An Abstract Factorization Theorem and Some Applications
aa r X i v : . [ m a t h . R A ] F e b AN ABSTRACT FACTORIZATION THEOREM AND SOME APPLICATIONS
SALVATORE TRINGALI
Abstract.
We combine the language of monoids with the language of preorders to formulate an ab-stract factorization theorem with several applications. In particular, this leads to (i) a generalization ofP.M. Cohn’s classical theorem on “atomic factorizations” from cancellative to Dedekind-finite monoids(and, hence, to a variety of rings that are not domains); (ii) a monoid-theoretic proof that every moduleof finite uniform dimension over a (commutative or non-commutative) ring R is isomorphic to a directsum of finitely many indecomposable R -modules (in fact, we obtain the result as a special case of ageneral decomposition theorem for the objects of certain categories with finite products, where the in-decomposable R -modules are characterized as the atoms of a certain “monoid of modules”). Also, werecover and extend an existence theorem of D.D. Anderson and S. Valdes-Leon on “irreducible factor-izations” in commutative rings [RMJM 1996]; a refinement of Cohn’s theorem to “nearly cancellative”monoids due to Y. Fan et al. [JA 2018]; and a characterization theorem of A.A. Antoniou and the authorabout atomic factorizations in certain “monoids of sets” [PJM 202?]. Introduction
Let H be a monoid (see Sect. 2.2 for notation and terminology). As usual, we let a principal rightideal of H be a subset of H of the form aH := { ax : x ∈ H } with a ∈ H ; and we say that H satisfies theascending chain condition (ACC) on principal right ideals (ACCPR) if there is no infinite sequence ofprincipal right ideals of H that is (strictly) increasing with respect to inclusion. The ACC on principalleft ideals (ACCPL) and the ACC on principal ideals (ACCP) are defined in a similar way, with principalright ideals replaced, resp., by principal left ideals — that is, subsets of H of the form Ha := { xa : x ∈ H } with a ∈ H — and principal ideals — that is, subsets of H of the form HaH := { xay : x, y ∈ H } with a ∈ H . The ACCPR, the ACCPL, and the ACCP (one and the same condition in the commutativesetting) have been the subject of extensive research and are famous for playing a critical role in the studyof the “arithmetic” of monoids and rings.More in detail, let an atom of H be a non-unit a ∈ H such that a = xy for all non-units x, y ∈ H ; andlet an irreducible of H be a non-unit a ∈ H such that a = xy for all non-units x, y ∈ H with HaH = HxH and
HaH = HyH . It is a classical theorem commonly attributed to P.M. Cohn that every non-unit ina cancellative monoid satisfying the ACCPR and the ACCPL factors as a finite product of atoms, see[10, Proposition 0.9.3]; an extension to “nearly cancellative” monoids was recently obtained by Y. Fan etal. in [15, Lemma 3.1(1)] (the commutative case) and [16, Theorem 2.28(i)]. In a similar vein, it was firstobserved by D.D. Anderson and S. Valdes-Leon in [2, Theorem 3.2] that every non-unit of a commutative
Mathematics Subject Classification.
Primary 06F05, 13A05, 13F15, 16E99. Secondary 13P05, 16D70, 20M50.
Key words and phrases.
ACC, artinian, ascending chain conditions, atoms, DCC, descending chain condition, factoriza-tion, irreducibles, monoids, monoids of modules, noetherian, orders, power monoids, preorders, quasi-orders, rings.
