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.