Decompositions in lattices and some representations of algebras

Andrzej Walendziak. Decompositions in lattices and some representations of algebras. 2001. <http://eudml.org/doc/286022>.

@book{AndrzejWalendziak2001,
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 ⟨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.},
author = {Andrzej Walendziak},
keywords = {generalization of modularity; decomposition theory of lattices; Kurosh-Ore theorem; Schmidt-Ore theorem; weak direct representations; congruences; congruence distributive algebras},
language = {eng},
title = {Decompositions in lattices and some representations of algebras},
url = {http://eudml.org/doc/286022},
year = {2001},
}

TY - BOOK
AU - Andrzej Walendziak
TI - Decompositions in lattices and some representations of algebras
PY - 2001
AB - 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.
LA - eng
KW - generalization of modularity; decomposition theory of lattices; Kurosh-Ore theorem; Schmidt-Ore theorem; weak direct representations; congruences; congruence distributive algebras
UR - http://eudml.org/doc/286022
ER -