Abbreviation: MLat
A \emph{modular lattice} is a lattice $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the
\emph{modular identity}: $((x\wedge z) \vee y) \wedge z = (x\wedge z) \vee (y\wedge z)$
A \emph{modular lattice} is a lattice $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the
\emph{modular law}: $x\le z\Longrightarrow (x\vee y) \wedge z\le x\vee (y\wedge z)$
A \emph{modular lattice} is a lattice $\mathbf{L}=\langle L,\vee,\wedge\rangle $ such that $\mathbf{L}$ has no sublattice isomorphic
to the pentagon $\mathbf{N}_{5}$ <canvas id="c1" width="60" height="60"></canvas>
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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: $M_3$ <canvas id="c2" width="60" height="60"></canvas>
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is the smallest nondistributive modular lattice. By a result of 1)
this lattice occurs as a sublattice of every nondistributive
modular lattice.
| Classtype | variety |
|---|---|
| Equational theory | undecidable 2) 3) |
| Quasiequational theory | undecidable 4) |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | no |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | no |
| Definable principal congruences | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | no |
| Strong amalgamation property | no |
| Epimorphisms are surjective | no |
$\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)