Talk:Semigroup with involution

So
The extremely tedious section Semigroup_with_involution is trying to say that the only free semigroup with involution is basically the one where word reversal is the involution? JMP EAX (talk) 22:22, 23 August 2014 (UTC)
 * Actually, that wasn't the case. The rewritten section explains the distinction (correctly, I hope). To get example 6 you need to take the quotient of the free monoid with involution by identifying every letter with its own involution/inverse. 86.127.138.67 (talk) 08:36, 21 April 2015 (UTC)

Please make this article easier to understand by non-mathematicians!
I have a PhD in Computer Science, and I've certainly had reasonable exposure to math and logic over the years, but I find this article unbearably difficult to unpack. The main problem is the high density of jargon and the domain-specific notation. Obviously the reason for that is precision, which is important also. But the problem is that a definition of an unfamiliar term is often given in terms of 2-3 other unfamiliar terms, which in turn are defined in terms of more unfamiliar terms, etc.. I find myself diving through several layers of dependency definitions -- each with a different wikipedia page -- just to unpack one definition. And the problem with specialized notation is that it is not even clear how to look up the definition of a symbol such as "*" or $$\in$$ (which I do know, but I'm just using as an example). Ask yourself: What would a reader type into a google search, to find out what the $$\in$$ symbol means? Examples help a lot. And plain language definitions help enormously, even if they are imprecise, though of course they should be clearly noted as being imprecise. The key point in a plain language definition is to avoid domain-specific jargon. For example: "Roughly speaking, a foo is . . . ". And later: "More precisely, a foo is . . . " (with full rigor and jargon). As a case in point, I was just looking up the definition of "inverse relation" on wikipedia, and the explanation talked about a "semigroup with involution", so I had to look up that, which was defined in terms of a "semigroup", so I had to look up that page, which says that "A semigroup generalizes a monoid", so I had to look up "monoid". . . except that I gave up at that point. :( (Stack overflow?) There are two main use cases that a page like this should address, and they are different: (a) someone runs across an unfamiliar term and reads the page to get a rough idea of what that term means; and (b) someone wants to dig deeply and precisely into the meaning of the term.  The (possibly imprecise) plain language definition should be given first, free of jargon.  The gory details and jargon should come later.  I do think it is important to introduce the jargon that is used in the field, but it should be fairly clearly separated from first providing a layman's (approximate) definition, so that readers can get the gist of what the term is about before they face the prospect of a deeply nested recursive traversal through many pages of jargon-filled definitions. I hope the above suggestions are helpful and don't just sound like complaints. I know it is hard to write such things in widely understandable ways, and I very much appreciate the efforts of all editors who contribute. Thanks! -- DBooth (talk) 16:36, 17 April 2015 (UTC)
 * In general learning mathematics from Wikipedia is not a good idea if one is clueless about basics. I have to say that while it's possible to get a compsci PhD (even from MIT) without ever learning the definition of a semigroup, that doesn't imply mean the math pages here need a lot of changes. Semigroup theory is not commonly thought even to math PhDs, although they'd probably at least have heard of the definition. The page for semigroups does however formally define it. Your complaint doesn't seem have anything specific about this page, which should definitely not repeat the definition of a semigroup. 86.127.138.67 (talk) 06:05, 18 April 2015 (UTC)
 * Also the lead of this page contains two examples that anyone with PhD in any science should have little trouble understanding. 86.127.138.67 (talk) 06:09, 18 April 2015 (UTC)
 * Also, complaining is easy. I invite you try and explain the formula (xy)* = y*x* in plain English. Not even the WP:FA on groups does that. I've actually spent a good number of minutes trying to phrase that in words, but I think it's a pretty pointless exercise. 86.127.138.67 (talk) 06:54, 18 April 2015 (UTC)

Actually, while there are plenty of examples here for the basic notion, the section Semigroup_with_involution introduces plenty of sub-concepts and alas exemplifies them only with the (homonymous) notions from C*-algebras, which may be unfamiliar to someone who came here (say) from the set theory example that is the inverse relation. So that section of the article should be expanded by instantiating those notions from a more accessible incarnation of a *-semigroup (like those mentioned in the lead). Alas, given the relative obscurity of the topic, the only treatments appear in [a handful of] graduate-level textbooks, so it may require a bit of WP:OR to add more accessible examples to that section. 86.127.138.67 (talk) 13:18, 18 April 2015 (UTC)
 * Well, a few pages later, Lawson covered that, so... I've added it. Also, it occurred to me that the *-semigroup of binary relations is a rather trivial case of the full linear monoid on binary matrices (i.e. over F2) representing relations. 86.127.138.67 (talk) 14:51, 18 April 2015 (UTC)

Most difficult section: *-regular semigroups
The study of these appears to have been motivated by some additional properties of the Penrose-Moore inverse [and Mn(C)] that aren't shared by *-semigroups at large. This section is the most newbie-unfriendly because is [first] states things in terms of Green's relations so those are a prerequisite in order to grasp what's going on there. (The axiomatic definition that follows doesn't help much.) Besides that, it's rather unfortunate (if I got this right) that *-regular semigroup doesn't just mean a *-semigroups that's also a regular semigroup. Due to the Google-unfriendly name that *-regular semigroups have, it's hard to locate the relevant literature on them... 86.127.138.67 (talk) 09:45, 19 April 2015 (UTC)

As far as I can see a *-regular semigroup is purely and simply an inverse semigroup (the inversion need not be the only involutory anti-isomorphism but if a regular semigroup has an involutory anti-isomorphism then the inverses are unique and so it is an inverse semigroup, which is also a '-regular semigroup with the inversion ' as "the" *.

If I am wrong it would be good to mention a counter-example.

If correct then it should be clearly stated (and in consequence the Moore-Penrose inverse is just the inverse in an inverse semigroup, which should also be clearly stated). 121.210.225.141 (talk) 06:00, 6 August 2015 (UTC)