Is every bounded subposet of a lattice, a lattice?

Question

Let $L$ be a finite lattice. Suppose $P$ is a subposet of $L$ and that $P$ is bounded i.e. there exist $\mathbf{a},\mathbf{b}\in P$ such that for any $\mathbf{p}\in P$, $\mathbf{a}\preccurlyeq\mathbf{p}\preccurlyeq\mathbf{b}$. Must $P$ be a lattice?

Answer 1

Let $L$ be the lattice of subsets of the set $S=\{1,2,3,4\}$ and let $P=\{X\in L:|X|\ne2\}.$ Then $\{1\},\{2\}$ are elements of $P$ with upper bounds $\{1,2,3\},\{1,2,4\},\{1,2,3,4\}$ but no least upper bound in $P.$