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Mathematics
Set Theory, Logic, Probability, Statistics
Lattice on the closed unit circle?
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[QUOTE="nomadreid, post: 4396532, member: 112452"] Would either or both of these work as a lattice on the closed unit circle in the plane? (1) Using a linear order: Expressing points in polar coordinates (with angles 0≤θ<2π), define: (r,α) < (s,β) iff r<s or (r=s & α<β) (r,α) ≤ (s,β) iff (r,α) < (s,β) or (r=s & α=β) The meet and join are then just the inf and the sup, resp. (2) Non-linear partial order: (r,α) < (s,β) iff r<s (r,α) ≤ (s,β) iff (r,α) < (s,β) or (r=s & α=β) [U]The join[/U]: if (r,α) < (s,β), then (r,α) [itex]\vee[/itex] (s,β) = (s,β) For the cases r=s: (r,α)[itex]\vee[/itex](r,α) =(r,α) If α≠β, then (r,α)[itex]\vee[/itex](r,β) = ((r+1)/2, (α+β)/2) [U]The meet[/U]: if (r,α) < (s,β), then (r,α) [itex]\wedge[/itex] (s,β) = (r,α) For the cases r=s: (r,α)[itex]\wedge[/itex](r,α) =(r,α) If α≠β, then (r,α)[itex]\wedge[/itex](r,β) = (r/2, (α+β)/2) [/QUOTE]
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Lattice on the closed unit circle?
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