The GLB & LUB of Two Elements: Non-Unique Possibilities?

[Gear300]

The greatest lower bound and least upper bound of two elements a, b in a lattice do not have to be unique, do they? It could be the case that two equivalent or non-comparable glb or lub exist, right?

 

[theorem4.5.9]

They are unique in a lattice. By definition (according to wikipedia) a lattice a po-set where any two elements have a unique glb and lub (or infimum and supremum respectively).

http://en.wikipedia.org/wiki/Lattice_(order )

 

[Gear300]

Thanks.

 

[dcpo]

In a poset the glb and lub must be unique by antisymmetry.

 

[blue_raver22]

Yes, it is possible for the greatest lower bound (glb) and least upper bound (lub) of two elements in a lattice to not be unique. This means that there can be more than one possible value for the glb or lub of a and b. This can occur when the elements a and b are equivalent or non-comparable in the lattice.

For example, in a lattice of real numbers, the glb of 2 and 3 could be either 2 or 3, as both values are equally valid lower bounds for the set {2, 3}. Similarly, the lub of 2 and 3 could be either 2 or 3, as both values are equally valid upper bounds for the set {2, 3}.

The existence of non-unique glb and lub values does not violate the properties of a lattice, as the glb and lub are defined as the greatest and least elements that satisfy certain conditions. Therefore, as long as there are elements that satisfy these conditions, there can be multiple glb and lub values for a given set of elements.

In conclusion, the glb and lub of two elements in a lattice may not be unique, but this does not affect the overall structure and properties of the lattice. It is important to consider all possible values when determining the glb and lub, especially in cases where the elements are equivalent or non-comparable.