Modular lattice

In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:

where  ≤  is the partial order, and  ∨  and  ∧ (called join and meet respectively) are the operations of the lattice. For an intuition behind the modularity condition see and below.

Modular lattices arise naturally in algebra and in many other areas of mathematics. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice.

Every distributive lattice is modular.

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

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(μx) = (λμ)x for vector spaces connects multiplication in the field and scalar multiplication.

The restriction x ≤ b is clearly necessary, since it follows from x ∨ (a ∧ b) = (x ∨ a) ∧ b. In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law. (To see this, just pick non-maximal b and let x be any element strictly greater than b.)

It is easy to see that x ≤ b implies x ∨ (a ∧ b) ≤ (x ∨ a) ∧ b in every lattice. Therefore, the modular law can also be stated as

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