# Decompositions in lattices and some representations of algebras

- 2001

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topAndrzej 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 -

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