Vector lattices and sequence spaces

An equivalence functor between local vector lattices and vector lattices

We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-...

Finite Vector Spaces and Certain Lattices

The Galois number Gn(q) is defined to be the number of subspaces of the n-dimensional vector space over the finite field GF (q). When q is prime, we prove that Gn(q) is equal to the number Ln(q) of n-dimensional mod q lattices, which are defined to be lattices (that is, discrete additive subgroups of n-space) contained in the integer lattice Z and having the property that given any point P in t...

Some Basic Properties of Vector Sequence Spaces

Subspace Lattices of Finite Vector Spaces Are 5-generated

Let n ≥ 3. 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-di...

Sublattices of lattices of convex subsets of vector spaces

For a left vector space V over a totally ordered division ring F, let Co(V ) denote the lattice of convex subsets of V . We prove that every lattice L can be embedded into Co(V ) for some left F-vector space V . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Ω) = {X ∩Ω | X ∈ Co(V )}, for some f...