Direct decompositions of non-algebraic complete lattices

Direct decompositions of non-algebraic complete lattices

For a given complete lattice L, we investigate whether L can be decomposed as a direct product of directly indecomposable lattices. We prove that this is the case if every element of L is a join of join-irreducible elements and dually, thus extending to non-algebraic lattices a result of L. Libkin. We illustrate this by various examples and counterexamples.

Direct Decompositions of Atomistic Algebraic Lattices

Direct Decompositions of Algebraic Systems

Preliminaries. An operator group with a principal series can obviously be written as a direct product of finitely many directly indecomposable admissible subgroups, and the classical WedderburnRemak-Krull-Schmidt theorem asserts that this representation is unique up to isomorphism. Numerous generalizations of this theorem are known in the literature. Thus it follows from the results in Baer [l;...

Direct Product Decompositions of Lattices , Closures andRelation

In this paper we study the direct product decompositions of closure operations and lattices of closed sets. We characterize the direct product decompositions of lattices of closed sets in terms of closure operations, and nd those decompositions of lattices which correspond to the decompositions of closures. If a closure on a nite set is represented by its implication base (i.e. a binary relatio...

Decompositions of Banach Lattices into Direct Sums

We consider the problem of decomposing a Banach lattice Z as a direct sum Z = X @ Y where X and Y are complemented subspaces satisfying a condition of incomparability (e.g. every operator from Y to X is strictly singular). We treat both the atomic and nonatomic cases. In particular we answer a question of Wojtaszczyk by showing that L1 fflL2 has unique structure as a nonatomic Banach lattice. O...