## Modular lattices

Abbreviation: MLat

### Definition

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

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

### Definition

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

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

### Definition

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> <script> unit=20; labelnodes=false; function node(x,y,t,r,nodecolor){ nodes[t]=[];nodes[t][0]=x;nodes[t][1]=y;if(r==undefined)r=(labelnodes?6:3);nodes[t][2]=r; if(nodecolor==undefined)nodecolor="black";nodes[t][3]=nodecolor; } function edge(i,j,edgecolor){ if(edgecolor==undefined)edgecolor="black";nodecolor=nodes[i][3]; x=nodes[i][0];y=nodes[i][1];z=nodes[j][0];w=nodes[j][1];r=nodes[i][2]; c.strokeStyle=edgecolor;c.beginPath();c.moveTo(unit*x,c.canvas.height-unit*y);c.lineTo(unit*z,c.canvas.height-unit*w);c.stroke(); c.strokeStyle=nodecolor;c.fillStyle="white";c.beginPath();c.arc(unit*x,c.canvas.height-unit*y,r,0,6.3,true);c.fill();if(r!=0)c.stroke(); if(labelnodes){c.fillStyle=nodecolor;c.fillText(i,unit*x-2.7,c.canvas.height-unit*y+3.5);} } nodes=new Array;c=document.getElementById('c1').getContext('2d');c.translate(10,-4); node(1,2,"4"); node(0,0.66,"1");node(0,1.33,"2");node(2,1,"3"); node(1,0,"0"); edge(4,2);edge(4,3); edge(2,1); edge(1,0);edge(3,0); edge(0,0); </script>

##### Morphisms

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:

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

### Examples

Example 1: $M_3$ <canvas id="c2" width="60" height="60"></canvas> <script> nodes=new Array;c=document.getElementById('c2').getContext('2d');c.translate(10,-4); node(1,2,"4"); node(0,1,"1");node(1,1,"2");node(2,1,"3"); node(1,0,"0"); edge(4,1);edge(4,2);edge(4,3); edge(1,0);edge(2,0);edge(3,0); edge(0,0); </script> is the smallest nondistributive modular lattice. By a result of 1) this lattice occurs as a sublattice of every nondistributive modular lattice.

### Properties

Classtype variety undecidable 2) 3) undecidable 4) undecidable no unbounded yes yes no no no no no no no no no

### Finite members

$\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)

### References

1) Richard Dedekind, \emph{\“Uber die von drei Moduln erzeugte Dualgruppe}, Math. Ann., \textbf{53}, 1900, 371–403
2) Ralph Freese, \emph{Free modular lattices}, Trans. Amer. Math. Soc., \textbf{261}, 1980, 81–91
3) Christian Herrmann, \emph{On the word problem for the modular lattice with four free generators}, Math. Ann., \textbf{265}, 1983, 513–527
4) L. Lipshitz, \emph{The undecidability of the word problems for projective geometries and modular lattices}, Trans. Amer. Math. Soc., \textbf{193}, 1974, 171–180
5) Peter Jipsen, Nathan Lawless, \emph{Generating all finite modular lattices of a given size}, Algebra Universalis, \textbf{74}, 2015, 253–264