The graphs of join-semilattices and the shape of congruence lattices of particle lattices

Commentationes mathematicae Universitatis Carolinae

Pavel R\xc5\xaf\xc5\xbei\xc4\x8dka

关键词: join-semilattice;    lattice;    join-irreducible;    dependency;    chain condition;    particle;    atomistic;    congruenceDOI: DOI 10.14712/1213-7243.2015.214AMS Subject Classification: 06A12 06A15 06B10 06F30 PDF;   

DOI  :  10.14712/1213-7243.2015.214

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

We attach to each $\\langle 0,\\vee \\rangle$-semilattice $\\boldsymbol S$ a graph $\\boldsymbol G_{\\boldsymbol S}$ whose vertices are join-irreducible elements of $\\boldsymbol S$ and whose edges correspond to the reflexive dependency relation. We study properties of the graph $\\boldsymbol G_{\\boldsymbol S}$ both when $\\boldsymbol S$ is a~join-semilattice and when it is a~lattice. We call a~$\\langle 0,\\vee \\rangle$-semilattice $\\boldsymbol S$ \\emph{particle\\/} provided that the set of its join-irreducible elements satisfies DCC and join-generates $\\boldsymbol S$. We prove that the congruence lattice of a particle lattice is anti-isomorphic to the lattice of all hereditary subsets of the corresponding graph that are closed in a certain zero-dimensional topology. Thus we extend the result known for principally chain finite lattices.

【 授权许可】

CC BY