As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem to 'only' cover groups, rings and fields. Why is this the case? It seems to me you'd want to study simpler structures like semigroups, too. Especially looking at Wikipedia, there seems to be a huge zoo of different kinds of semigroups.

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    $\begingroup$ Well, the thing is, semigroups are not simpler that groups! $\endgroup$ – Mariano Suárez-Álvarez Jan 6 '11 at 14:07
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    $\begingroup$ I am also a computer science graduate. (Although I am now determined to convert to mathematics.) It is true that semigroups are important to computer science. But I think the structures groups/rings/fields are of more importance to mathematicians. Also, if one understands the way of thinking behind these structures, it would not difficult for he or she to understand other structures, such as semigroups. For anyone, majored in computer science or mathematics or any other, abstract algebra tells people to think in the way of "structures". It is the way of thinking that is most benificial. $\endgroup$ – ShinyaSakai Feb 13 '12 at 19:28

There is not that much substantial to say about general semigroups at an introductory level, in comparison to groups, say, and what there is to say at this level (e.g. the theory of Green's relations) tends to rely on a prior understanding of groups in any case. (One thing I have in mind here --- but I am speaking as a non-expert --- is that semi-groups are much closer to general universal algebra than groups, and universal algebra is not, I would say, appropriate as a topic for a first course in abstract algebra.)

The theory of groups lends itself well to a first course, because the axioms are fairly simple but lead fairly quickly to non-trivial theorems, such as the Sylow theorems. Groups (especially group actions) are also ubiquitous, and the kind of counting arguments which one can make are very useful to learn. (It is the inability to make these sorts of counting arguments that causes so many problems when you investigate semi-groups, in comparison to the case of groups.)

The theory of fields is particularly useful in number theory, and Galois theory also provides a nice tie-in with the theory of groups, which in fact served as Galois's motivation for introducing groups in the first place. The theory of fields (especially the part to do with finding roots of polynomials) is also the part of abstract algebra that is closest to high school algebra, so it is not surprising that there is a substantial focus on it.

The theory of rings normally appears both because it is a precursor to field theory (fields are particular kinds of rings, and the polynomial rings $F[x]$ also play an important role in the study of fields), and because it includes many basic examples from mathematics, such as matrix rings, the integers, quaternions, and so on. Rings also play an important role in the study of group representations (via the appearance of group rings), which, even if they don't appear in a first course, are just over the horizon.

Summary: So overall, I think the answer is that groups, rings, and fields are the parts of algebra that are most closely connected to the basic core topics of mathematics, and are also closely integrated with one another. (Groups not immediately obviously so, but because of Galois theory and group rings, for example.) The theory of semigroups, by contrast, doesn't play much role in the rest of mathematics, and the theory that does exist is more complicated than the theory of groups (despite the axioms being simpler).

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    $\begingroup$ I would also venture to guess that groups, rings, and fields have the strongest connections outside of mathematics, for example to physics and chemistry. $\endgroup$ – Qiaochu Yuan Jan 6 '11 at 11:58
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    $\begingroup$ I think that saying that there is "not that much substantial to say" about semigroups is rather sweeping; I would agree that there is not much to be said (or not toof ar that one can go) at an elementary and introductory level, that can rival how far you can go with groups on a first go and with little in the way of background. But there is a lot about semigroups that does not really come from groups (e.g., Isbell's Zig-Zag Lemma, which in the context of all groups reduces to a triviality). That said, I would not cover it in an intro abstract algebra course. $\endgroup$ – Arturo Magidin Jan 6 '11 at 20:17
  • $\begingroup$ You're missing a right parenthesis in the first paragraph. $\endgroup$ – Brennan Vincent Jan 6 '11 at 21:00
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    $\begingroup$ @Arturo: Dear Arturo, Thanks for your comment. You're correct that my statement is too sweeping, and I'll edit to better reflect the meaning (which was that is not that much substantial to say at an introductory level). Best wishes, $\endgroup$ – Matt E Jan 6 '11 at 22:06
  • $\begingroup$ This answer have a pedagogical point of view and I'm sorry if I don’t trust in it. Of course the path of learning is important in book writing but there is a lot of research done with these structures, and I don't think researcher were thinking on teaching in each one. $\endgroup$ – Ernesto Iglesias Feb 13 '17 at 13:35

Historically, the first "modern algebra" textbook was van der Waerden's in 1930, which followed the groups/rings/fields model (in that order).

As far as I know, the first paper with nontrivial results on semigroups was published in 1928, and the first textbook on semigroups would have to wait until the 1960s.

