Talk:Semigroup action

Merger
FYI, I have merged the entire contents of this article into semiautomaton. At this time, WP has five distinct articles (e.g. transformation semigroup, transition monoid, state transition system, etc.) all talking about exactly the same thing, all using slightly different notations and language. I'm trying to consolidate/uniformize all of these. If there are objections or comments, please leave them at Talk:Semiautomaton linas 14:23, 24 April 2007 (UTC)


 * Five articles was certainly too many. However, there is a definite distinction between a semigroup action and a semiautomaton, so I've partially demerged. Further information at Talk:semiautomaton. Geometry guy 16:40, 14 September 2008 (UTC)
 * This article is written better than semiautomaton. However, every book I've looked in discusses the two notions together. To quote from Grätzer (in a similar context): "the word equivalent will not be defined". VasileGaburici (talk) 17:33, 14 September 2008 (UTC)
 * :-) However, in mathematics, we do try to define what we mean by equivalent, and there is no equivalence here! I suggest we have the discussion at Talk:semiautomaton. Geometry guy 17:39, 14 September 2008 (UTC)
 * George Grätzer is a mathematician, and the quotation is from "General Lattice Theory" where it's used in the context of the two definitions of a lattice (algebraic & order). Indeed the two notions debated here (semiautomaton/semigroup action) are not entirely equivalent. VasileGaburici (talk) 18:26, 14 September 2008 (UTC)
 * Then no doubt he tried, but decided it wasn't illuminating enough to share with his readership :-) Geometry guy 18:49, 14 September 2008 (UTC)

Well, we are back to having at least five different articles all of which say almost nearly the same thing, (but maybe not quite), using almost the same definitions (but maybe not quite) with no overview whatsoever between what makes them similar, and what makes them different. I find this irksome; it's certainly quite tedious to read through these, and try to compare them, to tease out the differences, and go "ah ha, I see now!". 67.198.37.16 (talk) 01:18, 16 October 2020 (UTC)

Original definitions?
Who exactly makes the distinction between M-act and S-act? Kilp, Knauer & Mikhalev do not; Howie's "Fundamentals..." does not mention the notions at all (as far as I can tell). Where are your definitions from? VasileGaburici (talk) 18:51, 14 September 2008 (UTC)


 * I got them from the monoid article. But how can Kilp et al. define an S-act in the same way as an M-act when a semigroup has no identity element?
 * Howie is just there for background. Geometry guy 18:54, 14 September 2008 (UTC)
 * K, K & M do not define S-act (as you understand it) at all. They simply don't consider the notion worth defining, so only M-act (as you call it) is defined, but they call it S-act. Please don't consider definitions from other Wikipedia pages as authoritative. I had to fix problems like that before, e.g. there were two distinct pages about semi-Thue system and string rewriting based on an unsourced (and vague) definition for the latter... VasileGaburici (talk) 19:10, 14 September 2008 (UTC)
 * I have spent the day trying to sort out some of the mess in Category:Semigroup theory, including in particular the disaster of an article that was semiautomaton and in return I am getting implied insults, like "hack job" and the implication that I copy Wikipedia pages on trust without having any idea what I am doing. Thankfully, you had the kindness to note in a few places that this article is much better written than semiautomaton was. I hope you might take that as a sign that I do actually know what I am talking about when it comes to mathematics and the state of Wikipedia.
 * I know only too well that Wikipedia pages are not authoritative. I also know perfectly well what an action is, be it of a semigroup, a monoid or a group. Semigroup actions certainly exist in the literature, and they are not and cannot be defined as monoid actions are because there is no identity element in general.
 * The definition of a monoid act/action from monoid is the KK&M one (and utterly standard), so no problem there. The S-act and M-act terminology is awful, because S and M are just letters, and could be anything. KK&M no doubt call an M-act an S-act because their monoids are denoted by the letter S. Indeed many authors use the term semigroup for a monoid. I kept some of the terminology from semiautomaton and its redirects as a courtesy to previous editors, assuming their good faith, and having found similar terminology myself in the literature. If any of these multiple synonymous terms are OR, they can go, but I'm afraid most of them probably are used somewhere.
 * This is a work in progress. Take a look at syntactic monoid for example to see just how bad things are. Geometry guy 20:17, 14 September 2008 (UTC)


