# On the representation of lattices by subgroup lattices

Research paper by V.B. Repnitskii

Indexed on: 01 Jan '97Published on: 01 Jan '97Published in: Algebra universalis

#### Abstract

Whitman's condition in a lattice L means that, for any elements $$a, b, c, d \in L, a \wedge b \leq c \vee d$$ implies either $$a \ wedge b \leq c$$ or $$a\wedge b \leq d$$, or $$a \geq c \vee d$$, or $$\leq bc \vee d$$. We prove that any lattice satisfying Whitman’s condition can be embedded in the subgroup lattice of a free group of an arbitrary non-soluble group variety. Some interesting corollaries (both on embeddings in lattices of subgroups and others) are examined.