There is also a slight problem with the notion of "simpler". It is true that semigroups have fewer axioms than groups, and as such should be more "ubiquitous". However, the theory of semigroups is also in some sense "more complex" than the theory of groups, just as the theory of noncommutative rings is harder than that of commutative rings (even though commutative rings are "more complex" than rings because they have an extra axiom) and the structure theory of fields is simpler than that of rings (fewer ideals, for one thing). Groups have the advantage of being a good balance point between simplicity of structure and yet the ability to obtain a lot of nontrivial and powerful results with relatively little prerequisites: most 1-semester courses, even at the undergraduate level, will likely reach the Sylow theorems, a cornerstone of finite group theory. Semigroups require a lot more machinery even to state the isomorphism theorems (you need to notion of congruences).

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    $\begingroup$ it's clear that you don't believe it either, but the idea that one structure is "more complex" than another because it has "more axioms" is kind of silly. It is vaguely reasonable that relational structures with more relations are more complicated: i.e., rings are more complicated than groups which are more complicated than sets. But for structures in a fixed language, the more axioms you require, the smaller the class of models becomes. So commutative groups are simpler than groups which are simpler than monoids which are simpler than semigroups which are simpler than magmas... $\endgroup$ – Pete L. Clark Jan 6 '11 at 22:33
  • $\begingroup$ @Pete: Yes; I did use scare quotes on purpose. (-: The fewer axioms, the more models, but the less you have to "grab a hold of" when studying them (hence, 'the structure theory of fields' is easier than 'the structure theory of rings'; and we have a classification theorem for finitely generated abelian groups, but only the coclass theorems for finite $p$-groups by way of comparison). $\endgroup$ – Arturo Magidin Jan 6 '11 at 23:51

Historical inertia. A relatively small number of people were responsible for more or less deciding the modern abstract algebra curriculum around the beginning of the 20th century, and their ideas were so influential that their choice of topics is rarely questioned, for better or for worse. See, for example, Section 9.7 of Reid's Undergraduate commutative algebra.

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    $\begingroup$ You make it sound like some dark cabal is out to get us! :) $\endgroup$ – Mariano Suárez-Álvarez Jan 6 '11 at 14:07
  • $\begingroup$ Well, there was already a great answer to the question "why should abstract algebra books mostly deal with groups, rings, and fields" so I thought a perspective on the question exactly as it was written ("why do...") would be valuable. $\endgroup$ – Qiaochu Yuan Jan 6 '11 at 14:24
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    $\begingroup$ Here is a specific example. Gian-Carlo Rota (I think it was) once wrote that it was very annoying that the typical first-year algebra curriculum does not mention tensor products, but in various other courses it is more or less taken for granted that you know what a tensor product is. And he attributes this to historical inertia in the choice of topics. $\endgroup$ – Qiaochu Yuan Jan 6 '11 at 14:25

Let me mention that my first (undergraduate!) course in algebra did start with the study of monoids.

Actually even that is not true. It started with a categorical perspective, especially the "quotient principle": when one map factors through another. At some point early on adjoint functors were introduced. And then, maybe 2-3 weeks into the course, we started talking about monoids. I especially remember the free monoid on the set $X$, which was referred to as the James construction. A childhood friend -- who is, by one of life's little coincidences, also a mathematician; at the time he was an undergrad two years ahead of me -- came to visit me in college about a month into the quarter and he was very surprised at what he saw in the course. Finally we started in on groups about five weeks in. (The instructor, Arunas Liulevicius, was famous for doing this sort of thing and sort of making it work due in part to his great enthusiasm and friendly manner.)

So the concept of monoids has been with me for essentially my entire mathematical career. I do think of this as a (slight) advantage, since a lot of people seem simply not to have this word in their vocabulary and go so far as to snicker when someone else says it. This is ridiculous because they are a natural structure which comes up all the time, and for instance the group completion of a commutative monoid should be part of every mathematician's toolkit.

However, I agree that the general study of monoids seems not to be as rewarding as that of groups, rings, fields (or modules). As evidence towards this, I have some lecture notes on semigroups and monoids which I wrote up a couple of years ago. But they didn't seem to be going anywhere interesting any time soon, so I stopped at $11$ pages and may or may not get back to them later.

All in all, this may be a situation in which it is good to give the definition, a few easy examples of the definition, and then move on and allow the student an opportunity to see the structure recur in their later mathematical experience.

