Abbreviation: MLat

A modular lattice is a lattice $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the

modular identity: $((x\wedge z) \vee y) \wedge z = (x\wedge z) \vee (y\wedge z)$

A modular lattice is a lattice $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the

modular law: $x\le z\Longrightarrow (x\vee y) \wedge z\le x\vee (y\wedge z)$

A modular lattice is a lattice $\mathbf{L}=\langle L,\vee,\wedge\rangle $ such that $\mathbf{L}$ has no sublattice isomorphic to the pentagon $\mathbf{N}_{5}$

Let $\mathbf{L}$ and $\mathbf{M}$ be modular lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$

Example 1: $M_3$ is the smallest nondistributive modular lattice. By a result of ¹⁾ this lattice occurs as a sublattice of every nondistributive modular lattice.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &4\\ f(6)= &8\\ f(7)= &16\\ f(8)= &34\\ f(9)= &72\\ f(10)= &157\\ f(11)= &343\\ f(12)= &766\\ f(13)= &1718\\ f(14)= &3899\\ f(15)= &8898\\ f(16)= &20475\\ f(17)= &47321\\ f(18)= &110024\\ f(19)= &256791\\ f(20)= &601991\\ f(21)= &1415768\\ f(22)= &3340847\\ f(23)= &7904700\\ f(24)= &18752942\\ f(25)= &\\ f(26)= &\\ \end{array}$⁵⁾

Distributive lattices

Complete modular lattices

Semimodular lattices

Geometric lattices