Talk:Semigroup

Closedness

I happen to think this article is pretty ambiguous about closedness of the set under its operation. It was only upon getting to the mention of magmas where closedness was mentioned, that it became apparent the set would be closed under its operation. By way of adding clarity - the positive integers without zero are a semigroup, being a monoid with the identity deleted. However the nonzero integers are ALSO a monoid with the identity removed. But they're not a semigroup because the sum of any element and its inverse would not be in the set. — Preceding unsigned comment added by 92.24.18.49 (talk) 16:13, 24 August 2022 (UTC)

"Totality" for group-like structures

The chart of group-like structures has a column for "totality" which links to the idea of a total function. But it's not quite clear how this applies to algebras. The fact that categories are classified as not possessing "totality" only perplexes me. Can someone illuminate this in the article?

In fact, it might be good to give the reader a tour of that little chart. The adjacent section says almost nothing directly about the chart. Ezrakilty (talk) 03:23, 16 March 2011 (UTC)

I think it would be better to get rid of the table. In any case, this is not the article in which to explain it. As for "totality", this is intended to mean that the operation can be applied to any ordered pair of elements. In a category, the "elements" are the morphisms, and the operation is composition of morphisms, so totality fails if there is more than one object. --Zundark (talk) 09:01, 16 March 2011 (UTC)

Semigroupoid=semicategory?

The section on generalizations mentions semigroupoids, but the accompanying table mentions "semicategory" at what would appear to be the corresponding place.

While I'm mentioning apparent inconsistencies, the section on semigroup applications in PDE's seems to have some as well. I think that H² should be L² and that A in the final set of equations should be D. Marc van Leeuwen (talk) 13:59, 21 September 2013 (UTC)

Making the article accessible

I have an engineering degree from MIT, and this article is still pretty much incomprehensible at first read. A visualization or an elementary worked-through example or two would be enormously helpful. Having lots of prose with mathematical symbols makes it very hard to unpack. -- Beland (talk) 19:55, 19 December 2014 (UTC)

Well, we could copy a visual example from Semigroup_with_two_elements. The catch is that methods of semi-group theory are in general rather different from those of group theory. So it's a strange new world the first time you read through it in any presentation. 86.127.138.67 (talk) 07:34, 12 April 2015 (UTC)

Actually since this page contains little more than definitions, can you clarify where you got lost? 86.127.138.67 (talk) 10:08, 12 April 2015 (UTC)

Yes, please help make this article easier to understand for those who are not math specialists! I have a PhD in Computer Science, and I've certainly had reasonable exposure to math and logic over the years, but I too 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 brought me here), 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) 16:29, 17 April 2015 (UTC)

I rewrote the lead. Let me know what you think. 86.127.138.67 (talk) 10:22, 18 April 2015 (UTC)

I like it. I split the last paragraph, which was quite large, into its three natural parts. I also added the "flip-flop" monoid to the examples section - it is very simple and easy to understand even for complete beginners, it is very different than the other examples, and it is fundamentally important for the structure of finite semigroups due to its central role in the Krohn-Rhodes theorem. — Preceding unsigned comment added by StormWillLaugh (talk • contribs) 19:06, 17 February 2020 (UTC)

Other uses of the term

Semigroup is sometimes employed for what we (and most extant mathematicians) call a monoid; e. g., as in "numeric(al) semigroup" employed as synonym for "submonoid of $(\{0,1,2,3,\ldots \},+)$"). I do not know whether historically there were furter deviations from our definition. However, I think any alternative usage (within algebra) should be mentioned in a short paragraph; and the usage "synonymous with monoid" probably even should be mentioned in the lead. JoergenB (talk) 16:58, 4 November 2023 (UTC)