  In recent years it has been noted that a number of combinatorial structures
such as real and complex hyperplane arrangements, interval greedoids, matroids
and oriented matroids have the structure of a finite monoid called a left
regular band. Random walks on the monoid model a number of interesting Markov
chains such as the Tsetlin library and riffle shuffle. The representation
theory of left regular bands then comes into play and has had a major influence
on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection
between algebraic and combinatorial invariants of a left regular band by
showing that certain homological invariants of the algebra of a left regular
band coincide with the cohomology of order complexes of posets naturally
associated to the left regular band.
  The purpose of the present monograph is to further develop and deepen the
connection between left regular bands and poset topology. This allows us to
compute finite projective resolutions of all simple modules of left regular
band algebras over fields and much more. In the process, we are led to define
the class of CW left regular bands as the class of left regular bands whose
associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and
important class of examples is a left regular band structure on the face poset
of a CAT(0) cube complex. Also, the recently introduced notion of a COM
(complex of oriented matroids or conditional oriented matroid) fits nicely into
our setting and includes CAT(0) cube complexes and certain more general CAT(0)
zonotopal complexes. A fairly complete picture of the representation theory for
CW left regular bands is obtained.
