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Numerical Semigroups

J. C. Rosales and P. A. García-Sánchez
Publication Date: 
Number of Pages: 
Developments in Mathematics 20
[Reviewed by
Marion Cohen
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I first became interested in this wonderful book because of its title. It seemed to connect with a problem I’m working on — briefly, find all binary operations # (“generalized addition”) on N (the set of natural numbers) such that both # and the induced binary operation (“generalized multiplication”), x*y = x # x # x…, with y x’s, are associative. (I call it the “characterizing associative arithmetics” problem.) I’ve found necessary conditions but can’t prove sufficiency. Indeed, * can be expressed via what I call its “generalized times tables” (that is, the sequences x*1, x*2, x*3…., a sequence for each x). Such sequences have to take a certain form; for example, they must be “eventually periodic”, the entries “before periodicity,” as well as within each period, must be distinct, and each period forms a (finite) cyclic group with respect to the operation #. Also, x*y must be congruent to the ordinary product xy modulo the eventual-period length. The snag to the attaining of a structure theorem occurs because the sequences must be well-defined — and the obstacles to such defining turn out to be semigroups of natural numbers. I figured that these semigroups were examples of either the “numerical semigroups” of the title, or something closely related.

Also, the single word “semigroups” caught my attention. I remember first learning about semigroups, all too briefly, back in my first Abstract (then called Modern) Algebra course. They were barely mentioned; groups quickly barged in. I wanted to know more about semigroups. On my back burners I probably assumed that additive semigroups of natural numbers were easy to classify (as are groups of integers). But when, in my work on my problem, I investigated, I saw (and this book bears it out) that that is not the case.

This book is a monograph, a summary of papers written on a rather specialized subject. Almost half of the bibliography (which has cardinality 109) is by one or both of the book’s authors. In the body of the book, each topic (in particular, in the introductions to the chapters) is accompanied by explanations as to which reference they appear in. Conversely, the impression is that most of the important results in the references are given and proven in the book, or at least mentioned. What follows are the highpoints (at least according to my perceptions) of the content of the book:

Chapter I, of ten chapters, is titled “Notable elements”. First, monoids are defined (semigroups with identity), then the stars of our show, numerical semigroups — submonoids of N which contain the identity and have finite complement. Here, Lemma 2.1 (p. 7) already gives us somewhat of a picture of what a numerical semigroup looks like: A nonempty subset of N generates a numerical semigroup if and only if the subset has greatest common divisor 1. From this follows Lemma 2: Any nontrivial submonoid of N is isomorphic to a numerical semigroup. (Just divide by its gcd.) Using “what is probably the most versatile tool in numerical semigroup theory” (p. 8), namely for every numerical semigroup S and every positive integer n, Ap(S,n) = {s ∈ S : s–n ∉ S}, Theorem 2.7 (p. 8) is proven: “Every numerical semigroup admits a unique minimal [with respect to set inclusion] system of generators. This minimal system… is finite.” Another important concept is that of Frobenius number F(S) of a given numerical semigroup S; it’s the largest number not in S. Other “notable elements” include, for every numerical semigroup S: (1) the multiplicity m(S), which is the lowest of the above-mentioned generators, (2) the embedding dimension e(S), the cardinality of the minimal set of generators, and (3) the genus g(S), the cardinality of the set of numbers not in S.

One interesting result is that e(S) ≤ m(S), and those semigroups S for which equality holds are aptly called semigroups with maximal embedding dimension — the subject of Chapter 2.These are characterized in various ways, involving the minimal generators, F(S), g(S), and so on. Also, on p. 22, after several harbinger-theorems, appears the tantalizing remark, “maximal embedding dimension numerical semigroups can be used to represent the whole class of numerical semigroups.” Also in that chapter are defined the concepts of Arf numerical semigroup, and a subclass of these called saturated; these have maximal embedding dimension as well as other interesting properties. (Arf means that if x, y, z, listed in descending order, are in S, then x+y–z is also in S; “subtraction” is defined, since x +y > z. The definition of saturated is too long to include here.)

