I need to enlight some points on this exercise. Say me where I wrong and what's the correct answer:
Given $T = \{(2,3),(4,1),(1,1),(7,1),(2,0),(0,4)\}$ and the relation $$(a,b)\rho(c,d)\Leftrightarrow |a-b|\leq|c-d|$$
- Show the Hasse diagram of $(T, \rho)$
- Is this a lattice?
- Is distributive?
- Is complemented?
- Has a boolean sub-lattice? If has, show it.
- This is a chain! $(7,1)$ is the max element, (1,1) the min.
- Sure a chain has max and min and so is totally ordered, so it's a lattice.
- I know the properties for a distributive lattice, but how I use in given case?
I think it isn't a complemented lattice. Being a chain, for two element $a$ and $b$ at the center of the chain $a \wedge b\neq min(a,b)$ and so $a \vee b\neq max(a,b)$ (where $\wedge$ is the infinum, and $\vee$ the supremum).
e.g.: $(a,b)=(4,1)$ and $(c,d)=(2,0)$, so $(4,1)\wedge(2,0)$ is not $(1,1)$ (what's the infinum? $(2,0)$ itself or $(2,3)$). Same for supremum. Am I wrong?
- If I was right for question 4 (uncomplemented lattice) than the lattice can't be boolean, neither have boolean sublattice
Best regards