G. Bergman. (2016)cite arxiv:1602.00034Comment: Comments: 50 pages. Main change from version 2: Lemma 26 strengthened to include contractibility statement which in previous version was noted as "likely" in paragraph following that result. Several small typoes also corrected.
DOI: 10.2140/agt.2017.17.439
Abstract
If $L$ is a finite lattice, we show that there is a natural topological
lattice structure on the geometric realization of its order complex $\Delta(L)$
(definition recalled). Lattice-theoretically, the resulting object is a
subdirect product of copies of $L$. We note properties of this construction and
of some variants thereof, and pose several questions. For $M_3$ the $5$-element
nondistributive modular lattice, $\Delta(M_3)$ is modular, but its underlying
topological space does not admit a structure of distributive lattice, answering
a question of Walter Taylor.
We also describe a construction of "stitching together" a family of lattices
along a common chain, and note how $\Delta(M_3)$ can be obtained as a case of
this construction.
Description
[1602.00034] Simplicial complexes with lattice structures
cite arxiv:1602.00034Comment: Comments: 50 pages. Main change from version 2: Lemma 26 strengthened to include contractibility statement which in previous version was noted as "likely" in paragraph following that result. Several small typoes also corrected
%0 Generic
%1 bergman2016simplicial
%A Bergman, George M.
%D 2016
%K composistion concept lattice order stitching topology
%R 10.2140/agt.2017.17.439
%T Simplicial complexes with lattice structures
%U http://arxiv.org/abs/1602.00034
%X If $L$ is a finite lattice, we show that there is a natural topological
lattice structure on the geometric realization of its order complex $\Delta(L)$
(definition recalled). Lattice-theoretically, the resulting object is a
subdirect product of copies of $L$. We note properties of this construction and
of some variants thereof, and pose several questions. For $M_3$ the $5$-element
nondistributive modular lattice, $\Delta(M_3)$ is modular, but its underlying
topological space does not admit a structure of distributive lattice, answering
a question of Walter Taylor.
We also describe a construction of "stitching together" a family of lattices
along a common chain, and note how $\Delta(M_3)$ can be obtained as a case of
this construction.
@misc{bergman2016simplicial,
abstract = {If $L$ is a finite lattice, we show that there is a natural topological
lattice structure on the geometric realization of its order complex $\Delta(L)$
(definition recalled). Lattice-theoretically, the resulting object is a
subdirect product of copies of $L$. We note properties of this construction and
of some variants thereof, and pose several questions. For $M_3$ the $5$-element
nondistributive modular lattice, $\Delta(M_3)$ is modular, but its underlying
topological space does not admit a structure of distributive lattice, answering
a question of Walter Taylor.
We also describe a construction of "stitching together" a family of lattices
along a common chain, and note how $\Delta(M_3)$ can be obtained as a case of
this construction.},
added-at = {2021-03-10T11:31:36.000+0100},
author = {Bergman, George M.},
biburl = {https://www.bibsonomy.org/bibtex/2a1c10c5812a09bebf2cbe11092b05cb1/stumme},
description = {[1602.00034] Simplicial complexes with lattice structures},
doi = {10.2140/agt.2017.17.439},
interhash = {501fbde6b4b8872e9a4c43b79368dca6},
intrahash = {a1c10c5812a09bebf2cbe11092b05cb1},
keywords = {composistion concept lattice order stitching topology},
note = {cite arxiv:1602.00034Comment: Comments: 50 pages. Main change from version 2: Lemma 26 strengthened to include contractibility statement which in previous version was noted as "likely" in paragraph following that result. Several small typoes also corrected},
timestamp = {2021-03-10T11:31:36.000+0100},
title = {Simplicial complexes with lattice structures},
url = {http://arxiv.org/abs/1602.00034},
year = 2016
}