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Lattics
Lattics (Windows)

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How to Find Lattics

Lattics

A lattice is a partially ordered set, containing elements that have both a least upper bound (also called a supremum or join) and a greatest lower bound (also called an infimum or meet). These are essentially the same as the two orderings for a set. For example, the Cartesian square of natural numbers is partially ordered by divisibility, with the supremum being the least common multiple and the infimum being the greatest common divisor.

How to find a lattice

In order to determine whether a poset is a lattice, you can use a Hasse diagram to identify which pairs of elements have a least upper bound and which are incomparable. For example, if the partial ordering on the left is d,e,f,g, then b and c are incomparable because there is no upward path from b to e, and no downward path from c to f.

Using this Hasse diagram, it is also easy to tell if a poset is a lattice when each pair of elements has both a least upper bound and a greatest lower bound. To check, draw the Hasse diagram again and look for arrows that are both downward and upward from one element to another.

A poset is called a complete lattice if all its subsets have both a join and a meet. In general, bounded lattice homomorphisms can preserve only finite joins and meets, but complete lattice homomorphisms are required to preserve arbitrary joins and meets.

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