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I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of two sets is their product), satisfying the condition that there is a unique minimal element in this finite poset (to make sure multiplication is well-defined), and also satisfying the condition that there is a unique maximal element in this semigroup if we want it to have an identity. (Unless my reasoning is wrong?)

I was wondering if there is a standard representation theory for such semigroups covered in a paper somewhere. I've thought about it a bit and reduced it to a combinatorial problem, but I'd rather not spend time on it since it is almost certainly a well-known (and easy probably) result.

Is there a more general representation theory for bands available? What about representation theory of rectangular bands? (which I presume is slightly more challenging, unless the problem above grows much more difficult as the poset grows more complicated, which is also a possibility).

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3 Answers

up vote 4 down vote accepted

Firstly, if you're only interested in finite semigroups, I suggest amending the title of the question ;)

Commutative semigroups in which ever element is idempotent are called semilattices, and are a special case of the more general notion of inverse semigroup. They have been much studied, although their infinite-dimensional representation theory seems slightly less well mapped out. In any case, the correspondence you describe between semilattices as semigroups and semilattices as certain kinds of poset is indeed well known.

If you're only interested in finite semilattices, then I think the paper to look at is one by Solomon where he gives explicit formulas for the central idempotents that generate the semigroup ring of a given finite (semi)lattice. Unfortunately I've forgotten the exact reference, but I think similar versions or improvements are discussed in

C. Greene, On the M\"obius algebra of a partially ordered set, Adv. in Math. 10 (1973), 177--187.

G. Etienne, On the M\"obius algebra of geometric lattices, Eur. J. Combin. 19 (1998), 921--933.

(Off the top of my head, for representations of more general finite semigroups, see recent work of Benjamin Steinberg, which is where I first became aware of some of the older work.)

I would have thought that the representation theory of rectangular bands should actually be much easier than that of general bands, since every rectangular band can be rewritten as $L \times R$ for index sets $L, R$ and with multiplication defined by $(l_1,r_1)\cdot (l_2,r_2)=(l_1,r_2)$.

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Oh right thanks, finding idempotents in the semigroup algebra for these semi-lattices is exactly what I'd ideally like to find out (it would be relevant to a problem I'm working on about representations of reductive groups over finite rings). Thanks very much. Right yeah rectangular bands probably has an "easier" representation theory than that of the general band; I was comparing it with the representation theory of semi-lattices (sorry bad wording) and presuming it was harder (since rectangular bands are non-commutative), but that is probably also false. –  Vinoth Nov 29 '09 at 22:23
    
Would anyone who downvoted like to suggest how the answer could be improved, or where it is deficient? –  Yemon Choi Nov 11 '11 at 22:23
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The most natural representations of semilattices are by subspace arrangement: assign a subspace of a vector space to each element, intersection should correspond to intersections. That has been considered here, for example: Sapir, Mark V., Scheinerman, Edward R. Irrepresentability of short semilattices by Euclidean subspaces. (English summary) Algebra Universalis 31 (1994), no. 4, 599--607. Since idempotent matrices correspond to subspaces (projections), these kind of representations are very close to the ordinary ones. The theory of linear representations of finite inverse semigroups have been started by Munn and Ponizovsky in the 50s. See, for example, Ponizovskiĭ, I. S. On matrix representations of associative systems. Mat. Sb. N.S. 38(80) (1956), 241--260., Munn, W. D. Matrix representations of semigroups. Proc. Cambrdige Philos. Soc. 53 (1957), 5--12.

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The best place to read about the representation theory of bands, as well as find exciting applications, is in Ken Brown's beautiful survey paper K. Brown, Semigroup and ring theoretical methods in probability, Representations of finite dimensional algebras and related topics in Lie theory and geometry, 3–26, Fields Inst. Commun., 40, Amer. Math. Soc., Providence, RI, 2004. Of course the representation theoretic results about bands can be found scattered in the semigroup theory literature as special cases of more general results, but Ken rediscovered it himself and wrote it down beautifully.

I do agree with Mark, though, that representing semigroups by partial linear maps should lead to an even more fascinating theory.

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