TITLE: Poset-troids …?
QUESTION [10 upvotes]: In many respects,

spanning tree : graph :: linear extension : poset

For instance, the number of spanning trees/linear extensions is a measure of the "richness" or "complexity" of the graph/poset. Also, the collection of all the spanning trees/linear extensions determines the graph/poset.
(Okay if the graph is not connected we should say "spanning forest" instead of "spanning tree"; but the tree terminology is more common.)
The collection of spanning trees of a graph (thought of as subsets of edges) is a prototypical example of a matroid. Indeed, the notion of matroid can be seen as an abstraction of this kind of collection.
Question: Has anyone ever defined/investigated a "matroid-like" abstraction of the collection of linear extensions of a poset? E.g., collections of permutations of size $n$ satisfying some compatibility condition?
A wildly optimistic guess would be that these could be related to the full flag variety in the way that matroids are related to Grassmannians; however, I take it that "flag matroids" already have an established meaning in this context, which at first glance does not appear to be related to what I am asking (see e.g. Cameron, Dinu, Michałek, and Seynnaeve - Flag matroids: algebra and geometry).

REPLY [3 votes]: I'm not aware of anything exactly along the lines of your question.  It really depends on what properties you're hoping to generalize.
Here's one direction in which one can generalize.  Promotion and evacuation are operations on the set of linear extensions of a poset.  These operations can be formulated in a more general setting, as Stanley explains in his paper.
Your mention of the relationship between matroids and Grassmannians suggests that you might find the concept of Coxeter matroids interesting, but I don't think Coxeter matroids can plausibly be considered to be generalizations of the set of linear extensions of an arbitrary poset.