Over the last 15 years, it has been noted that many combinatorial structures, such as real and complex hyperplane arrangements, interval greedoids, matroids, oriented matroids, and others have the structure of a left regular band, a certain kind of finite monoid. The representation theory of the associated monoid has had a major influence on understanding these objects along with related structures such as finite Coxeter groups and various Markov processes. In return, this has spurred a deeper development of the representation theory and cohomolgy theory of left regular bands and more general classes
of finite monoids. In particular, the Ext modules between simple LRB modules over a field turn out to be intimately related to the cohomology
of the order complex of the poset of principal right ideals of the LRB and other related simplicial complexes.
These fit into the wider class of LRBs all of whose retractions (certain intervals in the poset) are isomorphic to face posets of regular CW complexes. For this class of LRBs, we can compute a quiver presentation, the global dimension of the algebra and have an analogue of the Zaslavsky Theorem
on counting faces of hyperplane arrangements. Finally, a surprising connection to LeRay numbers and partially commutative LRBs will be discussed.