Is a Lattice just a linearly ordered set?

Question

The definition of Lattice, I was introduced to recently is a poset in which every two elements had a supremum and infimum. Doesn't this just mean that the poset is a chain and thus a linearly ordered set? Is there a problem with this definition of Lattice or my logic is flawed?

Answer 1

For a set to be linearly ordered, every two elements must be comparable. Not every two elements are necessarily comparable in a lattice. For example, the lattice of subsets of a set.

Answer 2

No. The poset of subsets of any set, $\mathcal{P}(X)$, has supremum given by union and infimum given by intersection, so it is a lattice; but (provided $X$ has at least two distinct elements) it is not linearly ordered, since $\{x\}$ and $\{y\}$ are incomparable if $x \neq y$.

However, a simple strengthening of the lattice condition is indeed equivalent to the order being linear. If a partial order $P$ has the property that for any two elements $x,y$, there is a supremum $x \vee y$, and additionally $x \vee y \in \{x,y\}$, then $P$ is a linear order. (Proof: to see that $x,y$ are comparable, we know that either $x \vee y = x$, in which case $x \geq y$, or $x \vee y = y$, in which case $y \geq x$.) Conversely, any linear order has this property, taking the pairwise supremum to just be whichever element is larger. And, of course, all the same goes through for the dual property stated in terms of infima.