Decompositions in lattices and some representations of algebras

Volume 392 / 2001

Andrzej Walendziak Dissertationes Mathematicae 392 (2001), 1-83 MSC: 06B05, 06B10, 06C05, 06C10, 06D20, 08A05, 08A30, 13E05, 13E10. DOI: 10.4064/dm392-0-1


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.


  • Andrzej WalendziakInstitute of Mathematics and Physics
    University of Podlasie
    08-110 Siedlce, Poland

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