This talk will be a ‘sales pitch’ for a rapidly growing part of combinatorial semigroup theory. The subject starts with ‘random walk on the chambers of a hyperplane arrangement’ this includes things like the analysis of riffle shuffles and the Tsetlin Library. There is a huge collection of examples with a more or less complete theory. Ken Brown realized that it all abstracts to random walk on idempotent semigroups (especially left regular bands). This in turn has been developed by Ben Steinberg and John Rhodes (with many others). In Parallel, Marcello Aguiar and Swapneil Mahajan have based an enormous abstraction of ‘algebra’ on these developments. This has resulted in an impressive set of four volumes (!) developing concepts like ‘Hopf’ or ‘Lie’ or even ‘Associative’ or ‘Descents’ in real generality. When specified to the ‘type A braid arrangement’ It gives the classical concepts. The generalizations seem useful in quite down to earth ways. All I can do in a single talk is to try to sketch the subject and some examples. I’ve worked on it since the beginning and I’m working on it now (we can fit the popular ‘overhand shuffle’ into this framework and it offers the first steps forward). I’ll try. to say it all in ‘mathematical English’