Abstract
Many basic concepts used throughout Algebra have a natural home in Partially Ordered Sets (hereafter called “posets”). Aside from obvious poset residents such as Zorn’s Lemma and the well-ordered sets, some concepts are more wider roaming. Among these are the ascending and descending chain conditions, the general Jordan-Hölder Theorem (seen here as a theorem on interval measures of certain lower semillattices), Galois connections, the modular laws in lattices, and general independence notions that lead to the concepts of dimension and transcendence degree.
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Notes
- 1.
In the literature on binary relations, the term “antisymmetric” often replaces its equivalent “antireflexive”.
- 2.
This is just metaphorical language, nothing more.
- 3.
Even an appropriate feeling of guilt is not discouraged. Who knows? Each indulgence in Zornification might revisit some of you in another life.
- 4.
We shall see very soon that a well-ordered poset is simply a chain with the descending chain condition (see p. 44 and Corollary 2.3.6).
- 5.
The definition of cardinal number appears on p. 18, and \(\aleph _0\) is defined to be the cardinality of the natural numbers in the paragraphs that follow.
- 6.
In a great deal of the literature, sets of pairwise incomparable elements are called independent . Despite this convention, the term “independent” has such a wide usage in mathematics that little is served by employing it to indicate the property of belonging to what we have called an antichain. However, some coherent sense of the term “independence” is exposed in Sect. 2.6 later in this chapter.
- 7.
Usually authors feel that the two poset relations should always have distinguished notation—that is, one should write \((P_1,\le _1)\) and \((P_2,\le _2)\) instead of what we wrote. At times this can produce intimidating notation that would certainly finish off any sleepy students. Of course that precaution certainly seems to be necessary if the two underlying sets \(P_1\) and \(P_2\) are identical. But sometimes this is a little over-done. Since we already have posets denoted by pairs consisting of the set \(P_i\) and a symbol “\(\le \)”, the relation “\(\le \)” is assumed to be the one operating on set \(P_i\) and we have no ambiguity except possibly when the ground sets \(P_i\) are equal. Of course in the case the two “ground-sets” are equal we do not hesitate for a moment to adorn the symbol “\(\le \)” with further distinguishing emblems. This is exactly what we did in defining the dual poset. But even in the case that \(P_1=P_2\) one could say that in the notation, the relation “\(\le \)” is determined by the name \(P_i\) of the set, rather then the actual set, so even then the “ordered pair” notation makes everything clear.
- 8.
Many books present an equivalent assertion “any poset has a linear extension”. The proof is an elementary induction for finite posets. For infinite posets it requires some grappling with Zorn’s Lemma and ordinal numbers.
- 9.
This isomorphism explains why it is commonplace to do an induction proof with respect to the second of these examples beginning with 1 rather than the first, which begins with 0.
In enumerative combinatorics, for example, the “natural numbers” \({\mathbb N}\) are defined to be all non-negative integers, not just the positive integers (see Enumerative Combinatorics, vol 1, p. 1. by R. Stanley) [1].
- 10.
There are variations on this theme: In an integral domain a non-unit a is said to be irreducible if and only if \(a = bc\) implies one of b or c is a unit. Let D be an integral domain in which each non-unit is a product of finitely many irreducible elements, and let U be its group of units. Let \(D^*/U\) be the collection of all non-zero multiplicative cosets Ux. Then for any two such cosets, Ux and Uy, either every element of Ux divides every element of Uy or else no element of Ux divides any element of Uy. In the former case write \(Ux\le Uy\). Then \((D^*/U,\le )\) is a poset. If D is a unique factorization domain, then, as above, \((D^*/U,\le )\) is locally finite for it is again a product of chains (one factor in the product for each association class Up of irreducible elements).
One might ask what this poset looks like when D is not a unique factorization domain. Must it be locally finite? It’s something to think about.
- 11.
In Aigner’s book (see references), \({\mathbf L}_{<\infty }(V, q)\) is denoted \(\mathcal{L}(\infty , q)\) in the case that V has countable dimension over the finite field of q elements. This makes sense when one’s plan is to relate the structure to certain types of generating functions (the q-series). But of course, it is a well-defined locally finite poset whatever the dimension of V.
- 12.
Its cardinality \(|\Pi _n |\) is called the n th Bell number and will reappear in Chap. 4 in the context of permutation characters.
- 13.
Here, the set of lines, \(\mathcal{L}\), is simply a family of subsets of the set of points, \(\mathcal{P}\). A subspace is a set S of points, with the property that if a line \(L\in \mathcal{L}\) contains at least two points of S, then \(L\subseteq S\). Thus the empty set and the set \(\mathcal{P}\) are subspaces. From the definition, the intersection over any family of subspaces, is a subspace. The subspace generated by a set of points X is defined to be the intersection of all subspaces which contain X.
- 14.
Note that \((ch[x,y],\subseteq )\) is not quite the same as \((ch([x,y],\le )\) since the latter may contain chains which, although lying in the interval [x, y], do not contain x or y.
- 15.
This adjective “algebraic” does not enjoy uniform usage. In Universal Algebras, elements which are the join of finitely many atoms are called algebraic elements (perhaps by analogy with the theory of field extensions). Here we are applying the adjective to an interval, rather than an element of a poset.
- 16.
Since the adjective “algebraic” entails the existence of a finite unrefinable chain, the height of a algebraic interval is always a natural number. The term “height” is used here instead of “length” which is appropriate when all unrefineable chains have the same length, as in the semimodular lower semilattices that appear in the Jordan-Hölder Theorem.
- 17.
The graduate student has probably encountered arguments like this many times, where a sequence with certain properties is said to exist because after the first n members of the sequence are constructed, it is always possible to choose a suitable \(n+1\)-st member. This has an uncomfortable feel to it, for the sequence alleged to exist must exemplify infinitely many of these choices—at least invoking the Axiom of Choice in choosing the \(x_i\). But in a sense it appears worse. The sets are not just sitting there as if we had prescribed non-empty sets of socks in closets lined up in an infinite hallway (the traditional folk-way model for the Axiom of Choice). Here, it as if each new closet was being defined by our choice of sock in a previous closet, so that it is really a statement about the existence of infinite paths in trees having no vertex of degree one. All we can feebly tell you is that it is basically equivalent to the Axiom of Choice.
- 18.
This conclusion reveals the incipient presence of the Axiom of Choice/Zorn’s Lemma in the argument of the first paragraph of the proof of Lemma 2.3.4.
- 19.
The prefix “semi-” is justified for several reasons. The term “modular function” has quite another meaning as a certain type of meromorphic function of a complex variable. Secondly the function in question is defined in the context of a semimodular lower semilattice. So why not put in the “semi”? We do not guarantee that every term coined in this book has been used before.
- 20.
We beg the reader to notice that in the case of groups there is no need for Zassenhaus’ famous “butterfly lemma”, nor the need to prove that subnormal subgroups of a finite group form a lattice. A lower semilattice will do. One of the classic homomorphism theorems, provides the semimodular function from \(\mathcal{C}ov_P\) to finite simple groups. The result is then immediate from Theorem 2.4.2, Eq. (2.13), where the interval measure displays the multiset of “chief factors” common to all saturated chains in P.
References
Oxley J (2011) Matroid theory, Oxford graduate texts. vol 21, Oxford University Press, Oxford
Stanley RP (1997) Enumerative Combinatorics, vol 1, Cambridge studies in advanced mathematics, vol 49, Cambridge University Press, Cambridge
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Shult, E., Surowski, D. (2015). Basic Combinatorial Principles of Algebra. In: Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-19734-0_2
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