Greatest lower bound in a 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]