Greatest lower bound in a poset

https://arbital.com/p/poset_greatest_lower_bound

by Patrick Stevens Sep 24 2016 updated Sep 24 2016

The greatest lower bound is an abstraction of the idea of the greatest common divisor to a general poset.


[summary: In a Partially ordered set, the greatest lower bound of two elements $~$x$~$ and $~$y$~$ is the "largest" element which is "less than" both $~$x$~$ and $~$y$~$, in whatever ordering the poset has.]

In a Partially ordered set, the greatest lower bound of two elements $~$x$~$ and $~$y$~$ is the "largest" element which is "less than" both $~$x$~$ and $~$y$~$, in whatever ordering the poset has. In a rare moment of clarity in mathematical naming, the name "greatest lower bound" is a perfect description of the concept: a "lower bound" of two elements $~$x$~$ and $~$y$~$ is an object which is smaller than both $~$x$~$ and $~$y$~$ (it "bounds them from below"), and the "greatest lower bound" is the greatest of all the lower bounds.

Formally, if $~$P$~$ is a set with partial order $~$\leq$~$, and given elements $~$x$~$ and $~$y$~$ of $~$P$~$, we say an element $~$z \in P$~$ is a lower bound of $~$x$~$ and $~$y$~$ if $~$z \leq x$~$ and $~$z \leq y$~$. We say an element $~$z \in P$~$ is the greatest lower bound of $~$x$~$ and $~$y$~$ if:

[todo: examples in different posets] [todo: example where there is no greatest lower bound because there is no lower bound] [todo: example where there is no GLB because while there are lower bounds, none of them is greatest]


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