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Lattice on the closed unit circle?

  1. May 27, 2013 #1
    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)
    The join: 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)
    The meet: 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)
     
  2. jcsd
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