%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 }