Texts on the theory of partial algebras.

Question

In all the texts on universal algebra I have read, I have not seen any mention of partial operations. Partial operations are sometimes important in mathematics, like the operation of division on real numbers, or the operation of subtraction on naturals. Is there a book or paper somewhere that develops the theory of algebras with partial operations, like universal algebra texts do with algebras with total operations?

Answer 1

I've been interested in similar questions. I haven't come across any such book.

However, there is the notion of an inverse semigroup which model the algebra of partial maps on a space by partial compositions and there are books devoted to that. For example:

• Petrich, Inverse Semigroups

Amd

• Lawson, Inverse Semigroups: The Theory of Partial Symmetries

Unfortunately, I haven't looked at either (they are on my to-read list) so please don't take this as a recommendation. They can be categorified into Partial categories. There is also the paper:

• Ruy Exel, Partial actions of groups and actions of semigroups

Which might be worth taking a look at. It's available on the Arxiv as article 9511003