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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.