On Subdirect Decompositions of Finite Distributive Lattices

Subdirect decomposition of algebra is one of its quite general and important constructions. In this paper, some subdirect decompositions (including subdirect irreducible decompositions) of finite distributive lattices and finite chains are studied, and some general results are obtained.

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

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.

On a cancellation rule for subdirect products of lattice ordered groups and of GMV-algebras

The notion of internal subdirect decomposition can be defined in each variety of algebras. In the present note we prove the validity of a cancellation rule concerning such decompositions for lattice ordered groups and for GMV-algebras. For the case of groups, this cancellation rule fails to be valid.

Representable divisibility semigroups

By a divisibility semigroup we mean an algebra (S,., ∧) satisfying (Al) (S,.) is a semigroup; (A2) (S, ∧) is a semilattice; (A3) .A divisibility semigroup is called representable if it admits a subdirect decomposition into totally ordered factors.In this paper various types of representable divisibility semigroups are investigated and characterized, admitting a representation in general or even a special decomposition, like subdirect sums of archimedean factors, for instance.

The lattice of completely regular semigroup varieties

A completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice L (CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on L (CR) yields a subdirect decomposition of L (CR). Using these results we show that L (CR) is arguesian. This confirms the (tacit) conjecture that L (CR) is modular.