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 ⟨ℒ,ψ⟩-product of algebras. We give sufficient conditions for an algebra to be isomorphic to an ⟨ℒ,ψ⟩-product with simple factors and with directly indecomposable factors. Some applications to subdirect, full subdirect and weak direct products are indicated.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm392-0-1,
author = {Andrzej Walendziak},
title = {Decompositions in lattices and some representations of algebras},
series = {GDML\_Books},
year = {2001},
zbl = {0985.06003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm392-0-1}
}
Andrzej Walendziak. Decompositions in lattices and some representations of algebras. GDML_Books (2001), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm392-0-1/