From the definition, it’s easy to see that all finite intersections of numerical semigroups are themselves numerical semigroups. But not all numerical semigroups are intersections of two numerical semigroups properly containing it. The ones which aren’t are called irreducible; they’re the subject of Chapter 3. One main result is that “irreducible numerical semigroups are maximal in the set of numerical semigroups with fixed Frobenius number” (p. 33). Along the way we’re treated to important Lemma 4.1 (p. 33): if we adjoin, to any numerical semigroup, its Frobenius number, we get another numerical semigroup. The concepts of symmetric and pseudo-symmetric numerical semigroups are also important: there is space here to say only that the former are irreducible with odd F(S), the latter with even F(S).

One way to obtain numerical semigroups is to take the set of all integer solutions to any inequality of the form, ax mod b ≤ cx, where a, b, c, are positive integers. The resulting numerical semigroups are said to be “proportionally modular”; these are the subject of Chapter 4. Necessary and sufficient conditions are given for a numerical semigroup to be proportionally modular; one (p. 61) is that there exist positive rational numbers α and β such that X equals the submonoid of the positive rationals which is generated by the rationals in the closed interval [α,β]. Interesting number theory concepts then emerge, leading for example to Corollary 5.19, (p. 62): “every numerical semigroup of embedding dimension two is proportionally modular.” This chapter also studies the special case where c = 1, resulting in (just-plain) modular numerical semigroups.

If we divide any numerical semigroup by any positive integer, we get another numerical semigroup. Chapter 5 contains interesting consequences of this idea. For example, “every numerical semigroup is one half of infinitely many symmetric numerical semigroups” and “every numerical semigroup is one fourth of a pseudo-symmetric numerical semigroup” (p. 79). Also in this chapter a particularly simple characterization of proportionally modular numerial semigroups is found: they must be quotients of numerical semigroups whose minimal generators are two consecutive integers.) (p. 88)

Chapter 6, titled “Families of numerical semigroups closed under finite intersections and adjoin of the Frobenius number”, studies just those, calling them, for short, “Frobenius varieties”. Among Froebenius varieties are the set of all numerical semigroups which are (A) Arf, (B) saturated, and (C) defined by patterns (where pattern is a generalization of Arf). We are also shown a cool way to make the set of all numerical semigroups into a directed graph; the trick is the above mentioned fact that if S is a numerical semigroup, so is S ∪ F(S). Iterating the operation of adjoining the Frobenius number leads to paths in the graph which connect any numerical semigroup to N, the root of the graph. One use of this graph is “to determine all numerical semigroups with Frobenius number (or genus) less than a given amount” (p. 93).

Chapter 7, the most difficult for me, has to do with presentations, minimal presentations, and free monoids on given sets. Its main result gives “an upper bound for the cardinality of a minimal presentation in terms of the multiplicity of” any numerical semigroup (p. 105). It was also nice to find out that “the concepts of minimal [presentations] with respect to set inclusion and cardinality coincide for numerical semigroups” (p. 110).

Chapter 8, also difficult, is titled “The gluing of numerical semigroups”. Following the introduction, it begins (p.124): “The idea of gluing is the following. A set of positive integers A, which is usually taken as the set of generators of a monoid, is the gluing of A1 and A2 if {A1, A2} is a partition of A and the monoid generated by A admits a presentation in which some relators only involve generators in A1, other relators only involve generators in A2 and there is only one element in this presentation relating elements in A1 with elements in A2.” There are several characterizations of the gluing of two sets of positive integers; one of them is expressed in strictly familiar terms from number theory, such as gcd and lcm. Other results, whose explanations and proofs are too long for this review, include “Every free numerical semigroup is a complete intersection” (p. 134).

Chapter 9, “Numerical semigroups with embedding dimension three,” introduces the concept of “unique expression” of elements in a given numerical semigroup (meaning there is only one “linear” combination of the generators which will produce the element). Several results follow, among them one whose statement makes no mention of this new idea: “a numerical semigroup with embedding dimension three has at most type two” (p. 148).

Finally, Chapter 10 provides “the structure of a numerical semigroup”. First, we get theorems in terms of properties, such as “a monoid is isomorphic to a numerical semigroup if and only if it is finitely generated, quasi-Archimedean, torsion free and with only one idempotent” (p. 155). The chapter (along with the book) ends with an actual structure theorem; roughly, every numerical semigroup is isomorphic to a monoid of the form (N×G,+I), , where G is a finite cyclic group and +I  is an operation satisfying certain properties “beyond the scope of” this review.

