An introduction to lockless algorithms

* Git history is a poset (under the relationship "commit X is an ancestor of commit Y").
* The packages in a (sufficiently powerful) package manager are a poset (under the relationship "package X is a transitive dependency of package Y").
* In languages with OOP and inheritance, the set of classes (or classes and interfaces) is a poset (under the relationship "class/interface X is a direct or indirect subclass/subtype of class/interface Y").

In short: A poset is a thing that lets you have some objects come "logically before" other objects, for whatever definition of "logically before" is appropriate.

Then, a totally ordered set is just a poset with the additional requirement that for every X and Y, either X comes before Y or Y comes before X (or X = Y). In the DAG interpretation, this is what you would get if you require that the DAG forms a simple chain A -> B -> C ... -> Z, without any branching or whatnot allowed. So, to continue our examples:

* Git history is totally ordered, if you never create any branches or merges, never use detached HEAD mode, and never create a second root commit (i.e. a commit without parents, other than the first commit in the repository).
* Packages are pretty much never totally ordered, in real-world cases. But maybe, if you wanted to ensure a totally deterministic installation order, you could insert artificial dependencies to force the package manager to install everything in the exact order you want.
* In languages with single inheritance, each individual class's set of superclasses (not counting interfaces) is totally ordered. But the language as a whole is not, in most reasonable cases.