Salvatore Tringali monoid satisfying the ACCP factors into a finite product of irreducibles : This extends Cohn’s theorem ina different direction than the one taken by Fan et al., since every atom is obviously an irreducible and, asa partial converse, every irreducible in a cancellative commutative monoid is an atom (see Example 3.8(4)and Corollary 4.4 for a more comprehensive analysis of the relations between atoms and irreducibles).On the whole, the above results can be regarded as a far-reaching generalization of the FundamentalTheorem of Arithmetic — that every integer greater than one factors as a product of prime numbers (inan essentially unique way) — and lie in the foundations of a subfield of algebra known as factorizationtheory [18, 20]. But “factorization theorems” are common to many other fields. For instance:( f
1) It is a basic result in ring theory that every artinian or noetherian R -module is isomorphic to afinite direct sum of indecomposable R -modules, where R is a (commutative or non-commutative)ring and we recall that an R -module M is indecomposable if M is neither a zero R -module northe direct sum of two non-zero R -modules.( f
2) It is folklore in group theory (see, e.g., [28, Proposition 2.35]) that every permutation of a finite k -element set X factors into a functional composition of k or fewer transpositions.( f
3) It is been known since J.A. Erdos’ seminal paper [12] that every singular matrix in the multi-plicative monoid of the ring of n -by- n matrices with entries in a field factors as a finite productof idempotent matrices, and later work has revealed that the same holds with fields replaced bya much wider class of commutative rings (see [11, Sect. 1] for a historical overview and recentdevelopments).Roughly speaking, these results have all in common that they pertain to the existence of a factorizationof certain elements of a monoid into a finite product of certain other elements that, in a sense, cannotbe “broken up into smaller pieces”. However, there is to date no general theory of factorization that givesshape and substance to this idea, and it is the primary goal of the present paper to start and fill the gap.More in detail, the plan is as follows. First, we generalize the ordinary notions of unit, atom, andirreducible by pairing a monoid with a preorder (Definitions 3.1, 3.4, and 3.6). Next, assuming a nat-ural analogue of the ACCP holds (Definitions 3.9 and 3.10 and Remark 3.11(4)), we prove an abstractfactorization theorem (Theorem 3.12) with many “down-to-earth” applications, including the following:( a
1) A generalization of Cohn’s theorem (on atomic factorizations) from cancellative to Dedekind-finitemonoids and, hence, to a variety of rings (Corollaries 4.1 and 4.6).( a
2) An “object decomposition theorem” (Corollary 4.13) for certain categories with finite productsyielding as a special case a monoid-theoretic proof (Corollary 4.14) that every R -module of finiteuniform dimension over a ring R (so in particular, every artinian or noetherian R -module) isisomorphic to a direct sum of finitely many indecomposable R -modules (see Sect. 4.3 for details).( a
3) A “new” and, in a way, more conceptual proof of the aforementioned folk theorem on permutationsof finite sets, where we characterize the transpositions of a finite set as a sort of irreducibles as-sociated with the fixed points of a permutation (Example 3.15).From ( a As a matter of fact, [2, Theorem 3.2] is only stated for commutative rings (with or without non-trivial zero divisors).However, the proof carries over verbatim to commutative monoids. n Abstract Factorization Theorem and Some Applications H : The former are discussed in Sect. 4.1, while the latter is thecontent of Sect. 4.2 (see, in particular, Theorem 4.12).Further applications (especially to “idempotent factorizations” in matrix rings as outlined in item ( f Preliminaries.
In this section, we establish notations and terminology used all through the paper. Further notationsand terminology, if not explained when first introduced, are standard or should be clear from context.2.1.
Generalities.
We assume throughout that all relations are binary; all rings are non-zero, unital,and associative; and all modules are right modules. We will usually be rather casual about the distinctionbetween “small sets” (or simply “sets”) and “large sets” (or “classes”), but differentiating between thesetwo “types” will become relevant in Sect. 4.3, where, in essence, we need to guarantee that every categoryhas a skeleton: With this in mind, we set out from the beginning to use von Neumann-Bernays-Gödel(NBG) set theory as foundations for the present work. Alternatives are possible (we refer the interestedreader to [25, Sect. I.6] and [29] for more details), but the question goes far beyond the scope of the paperand hence we shall be content with our pick.We denote by N the (set of) non-negative integers, by Z the integers, and by R the real numbers. Forall a, b ∈ R ∪ {±∞} , we let J a, b K := { x ∈ Z : a ≤ x ≤ b } be the discrete interval between a and b . Unlessa statement to the contrary is made, we reserve the letters ℓ , m , and n (with or without subscripts orsuperscripts) for positive integers; and the letters i , j , and k for non-negative integers.Given a set X and an integer k ≥ , we use P ( X ) for the power set of X and X × k for the Cartesianproduct of k copies of X ; moreover, we write | X | for the size of X , by which we mean that | X | is thenumber of elements of X when X is finite, and is ∞ otherwise. If R is a relation on X , we say that x is R -equivalent to y , for some x, y ∈ X , if either x = y , or x R y and y R x (so that “being R -equivalent” isan equivalence on X ); here, u R v is, as usual, shorthand for ( u, v ) ∈ R .2.2. Monoids.
