Is this a lattice of $6$ elements?

Question

I am reading "Set Theory & General Topology" by Fuichi Uchida.

In this book, the author wrote all lattices($15$ lattices) of $6$ elements.

I wonder the following is also a lattice of $6$ elements, but the following is not in the list written by the author.

Is the following a lattice or not?

Answer 1

Call the two atoms (elements in the lower row) $a$ and $b$, and the two co-atoms (in the upper row) $x$ and $y$.

This is not a lattice because $a$ and $b$ do not have a supremum (join). In particular, $x$ and $y$ are both upper bounds for $a$ and $b$, but are incomparable.