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Irredundant Decomposition of Algebras into One-Dimensional Factors

Bogdan Staruch — 2016

Bulletin of the Section of Logic

We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of subdirect product. We obtain the star-decomposition of algebras into one-dimensional factors, which generalizes the known decomposition theorems e.g. for Abelian groups, linear spaces, Boolean algebras.

Decomposition of Congruence Modular Algebras into Atomic, Atomless Locally Uniform and Anti-Uniform Parts

Bogdan Staruch;  Bożena Staruch — 2016

Bulletin of the Section of Logic

We describe here a special subdirect decomposition of algebras with modular congruence lattice. Such a decomposition (called a star-decomposition) is based on the properties of the congruence lattices of algebras. We consider four properties of lattices: atomic, atomless, locally uniform and anti-uniform. In effect, we describe a star-decomposition of a given algebra with modular congruence lattice into two or three parts associated to these properties.

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