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