Multiset is a set that can have more than one of each member for example $\{1,3,3,9\}$ is a Multiset. Let $\mathbb{K}$ be the set of all multisets that has exactly $k$ members. ($k$ is a fixed constant) (The multisets in this question only include numbers). We define relation $\preccurlyeq$ between two multisets $A,B$ as $A\preccurlyeq b$ if and only if there exists an order of elements of A and B like $(a_1,a_2,a_3,...,a_k)$ , $(b_1,b_2,b_3,...,b_n)$ such that $\forall i\leq k ; a_i\leq b_i$ for example $\{1,3,3,9\} \preccurlyeq \{1,3,4,9\} $. Prove that $(\mathbb{K},\preccurlyeq)$ is a lattice.
My approach for this question: For every two multisets, we must introduce a glb and lub. At first, For every two multisets in $K$, I sort the members Decreasingly. then the GLB is the multiset that has $(min(a_1,b_1) , ... , min(a_k,b_k))$ and then for LUB I sort the members increasingly and he LUB is $(max(a'_1,b'_1),...max(a_k,b_k))$. I think my answer makes sense but I can't prove this formally.