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Edit. (Question was bumped again) The book is published, and called "Representation theory of finite monoids." Link to the book at publisher's page

Edit(4/1/15). Since this question just got bumped again, let me add that I am in the process of writing a book on the representation theory of monoids. In a sense I started writing this book because of this question (which was the first MO question I ever answered). Hopefully the book will be an answer to this question. I will make a link available shortly from my blog page.

Edit. Our new paper gives a close connection between monoid representation theory, poset topology and Leray numbers of simplicial complexes with classifying spaces of small categories thrown in. If browsing this paper doesn't convince you that monoid representation theory has something to it, then I don't know what will.

Edit. Our new paper gives a close connection between monoid representation theory, poset topology and Leray numbers of simplicial complexes with classifying spaces of small categories thrown in. If browsing this paper doesn't convince you that monoid representation theory has something to it, then I don't know what will.

Edit. (2/18/14) Since this question just got bumped, let me add the new paper http://arxiv.org/abs/1401.4250 which gives a general introduction to Markov chains and semigroup representation theory and new examples.