# 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:

• $z$ is a lower bound of $x$ and $y$, and
• for every lower bound $w$ of $x$ and $y$, we have $w \leq z$.

[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]