Pavel Růžička

Lattices of two-sided ideals of locally matricial algebras and the Γ-invariant problem

We develop a method of representation of distributive (V,0,1)-semilattices as semilattices of finitely generated ideals of locally matricial algebras. We use the method to reprove two representation results by G. M. Bergman and prove a new one that every distributive (0,1)-lattice is, as a semilattice, isomorphic to the semilattice of all finitely generated ideals of a locally matricial algebra. We apply this fact to solve the Γ-invariant problem.

A distributive semilattice not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group

Abstract We construct a distributive 0-semilattice which is not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group. The size of the semilattice is ℵ 2 .

Minimal generating sets of groups, rings, and fields

A subset X of a group (or a ring, or a field) is called generating, if the smallest subgroup (or subring, or subfield) containing X is the group (ring, field) itself. A generating set X is called minimal generating, if X does not properly contain any generating set. The existence and cardinalities of minimal generating sets of various groups, rings, and fields are investigated. In particular it is shown that there are groups, rings, and fields which do not have a minimal generating set. Among other result, the cardinality of minimal generating sets of finite abelian groups and of finite products of Zn rings is computed.

Chains, antichains, and complements in infinite partition lattices

We consider the partition lattice

Free trees and the optimal bound in Wehrung's theorem

\Pi (\lambda )

Characterization of Abelian groups with a minimal generating set

Π(λ) on any set of transfinite cardinality

Liftings of distributive lattices by locally matricial algebras with respect to the Idc functor

\lambda