%0 Journal Article %1 meszaros2020boolean %A Mészáros, Tamás %A Micek, Piotr %A Trotter, William T. %D 2020 %J Order %K boolean dimension lattice %N 2 %P 287--298 %R 10.1007/s11083-019-09505-3 %T Boolean Dimension, Components and Blocks %U https://doi.org/10.1007/s11083-019-09505-3 %V 37 %X We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if \$\backslashdim (C)\backslashle d\$for every component C of a poset P, then \$\backslashdim (P)\backslashle \backslashmax \backslashlimits \backslash\2,d\backslash\\$; also if \$\backslashdim (B)\backslashle d\$for every block B of a poset P, then \$\backslashdim (P)\backslashle d+2\$. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if ldim(C) = d for every component C of a poset P, then ldim(P) = d + 2; however, for every d = 4, there exists a poset P with ldim(P) = d and \$\backslashdim (B)\backslashle 3\$for every block B of P. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if bdim(C) = d for every component C of P, then bdim(P) = 2 + d + 4 · 2d; also if bdim(B) = d for every block of P, then bdim(P) = 19 + d + 18 · 2d.

@article{meszaros2020boolean, abstract = {We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if {\$}{\backslash}dim (C){\backslash}le d{\$}for every component C of a poset P, then {\$}{\backslash}dim (P){\backslash}le {\backslash}max {\backslash}limits {\backslash}{\{}2,d{\backslash}{\}}{\$}; also if {\$}{\backslash}dim (B){\backslash}le d{\$}for every block B of a poset P, then {\$}{\backslash}dim (P){\backslash}le d+2{\$}. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if ldim(C) = d for every component C of a poset P, then ldim(P) = d + 2; however, for every d = 4, there exists a poset P with ldim(P) = d and {\$}{\backslash}dim (B){\backslash}le 3{\$}for every block B of P. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if bdim(C) = d for every component C of P, then bdim(P) = 2 + d + 4 {\textperiodcentered} 2d; also if bdim(B) = d for every block of P, then bdim(P) = 19 + d + 18 {\textperiodcentered} 2d.}, added-at = {2021-06-23T09:36:48.000+0200}, author = {M{\'e}sz{\'a}ros, Tam{\'a}s and Micek, Piotr and Trotter, William T.}, biburl = {https://www.bibsonomy.org/bibtex/25bee3989e338953626b1ed75a3b89d6b/stumme}, day = 01, description = {Boolean Dimension, Components and Blocks | SpringerLink}, doi = {10.1007/s11083-019-09505-3}, interhash = {595578fda9e28eaca60cd11eebbcb19a}, intrahash = {5bee3989e338953626b1ed75a3b89d6b}, issn = {1572-9273}, journal = {Order}, keywords = {boolean dimension lattice}, month = jul, number = 2, pages = {287--298}, timestamp = {2021-06-23T09:36:48.000+0200}, title = {Boolean Dimension, Components and Blocks}, url = {https://doi.org/10.1007/s11083-019-09505-3}, volume = 37, year = 2020 }