# Greatest lower bound in a poset

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 $$w \leq z$$.

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

Parents:

• Partially ordered set

A set en­dowed with a re­la­tion that is re­flex­ive, tran­si­tive, and an­ti­sym­met­ric.