Representing some families of monotone maps by principal lattice congruences

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• 作者：Gábor Czédli
• 关键词：Key words and phrasesprincipal congruence ; lattice congruence ; ordered set ; order ; poset ; quasi ; colored lattice ; preordering ; quasiordering ; monotone map ; categorified lattice ; functor ; lattice category ; lifting diagrams
• 刊名：Algebra universalis
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：77
• 期：1
• 页码：51-77
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algebra;
• 出版者：Springer International Publishing
• ISSN：1420-8911
• 卷排序：77

文摘

For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result.