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458:. 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
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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
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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
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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?
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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
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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
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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. --
512:, 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).
673:"). 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.
523:: (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
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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.
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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
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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! --
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The section on generalizations mentions semigroupoids, but the accompanying table mentions "semicategory" at what would appear to be the corresponding place.
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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.
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While I'm mentioning apparent inconsistencies, the section on semigroup applications in PDE's seems to have some as well. I think that
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semigroup generalizes a monoid", so I had to look up "monoid" . . . except that I gave up at that point. (Stack overflow?)
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think it is important to introduce the jargon that is used in the field, but it should be fairly clearly separated from
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they face the prospect of a deeply nested recursive traversal through many pages of jargon-filled definitions.
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providing a layman's (approximate) definition, so that readers can get the gist of what the term is about
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important for the structure of finite semigroups due to its central role in the Krohn-Rhodes theorem.
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Actually since this page contains little more than definitions, can you clarify where you got lost?
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Knowledge. If you would like to participate, please visit the project page, where you can join
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composition of morphisms, so totality fails if there is more than one object. --
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is sometimes employed for what we (and most extant mathematicians) call a
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plain language definitions help enormously, even if they are imprecise
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