 * PS. Although I don't have access to a library today, I have been using the web to check things. 10 seconds googling for "Semigroup action arxiv" produced this article. See page 2. Geometry guy 20:25, 14 September 2008 (UTC)

Okay, good, you've clarified what KK&M say. So they distinguish the two notions, but use different terminology. That's fine. After all, a monoid is a semigroup, so one can consider semigroup actions of monoids. I think the article is in safe hands with you. Please excuse my letting of steam above. Geometry guy 20:36, 14 September 2008 (UTC)

I always find it remarkable how clearly Wikipedia editing reveals the lack of unambiguous definitions for many basic concepts. Are there still OR concerns here? Independent of that issue, I think it would be more accessible to simply say "monoid action" instead of M-act, and similar for semigroup actions. If there is a particular fine distinction that has to be made at some point, explain it in detail, but don't rely on the reader's familiarity with abbreviations. Writing that way has the side effect of mollifying any concerns about original terminology. &mdash; Carl (CBM · talk) 13:37, 15 September 2008 (UTC)


 * I agree that the definitions that User:Geometry guy has written are actually more "orthogonal" than what's being used in published sources (see the alternate definitions section for comparison). The only problem with User:Geometry guy's definitions is that you won't find them anywhere else... Granted, nobody is going to write a paper just clear up some definitions, so we're a bit of a gray area here. I don't have Barr & Wells "Category Theory for Computing Science" handy, but it also covers the definitions and the correspondence with semiautomata; I won't have access to that book at least 2-3 more weeks (and I've read it years ago), so if somebody wants to look-up the topic and synthesize some common ground with KK&M, that would be great. Clifford and Preston (1967) is too old, and their terminology is not used anymore... VasileGaburici (talk) 14:37, 15 September 2008 (UTC)


 * Speaking of lack of unambiguous definitions I can only agree with that. See Talk:semiautomaton for more examples... VasileGaburici (talk) 14:37, 15 September 2008 (UTC)


 * I've reworked the definition section so that it avoids using confusing and potentially OR terminology. The definitions of semigroup action and monoid action given can both be found in the literature. The terminology varies from author to author. Geometry guy 19:57, 15 September 2008 (UTC)

semigroup action vs. group action
What makes the differences is not explained. I was trying to find out the differences, but I was not able to find it. Jackzhp (talk) 20:19, 27 March 2011 (UTC)
 * This is covered in Group_action. Every group action of $$G$$ on $$X$$ is a monoid action of $$G$$ (regarded as a monoid) on $$X$$. --Beroal (talk) 14:01, 20 August 2011 (UTC)

Transformation semigroups
0. This chapter repeats stuff that is already given in the same article (e.g. "the semigroup S is a semigroup of transformations of a set X, i.e., a collection of functions from X to itself that is closed under composition. Thus S is a subsemigroup of the monoid of all transformations of X."), but omits the important stuff (how we get a transformation semigroup from a semigroup action, namely, that transformation semigroup is $$\{x\mapsto s\cdot x | s\in S\}$$). Is there a good textbook so I can rewrite the chapter referring to that textbook, i.e. without much original research?

1. IMHO this paragraph is off topic in this article:


 * Any semigroup can be realized as a transformation semigroup via an analogue of Cayley's theorem: after adjoining an identity element if necessary, the action of the semigroup on itself by (say) left multiplication is effective.

It links semigroups and transformation semigroups, while the chapter "Transformation semigroups" in the article "Semigroup action" is supposed to link semigroup actions and transformation semigroups. I want this moved to Transformation semigroup or Semigroup. Actually, it is already there.

--Beroal (talk) 14:42, 20 August 2011 (UTC)


 * done --Beroal (talk) 20:20, 23 August 2011 (UTC)

Incorrect link for transformation term?
I am definitely not an expert in this area, but I nevertheless think that the link for the term 'transformation' is not right. The link is currently transformation, but I think it should be transformation. — Preceding unsigned comment added by 207.182.200.34 (talk) 18:09, 5 May 2015 (UTC)