Timeline for Complete anti-chain lattices and the axiom of choice
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
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| Nov 20, 2012 at 18:46 | comment | added | Joseph Van Name | That is correct for any poset(not just lattices). A poset $X$ has the ascending chain condition if and only if the lower sets of $X$ are in a canonical one-to-one correspondence with the antichains in $X$. And this canonical one-to-one correspondence preserves order in posets satisfying ACC. | |
| Nov 20, 2012 at 13:01 | comment | added | Leo | Thanks, that clears some things up. It seems to me it is sufficient to require that a lattice $(L,\leq)$ has no infinite ascending chains in order for the one-to-one correspondence between anti-chains and down-sets to hold, right? Is it necessary also? (sorry, I'm too busy to think through this myself atm, I don't mean to you have you do my slave work, but I htought maybe you have some quick input) | |
| Nov 19, 2012 at 21:37 | history | answered | Joseph Van Name | CC BY-SA 3.0 |