## Decompositions in lattices and some representations of algebras

### Volume 392 / 2001

#### Abstract

We first investigate the properties of consistence, strongness and semimodularity, each of which may be viewed as a generalization of modularity. Chapters 2 and 3 present the decomposition theory of lattices. Here we characterize modularity in terms of the Kurosh–Ore replacement property. Next, we study $c$-decompositions of elements in lattices (the notion of $c$-decomposition is introduced as a generalization of those of join and direct decompositions). We find a common generalization of the Kurosh–Ore Theorem and the Schmidt–Ore Theorem for arbitrary modular lattices, solving a problem of G. Grätzer. The decomposition theory for lattices enables us to develop a structure theory for algebras. Some kinds of representations of algebras are studied. We consider weak direct representations of a universal algebra. The existence of such representations is considered. Finally, we introduce the notion of an $\langle {\cal L},\psi \rangle $-product of algebras. We give sufficient conditions for an algebra to be isomorphic to an $\langle {\cal L},\psi \rangle $-product with simple factors and with directly indecomposable factors. Some applications to subdirect, full subdirect and weak direct products are indicated.