ b ≤ b\n, replace\na\nwith\na ∧ b\nin the defining equation of the modular law to obtain:\n(a ∧ b) ∨ (x ∧ b) = ((a ∧ b) ∨ x) ∧ b\n. This shows that, using terminology from [universal algebra](https://bliptext.com/articles/universal-algebra), the modular lattices form a sub[variety](https://bliptext.com/articles/variety-universal-algebra) of the [variety](https://bliptext.com/articles/variety-universal-algebra) of lattices.\nTherefore, all homomorphic images, [sublattices](https://bliptext.com/articles/sublattice) and direct products of modular lattices are again modular.\n\n# Examples\nThe lattice of submodules of a [module over a ring](https://bliptext.com/articles/module-over-a-ring) is modular.\nAs a special case, the lattice of subgroups of an [abelian group](https://bliptext.com/articles/abelian-group) is modular.\nThe lattice of [normal sub[group](https://bliptext.com/articles/group-mathematics)s](https://bliptext.com/articles/normal-sub[group](https://bliptext.com/articles/group-mathematics)) of a [group](https://bliptext.com/articles/group-mathematics) is modular.\nBut in general the [lattice of all subgroups](https://bliptext.com/articles/lattice-of-subgroups) of a group is not modular.\nFor an example, the lattice of subgroups of the dihedral group of order 8 is not modular.\nThe smallest non-modular lattice is the \"pentagon\" lattice N5 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.\nFor this lattice,\nholds, contradicting the modular law.\nEvery non-modular lattice contains a copy of N5 as a sublattice.\n# [N5, the smallest non-modular lattice: x∨(a∧b) = x∨0 = x ≠ b = 1∧b =(x∨a)∧b. | upload.wikimedia.org/wikipedia/commons/5/5b/Smallest///nonmodular///lattice///2.svg]\n\n# Properties\nEvery [distributive lattice](https://bliptext.com/articles/distributive-lattice) is modular.\nproved that, in every finite modular lattice, the number of join-irreducible elements equals the number of meet-irreducible elements.\nMore generally, for every k, the number of elements of the lattice that cover exactly k other elements equals the number that are covered by exactly k other elements.\nA useful property to show that a lattice is not modular is as follows:\na, b, c ∈ G\n,\nSketch of proof: Let G be modular, and let the premise of the implication hold.\nThen using absorption and modular identity:\nFor 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.\nLet x = (a∧b) ∨ c, y = a ∧ (b∨c).\nFrom the modular inequality immediately follows that x ≤ y.\nIf we show that x∧b = y∧b, x∨b = y∨b, then using the assumption x = y must hold.\nThe rest of the proof is routine manipulation with infima, suprema and inequalities.\n\n# Diamond isomorphism theorem\nFor any two elements a,b of a modular lattice, one can consider the intervals [a ∧ b, b] and [a, a ∨ b].\nThey are connected by order-preserving maps\nthat are defined by φ(x) = x ∨ a and ψ(y) = y ∧ b.\nThe 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.\nThe example shows that this inequality can be strict in general.\nIn a modular lattice, however, equality holds.\nSince 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.\nThis result is sometimes called the **diamond isomorphism theorem** for modular lattices.\nA lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.\nThe diamond [isomorphism theorem](https://bliptext.com/articles/isomorphism-theorem) for modular lattices is analogous to the second [isomorphism theorem](https://bliptext.com/articles/isomorphism-theorem) in algebra, and it is a generalization of the [lattice theorem](https://bliptext.com/articles/lattice-theorem).\n\n# Modular pairs and related notions\nIn 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.\nAn element b of a lattice is called a **right modular element** if (a, b) is a modular pair for all elements a, and an element a is called a **left modular element** if (a, b) is a modular pair for all elements b.\nA 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**.\nThus, in an M-symmetric lattice, every right modular element is also left modular, and vice-versa.\nSince a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric.\nIn the lattice N5 described above, the pair (b, a) is modular, but the pair (a, b) is not.\nTherefore, N5 is not M-symmetric.\nThe centred hexagon lattice S7 is M-symmetric but not modular.\nSince N5 is a sublattice of S7, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices.\nM-symmetry is not a self-dual notion.\nA **[dual](https://bliptext.com/articles/duality-order-theory) modular pair** is a pair which is modular in the [dual](https://bliptext.com/articles/duality-order-theory) lattice, and a lattice is called [dual](https://bliptext.com/articles/duality-order-theory)ly M-symmetric or **M*-symmetric** if its [dual](https://bliptext.com/articles/duality-order-theory) is M-symmetric.\nIt can be shown that a finite lattice is modular if and only if it is M-symmetric and M*-symmetric.\nThe same equivalence holds for infinite lattices which satisfy the [ascending chain condition](https://bliptext.com/articles/ascending-chain-condition) (or the descending chain condition).\nSeveral less important notions are also closely related.\nA lattice is **cross-symmetric** if for every modular pair (a, b) the pair (b, a) is dually modular.\nCross-symmetry implies M-symmetry but not M*-symmetry.\nTherefore, cross-symmetry is not equivalent to dual cross-symmetry.\nA 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.\n# [The centred hexagon lattice S7, also known as D2, is M-symmetric but not modular. | upload.wikimedia.org/wikipedia/commons/1/1a/Centred///hexagon///lattice///D2.svg]\n\n# History\nThe definition of modularity is due to [Richard Dedekind](https://bliptext.com/articles/richard-dedekind), who published most of the relevant papers after his retirement.\nIn a paper published in 1894 he studied lattices, which he called dual groups as part of his \"algebra of [modules](https://bliptext.com/articles/module-mathematics)\" and observed that ideals satisfy what we now call the modular law.\nHe also observed that for lattices in general, the modular law is equivalent to its dual.\nIn 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.\nIn a digression he introduced and studied lattices formally in a general context.\nHe observed that the lattice of submodules of a module satisfies the modular identity.\nHe called such lattices dual groups of module type (Dualgruppen vom Modultypus).\nHe also proved that the modular identity and its dual are equivalent.\nIn the same paper, Dedekind also investigated the following stronger form of the modular identity, which is also self-dual:\nHe called lattices that satisfy this identity dual groups of ideal type (Dualgruppen vom Idealtypus).\nIn modern literature, they are more commonly referred to as [distributive lattices](https://bliptext.com/articles/distributive-lattice).\nHe gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type.\nA 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).","revisionCount":0},"hasMore":false,"nextCursor":null}