In reference to Sub-lattices and lattices.
In Wikipedia, it's given that-
A sublattice of a lattice L is a nonempty subset of L that is a lattice with the same meet and join operations as L.
That is if L is a lattice and $M ≠ {\displaystyle \varnothing } $ is a subset of L such that for every pair of elements a, b in M both a ∧ b and a ∨ b are in M, then M is a sublattice of L
In Discrete Mathematics and It's Application - Kenneth Rosen, $7^{th} \ Indian \ edition$, it's given that -
A sublattice of a lattice L is a subset $S \subseteq L$ such that if $a,b \in S, \ a \wedge b \in S \ and \ a \vee b \in S. $
I understood that $1^{st}$ condition of $M \subseteq L$ to be a sub-lattice is that- for every pair of elements a, b in M both a ∧ b and a ∨ b should present in M.
But is it necessary that M should have the same meet and join operations as L?
Yes, it is. This is implied in the second definition; the symbols $\land$ and $\lor$ are being used to refer to the operations in $L$.