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Boolean Dimension, Components and Blocks

Article

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 $$\dim (C)\le d$$ for every component C of a poset P, then $$\dim (P)\le \max \limits \{2,d\}$$; also if $$\dim (B)\le d$$ for every block B of a poset P, then $$\dim (P)\le d+2$$. 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 $$\dim (B)\le 3$$ 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 · 2d; also if bdim(B) = d for every block of P, then bdim(P) = 19 + d + 18 · 2d.

Keywords

Posets Boolean dimension

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© Springer Nature B.V. 2019

Authors and Affiliations

• Tamás Mészáros
• 1
• Piotr Micek
• 2
• William T. Trotter
• 3
1. 1.Institute of Mathematics, Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany
2. 2.Department of Theoretical Computer Science, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaGeorgia

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