if a set B has a least upper bound of an on an ordered relation <(a) then will it have least upper bound on an ordered relation <(b)

Question

given a set B ordered by a relation <(a) has a least upper bound property, does B have an least upper bound property if it is ordered by another ordered relation <(b).

Answer 1

Not necessarily. Let $B=\mathbb R$ and $<_a$ the standard order (which has the least upper bound property). Let $<_b$ be defined by $$x<_by\iff x<y\lor (x\notin\mathbb Q\land y\in\mathbb Q).$$ Then $\{\,x\in\mathbb Q\mid x^2<2\,\}$ has $42$ as upper bound, but no least upper bound. Also, the set $\mathbb R\setminus\mathbb Q$ has $-666$ as upper bound, but no least upper bound.