Lattice on the closed unit circle?

[nomadreid]

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)

 

[mmwave]

Both of these would work as lattices on the closed unit circle in the plane. A lattice is a partially ordered set in which any two elements have a unique greatest lower bound (meet) and a unique least upper bound (join). In both of these examples, the elements are ordered based on their distance from the origin (r) and their angle (α) in polar coordinates.

In the first example, a linear order is used to define the meet and join operations. The meet is the infimum (greatest lower bound) and the join is the supremum (least upper bound). This means that for any two elements, their meet will be the element with the smaller distance from the origin and the smallest angle, while their join will be the element with the larger distance from the origin and the larger angle. This follows the definition of a lattice, as any two elements will have a unique greatest lower bound and a unique least upper bound.

In the second example, a non-linear partial order is used to define the meet and join operations. The meet is still the infimum and the join is still the supremum, but the definition is slightly different. For the meet, if one element is less than the other, the meet will be the element with the smaller distance from the origin and the smaller angle. If the two elements are equal, the meet will be that element itself. For the join, if one element is less than the other, the join will be the element with the larger distance from the origin and the larger angle. If the two elements are equal, the join will be that element itself. This also follows the definition of a lattice, as any two elements will have a unique greatest lower bound and a unique least upper bound.

In conclusion, both of these examples would work as lattices on the closed unit circle in the plane. They both follow the definition of a lattice and provide unique meet and join operations for any two elements.