Lower bound in a partially ordered set ($\ell \le s\text{ for every }s\in S$)

A lower bound of a subset $S$ of a partially ordered set is an element $\ell$ no greater than every member of $S$. For minimization, a number below every feasible objective value is therefore a lower bound on the optimum; a relaxation often provides such a certificate.

// Target: mathematical-optimization.bigb

A lower bound in a partially ordered set which is greater than or equal to every other lower bound. For a pair it is written $a\wedge b$; an empty meet, when it exists, is the greatest element. In a complete lattice, every subset has a meet. In the open-set frame, this is the interior of the intersection, with finite meets given by the intersections themselves.