1. ## Singular endomorphism.

Hi: where could I find a definition of singular endomorphism? I found this term in a book on finite semigroups. I searched google but the references I found assume the reader knows the term. Regards.

2. Can you find the term singular homomorphism?

3. No. I can only find "singular homology". Perhaps I could disregard the meaning of the term. However, how do you explain this: I quote from "Tutorial - Computing with Semigroups in GAP": "Consider the set [n] = {1, 2, ..., n} with the usual total order. An endomorphism of [n] is a map f: [n] --> [n] such that i < j implies if <= jf. Let O_n denote the semigroup of singular endomorphisms of the chain [n]. It is a simple combinatorial observation that O_n has size...". And here it gives a formula to compute the size. According to this formula, O_1 = 0, O_2 = 2. However, I find that O_1 = 1 and O_2 = 3. For every n, I count one more endomorphism than the formula. So, the word "singular" above must be limiting the number of possible endomorphisms. Thanks for your reply and regards.

EDIT: the formula they give for the size of O_n is $\sum_{k=1}^{n-1}\left(\begin{array}{c}{n-1}&{k-1}\end{array}\right)\left(\begin{array}{c}n&k\end{ array}\right) - 1$. From this it could, perhaps, be possible to infer what is the set of mappings that is being considered. This is what I am trying to do.

4. I'm afraid I haven't studied semigroups at all. I hope to some day, but that's not yet. Anyone else have an idea?