I say all this as someone who personally prefers the term “magma”, by the way. Perhaps the next time Mathematical Reviews carries out one of their classification revisions, someone should lobby for MSC 20N02 to be called “Magmas”.

Wikipedia, which must be a respectable source if an outstanding mathematician like John Baez cites it, says a pseudogroup “is not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of homeomorphisms defined on open sets U of a given Euclidean space E or more generally of a fixed topological space S.”

Is this another mess like the two sorts of groupoids, and I just haven’t seen the other pseudogroups floating around in the literature?

The only instances that I know of where pseudogroups appear in the context of universal algebra apparently take over the usual topological concept of pseudogroups, although I can never quite imagine how any sort of group emerges on the other side of the universal-algebra grinder that breaks down the binary operation of groups into a weird mix of nullary, unary, and binary operations.

There’s obviously something behind your all-too-brief comment that I don’t understand, and maybe you can provide a link to some manifestation of pseudogroups that doesn’t involve higher-order topological operations like homeomorphisms, or at least presents them in a way that makes the reduction of the relevant set of operations on homeomorphisms to something simpler than a magma more transparent.

At a level far below homeomorphisms in the hierarchy of topological concepts, even some components of the Wikipedia definition of a pseudogroup as simple as inverses may not exist for any element of a smegma, where the range of leaky operations is absolutely unrestricted. For example, it’s perfectly possible to define a smegma that maps every real number onto Pamela Anderson, and once you arrive at Pamela Anderson, there’s no way back.

Since a magma only conforms to the sole condition of closure under a binary operation, the most obvious way to weaken conditions would allow partial closure, so that the result of a few pairings leaks out of the underlying set.

I can already see a whole theory of non-leaky cycles developing on subsets where repeated application of the “operation” never oozes out of the original domain, but before research on our newly invented mathematical object can begin in earnest, it needs a name, and in analogy with the magma that engendered it, I propose that we call it a smegma.