ACTA issues

## Subspace lattices of finite vector spaces are 5-generated

László Zádori

Acta Sci. Math. (Szeged) 74:3-4(2008), 493-499
6028/2009

 Abstract. Let \$n\geq3\$. From the description of subdirectly irreducible complemented Arguesian lattices with four generators given by Herrmann, Ringel and Wille it follows that the subspace lattice of an \$n\$-dimensional vector space over a finite field is generated by four elements if and only if the field is a prime field. By exhibiting a 5-element generating set we prove that the subspace lattice of an \$n\$-dimensional vector space over an arbitrary finite field is generated by five elements. AMS Subject Classification (1991): 06C05, 50D30, 14N20, 51D25 Keyword(s): Arguesian lattice, subspace lattice of a vector space, generating set of a subspace lattice Received February 4, 2008, and in revised form February 8, 2008. (Registered under 6028/2009.)