I am reading Gratzer's General Lattice Theory and one of the exercises is to give an example of a poset with maximal elements in which not every element is included in a maximal element. I am struggling to see how this is possible. An element that is incomparable to all other elements would be a maximal element by default, so how can an element that is comparable to another element not be included in a maximal element?
The only possibility I can see is if you had a finite chain together with an infinite chain. So you would have a maximal element in the finite chain, but no maximal element in the infinite chain.
Are there any other examples?
Take a chain of order type $\Bbb Z$ and add an incomparable element "on the side". The incomparable element is maximal; but no element in the $\Bbb Z$ part is below a maximal element.