I am reading "Introduction to Set Theory and Toplogy" by Kazuo Matsuzaka (in Japanese).
In this book, the definition of a complete lattice is the following:
Let $M$ be a partially ordered set.
If any non-empty subset of $M$ has a supremum and an infimum (in $M$), then $M$ is called a complete lattice.
The definition of a complete lattice in Wikipedia is the following (https://en.wikipedia.org/wiki/Complete_lattice):
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
I think the two definitions above are different.
If the two definitions above are different, then is Matsuzaka' definiton bad?
Let $M$ be a partially ordered set.
Suppose any non-empty subset of $M$ has a supremum and an infimum (in $M$).
Any element of $M$ is an upper bound of $\emptyset$ and any element of $M$ is a lower bound of $\emptyset$.
So, if $M$ has a maximum element, then $\emptyset$ has an infimum and if $M$ has a minimum element, then $\emptyset$ has a supremum.
If $M$ doesn't have a maximum element or a minimum element, then the Wikipedia's definition says $M$ is not a complete lattice and the Matsuzaka's definition says $M$ is a complete lattice.
Is there a partially ordered set $M$ such that any non-empty subset of $M$ has a supremum and an infimum (in $M$) and $M$ doesn't have a maximum element or a minimum element?
Best Answer
These two definitions are equivalent - at least, under the additional assumption that the lattice $M$ itself is nonempty. (Some texts require structures, such as lattices, to be nonempty - but not all do!) The point is that, switching sups and infs, $M$ itself can do what we want $\emptyset$ to do (so to speak): the supremum of the emptyset is the infimum of the whole lattice, and similarly the infimum of the emptyset is the supremum of the whole lattice.
Note that the "inf/sup switch" here is crucial: in a semilattice context we do indeed get two different notions which differ only by the requirement of a greatest/least element (depending on whether we're looking at complete upper or complete lower semilattices). But with both sups and infs, the two definitions in the OP coincide.