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  • $\begingroup$ On the other hand, specific types of monoids are useful in computer science and combinatorics, and monoids in categories other than Set are everywhere, e.g. monoids in Ab, monoids in Vect, monoids in Cat... $\endgroup$ – Qiaochu Yuan Jan 6 '11 at 16:05
  • $\begingroup$ Good old Arunas. I picked up some of his lecturing techniques. "this is a test of eyesight..." $\endgroup$ – Matthew Leingang Jan 6 '11 at 16:31
  • $\begingroup$ Hi, Matt! Yes, I say that in lectures myself too, although I have never been able to muster as much glee as he had when he said it. $\endgroup$ – Pete L. Clark Jan 6 '11 at 17:32
  • $\begingroup$ What does "this is a test of eyesight?" mean? $\endgroup$ – MJD Mar 1 '14 at 15:20
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    $\begingroup$ @MJD: It is what you say when you reveal that the answer to the question you've asked is already somewhere on the board. $\endgroup$ – Pete L. Clark Mar 2 '14 at 2:01

The bias against less-fashionable and/or "applied" algebra in older abstract algebra textbooks is partly a result of tradition, and partly due to historical prejudices - not only against applied algebra but also closely related fields, e.g. combinatorics. Younger mathematicians may not be aware that, not too long ago, work in computational algebra and combinatorics got little respect from many pure mathematicians. For example, such branches of math were so shunned that leading algebraic geometers kept it secret that they were gaining insights by passing Gröbner basis calculations to the computer science department.

Thankfully nowadays most of those prejudices are old history, with a couple generations of mathematicians having been raised in the age of the computer being exposed first-hand to many diverse applications of algebra (e.g. Gröbner bases, cryptography, coding theory). Further, various authors have championed related fields that deserve respect, e.g. Rota's school of combinatorics. The pioneers knew long ago that these fields would soon come to the fore. For example, below is a pertinent excerpt from a Zbl review of the first (1974) edition of Hungerford's Algebra by a leading researcher in semigroup theory (B. M. Schein).

The book would have certainly been one of the best manuals of algebra if it had been published fifteen or more years ago. However, nowadays it seems to be out-of-date. What is said in the book is said clearly and well and is read with intellectual pleasure. However, the title of the book is misleading for no really modern book on algebra can be considered even remotely complete and presenting an undistorted picture of algebra as it is, unless it contains chapters on lattices and semigroups. The present book lacks both chapters. It is patterned after the famous van der Waerden book but it disregards a major change which occurred after the revolution begun by van der Waerden. One cannot but agree with G. Birkhoff [Amer. Math. Monthly 80, 760-782 (1973)] when he says: "No longer do axioms and deductive systems, patterned after Euclid's Elements, seem so fundamental. Neither do groups or rings, with their subgroups, subrings and morphisms... Instead, the kinds of algebraic structures ... which are most relevant to digital computers and combinatorics are loops, monoids and lattices (or groupoids, semigroups and semilattices), which were largely ignored by most algebraists in 1930-1960".

If you study the history of mathematics you will soon realize that fashions come and go in mathematics just as they do in Paris or Milan. Try as we may, we are powerless to predict which little-known esoteric branch of math will soon come to the fore to play a crucial role in some application. Even Hardy's pure haven of number theory now has significant applications in cryptography. Physicists continue to marvel at the The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

So, in summary, don't let the prejudices of the past wrongly influence your studies of algebra and mathematics. Nowadays there can be no doubt that the the theory of semigroups, lattices, and combinatorics play fundamental roles in computer science (as well as mathematics).

See also below, from this deleted thread which, alas, cannot be undeleted due to a deficiency in platform design.

Speaking very roughly, the sort of "patterns" or structure that are abstracted in mathematics need not have any real-world counterparts. Let's consider a simple subject - elementary number theory. One pattern that lies at the foundation is that any nontrivial set of integers closed under subtraction has a very simple structure. Namely, it is the set of all multiples of its least positive element. Most all of the basic results of elementary number theory can be quickly derived from this simple fact (whose proof is a straightforward consequence of division with smaller remainder).

This simple pattern appears in many places, e.g. possible denominators $\,n\,$ of a fraction $\,q\,$ are also closed under subtraction $\,nq,kq\in\Bbb Z\,\Rightarrow\, (n\!-\!k)q = nq-kq\in\Bbb Z,\,$ so the least denominator divides every denominator (so it is unique). $ $ Similarly in groups, the set of $\,n\,$ such that $\,a^n = 1\,$ is closed under subtraction $\,a^n = 1 = a^k\,\Rightarrow\, a^{n-k} = a^n a^{-k} = 1,\,$ hence the least positive such $\,n,\,$ called the order of $\,a,\,$ divides every such $\,n.\,$

This simple structure plays such a fundamental role in group theory and number theory that mathematicians have abstracted it into various forms that makes it easily recognizable, viz. the theory of cyclic groups, and the ideal theory of Euclidean domains, whose ideals are all principal, i.e. the set of all multiple of the "smallest" element.