In the introduction, the authors say, “our goal has been to write a self-contained monograph on numerical semigroups that needs no auxiliary background other than basic integer arithmetic.” (p. 2) A tall order, and they pretty much succeed. Still, notions such as “generated by” might be difficult for someone who knows only “basic integer arithmetic”. Not to mention Chapters 7 and 8. But for the most part, nothing is assumed if it can be imparted in just a short sentence. For example, the well-known result that, if and only if a set of integers has greatest common divisor 1, then 1 can be expressed as a sum of integer multiples of those integers, is stated very early on in the book (p. 7) (although without proof, which I think is appropriate). And concepts well-known to upper undergraduate math majors are usually reviewed — for example, in the last chapter, the bare bones of group theory such as “the order of an element” (p. 167). Some theorems are even stated twice, first as motivation, then more formally.

With few exceptions, the writing is extremely clear, and the ideas well motivated. I was especially helped, on p. 96, by the sentence, “The idea of pattern on a numerical semigroup was introduced… in order to generalize the concept of Arf numerical semigroups.” Often the authors use the kind of wording used in textbooks, such as “we are going to show…” and “we will see that…” One very much appreciated feature of this book is its List of symbols (p. 177), which includes the page numbers on which each symbol is introduced; indeed, whenever I needed to recall a symbol, it was in there; likewise terms in the Index. The exercises at the end of the chapters are well taken, a mixture of simple but not trivial, difficult, and not yet solved by anybody. In general, without boring the reader or becoming too lengthy, the book does a very impressive job of being self-contained, and friendly to readers and students.

Most important, perhaps: this book is just the kind of thing I love! (In fact, as I was reading it, I often thought, “I’m in love!”) From the simple, self-contained ideas, such as “adjacent fractions” (p. 66), beautiful in themselves, whether or not they help study numerical semigroups, to stuff less self-contained like Toms decompositions of numerical semigroups (p. 83) — I could spend my life with this book, along with the papers in the 109-paper bibliography!

True, there are nitpicks, if one wants to be picky. Proofs by contradiction could often be clearer, in the sense that they could be stated as such. Definitions are not set out, and sometimes proofs appear before statements, sometimes after (usually after). Sometimes I felt the need for examples, as on p. 33, the first mention of the notion of irreducible numerical semigroup; I wanted (and provided) an example of a “reducible” one. And sometimes, if I’m correct, concepts could be simplified and made more mathematically sophisticated, as (p. 96) in the defining of “pattern”; I think of a pattern as simply an n-tuple of non-zero integers such that, etc. (Thus Arf semigroups have pattern (1, 1, –1).) Finally, I still have questions (maybe it’s just me): Why is the smallest generator of a numerical semigroup called its multiplicity? (p. 9) Likewise, why is the second-smallest generator called the ratio? (p. 17) Also (p. 44) I “proved” that all pseudo-Frobenius numbers are special gaps; where is the error is my “proof”?!

The ultimate of my questions, purely personal and not at all reflective of the quality of the writing: Will this book help me solve my “associative arithmetics problem”? Well, numerical semigroups are indeed related to the objects which I need to study. Many of the book’s results are reminiscent and supportive of my discoveries. For example, the Frobenius number of a given numerical semigroup is one less than where the eventual periodicity of my “generalized times-tables” begins. And in the very last section of the book, the structure theorem given for numerical semigroups (alpha-null copies of a cyclic group), suggest the eventual periodicity and the properties of the period. Other passages also ring bells. Will any of it pan out? Right now I need to send out this review; after that I’ll work on it (and other papers…) and time will tell.

But whatever happens, throughout all time from now on, because of this lovely book, I’ll know a little something about numerical semigroups, and that has certainly enhanced my mathematical life. Also, numerical semigroups have had many other applications, in particular in number theory and algebraic geometry, and that has enhanced the life of mathematics itself.

Marion Cohen‘s poetry book about the experience of math, Crossing the Equal Sign, was reviewed on this site.