Yet another question coming from the book Lattice Theory : Foundation, written by George Grätzer.
In the third set of exercises, he proposes the following one :
Find a subset $H$ of a lattice $L$ such that $H$ is not a sublattice of $L$ but $H$ is a lattice under the ordering of $L$ restricted to $H$.
I am confused since I thought that being a lattice under the ordering of its superset restricted to oneself was the definition of a sublattice.
Can anyone tell me what I am missing ?
Thanks again !
No, that's not the definition of a sub-lattice.
It has to verify that property, but moreover, for each pair of elements $a$ and $b$ in $L$, the elements $a \vee b$ and $a \wedge b$, as computed in $L$, must belong to $H$.
Consider, for example, the lattice $L$ in the following diagram:
If you take $H$ to be $L$ except the element which covers the bottom, then $H$ with order inherited from $L$ is a lattice, but not a sub-lattice because, if you name $a$ and $b$ to be the elements covered by the top, then $a \wedge b$ in $H$ is different than in $L$.