There are also many common divisibility patterns, e.g. the universal properties of gcd and lcm

$$\begin{align} & a\mid b,c \iff a\mid \gcd(b,c)\\ & b,c\mid a \iff {\rm lcm}(b,c)\mid a \end{align}\qquad$$

Thus whenever ones sees the pattern on the LHS, it may be replaced by the pattern on the RHS. This, combined with the basic laws of gcd and lcm, goes a long way towards solving many divisibility problems, e.g. see various posts here on gcds and on lcms.

In mathematics we strive to abstract out these fundamental patterns in their most general form. Doing so maximizes the chance that they will be easily recognizable in diverse contexts, and hence can be efficiently reused. Though the recognition and abstraction of these non-real-world patterns is probably not something that evolution has trained us for, one can become quite proficient at such. There have been studies of such symbolic reprogramming of our innate mental faculties, e.g. see the classic study Thought and choice in chess by the psychologist Adriaan de Groot. Just as grandmasters have trained their minds to efficiently recognize the many fundamental abstract patterns of pieces that arise in (human) chess games, so too have mathematicians trained their minds to recognize fundamental abstract mathematical patterns.

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Most books on abstract algebra that I've seen tend to start with or want their students to have some sort of number-theoretic background. Lattices, semilattices, Boolean Algebras, De Morgan Algebras, and the like probably have gotten ignored for this reason. Heck, I'd believe some people would consider all of those structures as part of logic, not mathematics (and there there exist disparaging comments about Lattice Theory from people like G. H. Hardy). Also, semigroups don't fit well, since most of the common number systems have more structure than that. The positive natural numbers form a commutative semigroup, but you don't see that in many texts either.

You hit the nail on the head about a huge zoo of semigroups. I tried to do some thinking on this on my own after reading Pete's link, and instead of talking about examples of semigroups, it seemed better to talk about classes of examples of semigroups. With some infinite semigroups I could form subsemigroups on the fly by remembering ordering properties of numbers, and then deleting the lower (or higher) end members (e. g. ([0, 1], $\ast$) is a semigroup with $\ast$ as multiplication, so then delete some interval of real numbers [x, 1] from [0, 1] and since multiplication always decreases, closure holds, and I think that implies association comes for free for such a subsemigroup (see the link). You can't get examples this easily with groups (and I doubt with rings or fields), so perhaps finding examples of groups, rings, and fields seems more impressive.

Additionally, the Schaum's Outline of Group Theory introduces groupoids, then semigroups, then monoids (though it doesn't use that name), and homomorphisms, isomorphisms get expressed in terms of a groupoid first, and Cayley's theorem get discussed for semigroups first also.

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Semigroups and monoids are at the heart of algebra. In fact, the majority of algebra is really monoid theory in different guises. Groups are but monoids whose element are invertible. Rings are monoids along with an "addition" operation (another monoid!) that comes along for the ride. Modules are the actions of such monoids. Algebras are monoids of modules. So, it is quite perplexing to me that the standard algebra text books don't lay the proper conceptual foundations for the student, which are surely in monoid theory. One rare book stands out as a shining exception:

Claude Chevalley: Fundamental Concepts of Algebra, Academic Press, 1957.

The moral is that if any one reads algebra with semigroup/monoid goggles, they will find them pretty much everywhere. For my own personal education and enjoyment, I have been collecting Notes on Semigroups as I read the traditional algebra, which I am happy to make available for general reading. (But, please note that these are written as personal notes for myself and they will change and expand as I go along.)

However, note that there are plenty of standard text books available on semigroups. Some widely used ones are:

Clifford and Preston, The algebraic theory of semigroups, AMS, 1961.

Howie, J. M, An introduction to semigroup theory, Academic Press, 1976.

Higgins, Techniques of Semigroup Theory, Oxford Science Publications, 1992

There are also some very nice algebraic automata theory books that deal with semigroups in considerable detail:

Eilenberg, S., Automata, languages and machines, Academic Press, 1976

Holcombe, W.M.L., Algebraic automata theory, Cambridge University Press, 1982

Pin, J.-E., Mathematical foundations of automata theory, online book for now.

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