Thursday, October 10, 2013

Complete lattice extensions

The natural numbers with infinity can be described as a complete lattice which can be extended by the integers with positive and negative infinity which can in turn be extended by the complete lattice of real numbers.

Given a real number such as 27/5 we can get an interval approximation for this number through the integers sublattice of (5,6). Likewise for a negative real number such as -5/4 we can get an approximation of (-2,-1). Since the sublattice of natural numbers doesn't contain any negative numbers the only approximation we can get there is (> 0).

Given the trivial lattice containing just the number zero we can get approximations of (< 0), (> 0), and equal to zero which are essentially just the sign of the number. Given the complete lattice of real numbers we can always get a standard part which doesn't even need to be expressed as an interval because the real numbers form a dense complete total order.

The surreal number $27/5+\frac{1}{\omega}$ has a standard part of 5, an integer part of (5,6) as described earlier, and a sign of (< 0). The infinite surreal numbers $\omega$ and $\sqrt{\omega}$ both have a standard part of (< Infinity). Complete lattice extensions can be used to construct ever more advanced total ordered number systems.

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