The Darmstadt school, particularly Christian Herrmann, studied the Gelfand Ponomarev papers carefully to see what they said about 4-generated modular lattices; see Ch. Herrmann, Rahmen und erzeugende Quadrupel in modularen Verbande, Algebra Universalis, 14(1982) 357-387. Some of the interesting results include:

1. The lattice of subspaces of every $n$-dimensional vector space, $3 \le n < \infty$, over a prime field is $4$-generated. Moreover, the quadruples generating the whole lattice are known. The dimension of each of the subspaces in the generating set can differ from $n/2$ by at most $2$.
2. The lattice of subgroups of $(\mathbb Z/p^k\mathbb Z)^n$ is four generated.

The hope was to classify the modular lattices generated by quadruples and so solve the word problem for $FM(4)$. But is now known that the word problem for free modular lattices is undecidable.

The Darmstadt school, particularly Christian Herrmann, studied the Gelfand Ponomarev papers carefully to see what they said about 4-generated modular lattices; see Ch. Herrmann, Rahmen und erzeugende Quadrupel in modularen Verbande, Algebra Universalis, 14(1982) 357-387. Some of the interesting results include:

1. The lattice of subspaces of every $n$-dimensional vector space, $3 \le n < \infty$, over a prime field is $4$-generated. Moreover, the quadruples generating the whole lattice are known. The dimension of each of the subspaces in the generating set can differ from $n/2$ by at most $2$.
2. The lattice of subgroups of $(\mathbb Z/p^k\mathbb Z)^n$ is four generated if $3 \le n < \infty$.

The hope was to classify the modular lattices generated by quadruples and so solve the word problem for $FM(4)$. But is now known that the word problem for free modular lattices is undecidable.

The Darmstadt school, particularly Christian Herrmann, studied the Gelfand Ponomarev papers carefully to see what they said about 4-generated modular lattices; see Ch. Herrmann, Rahmen und erzeugende Quadrupel in modularen Verbande, Algebra Universalis, 14(1982) 357-387. Some of the interesting results include:

1. The lattice of subspaces of every $n$-dimensional vector space, $3 \le n < \infty$, over a prime field is $4$-generated. Moreover, the quadruples generating the whole lattice are known. The dimension of each of the subspaces in the generating set can differ from $n/2$ by at most $2$.
2. The lattice of subgroups of $\mathbb Z/p^k\mathbb Z$ is four generated.

The hope was to classify the modular lattices generated by quadruples and so solve the word problem for $FM(4)$. But is now known that the word problem for free modular lattices is undecidable.

The Darmstadt school, particularly Christian Herrmann, studied the Gelfand Ponomarev papers carefully to see what they said about 4-generated modular lattices; see Ch. Herrmann, Rahmen und erzeugende Quadrupel in modularen Verbande, Algebra Universalis, 14(1982) 357-387. Some of the interesting results include:

1. The lattice of subspaces of every $n$-dimensional vector space, $3 \le n < \infty$, over a prime field is $4$-generated. Moreover, the quadruples generating the whole lattice are known. The dimension of each of the subspaces in the generating set can differ from $n/2$ by at most $2$.
2. The lattice of subgroups of $(\mathbb Z/p^k\mathbb Z)^n$ is four generated.

The hope was to classify the modular lattices generated by quadruples and so solve the word problem for $FM(4)$. But is now known that the word problem for free modular lattices is undecidable.