Interval containment order ($[l,u]⪯[l^{\prime },u^{\prime }]⟺l^{\prime }\le l\wedge u\le u^{\prime }$)

The partial order on nonempty closed integer intervals given by subset inclusion. The least upper bound of two such intervals is their interval hull $[\min (l,l^{\prime }),\max (u,u^{\prime })]$. The hull can contain points absent from both original intervals, so it need not equal their union.