We take a monoid to be a semigroup with an identity. Unless stated otherwise, monoidswill typically be written multiplicatively and need not have any special property (e.g., commutativity).We refer the reader to [23, Ch. 1] for basic aspects of semigroup theory.Let H be a monoid with identity H . An element u ∈ H is right-invertible (resp., left-invertible ) if uv = 1 H (resp., vu = 1 H ) for some v ∈ H . We use H × for the set of units (or invertible elements ) of H , namely, the elements of H that are both left- and right-invertible: This means that u ∈ H × if andonly if there is a provably unique v ∈ H , called the inverse of u (in H ) and denoted by u − , such that uv = vu = 1 H . It is well known that H × is a subgroup of H , and we say that H is • reduced if the only unit of H is the identity, i.e., H × = { H } ; • cancellative if xz = yz and zx = zy for all x, y, z ∈ H with x = y ; • Dedekind-finite if every left- or right-invertible element is a unit, or equivalently, if xy = 1 H forsome x, y ∈ H implies that at least one of x , y , and yx is a unit (in the sequel, we will often usethis equivalent formulation of Dedekind-finiteness without comment). Salvatore Tringali
Cancellative and Dedekind-finite monoids abound in nature and have been studied for a long time (oftenin disguise): Most notably, the non-zero elements of a domain form a cancellative monoid under multipli-cation, and every cancellative or commutative monoid is Dedekind-finite (see also Propositions 4.3 and4.11 and Remark 4.9).Given X , . . . , X n ⊆ H , we write X · · · X n for the the setwise product of X through X n , i.e., theset { x · · · x n : x ∈ X , . . . , x n ∈ X n } ⊆ H ; note that, if X i = { x i } for some i ∈ J , n K and there is nolikelihood of confusion, we will replace the set X i in the product X · · · X n with the element x i .In particular, we denote by X n the setwise product of n copies of a set X ⊆ H , and then we let Sgrp h X i H := [ n ≥ X n = [ n ≥ { x · · · x n : x , . . . , x n ∈ X } ⊆ H be the subsemigroup of H generated by X (note that Sgrp h X i H is not, in general, a sub monoid of H ).Accordingly, we say that H is a finitely generated (resp., k -generated ) monoid if H = { H } ∪ Sgrp h X i H for a finite (resp., k -element) set X ⊆ H ; and we write Sgrp h x , . . . , x n i H in place of Sgrp h X i H when X is a non-empty finite set with elements x , . . . , x n .A monoid congruence on H is an equivalence R on H such that, if x R u and y R v , then xy R uv . If R is a monoid congruence on H , we will write x ≡ y mod R in place of x R y and say that “ x is congruentto y modulo R ”. Consequently, we will use x y mod R to signify that ( x, y ) / ∈ R .2.3. Presentations.
In a couple of cases, we will consider a finitely generated monoid defined via gen-erators and relations. Therefore, we find it useful before proceeding to fix the notation and review somebasic facts about presentations, cf. [23, Sect. 1.5].Let X be a fixed set. We denote by F ( X ) the free monoid over X ; use the symbols ∗ X and ε X , resp.,for the operation and the identity of F ( X ) ; and refer to an element of F ( X ) as an X -word , or simply asa word if no confusion can arise. We recall that F ( X ) consists, as a set, of all finite tuples of elements of X ; and u ∗ X v is the concatenation of two such tuples u and v . Accordingly, the identity of F ( X ) is theempty tuple (i.e., the unique element of X × ), herein called the empty X -word .We take the length of an X -word u , denoted by k u k X , to be the unique non-negative integer k suchthat u ∈ X × k ; in particular, the empty word is the only X -word whose length is zero. Note that, if u isan X -word of positive length k , then u = u ∗ X · · · ∗ X u k for some uniquely determined u , . . . , u k ∈ X .Given z ∈ X , we let the z -adic valuation on X be the function v Xz : F ( X ) → N that maps ε X to anda non-empty X -word u ∗ X · · · ∗ X u n of length n to the number of indices i ∈ J , n K with u i = z .We shall systematically drop the subscript (resp., superscript) X from the above notation when thereis no serious risk of ambiguity. As a result, we will write ∗ instead of ∗ X and u ∗ k for the k th power of aword u ∈ F ( X ) , so that u ∗ := ε X and u ∗ k := u ∗ ( k − ∗ X u for k ∈ N + .With these premises in place, let R be a relation on the free monoid F ( X ) . We denote by R ♯ thesmallest monoid congruence on F ( X ) containing R ; formally, this means that R ♯ := \ (cid:8) ρ ⊆ F ( X ) × F ( X ) : ρ is a monoid congruence and R ⊆ ρ (cid:9) . In consequence, u ≡ v mod R ♯ if and only if there are z , z , . . . , z n ∈ F ( X ) with z = u and z n = v suchthat, for each i ∈ J , n − K , there exist X -words p i , q i , q ′ i , and r i with the following properties:(i) either q i = q ′ i , or q i R q ′ i , or q ′ i R q i ; (ii) z i = p i ∗ q i ∗ r i and z i +1 = p i ∗ q ′ i ∗ r i . n Abstract Factorization Theorem and Some Applications Mon h X | R i the monoid obtained by taking the quotient of F ( X ) by the congruence R ♯ .We write Mon h X | R i multiplicatively and call it a ( monoid ) presentation ; in particular, Mon h X | R i isa finite presentation if X and R are both finite sets. We refer to the elements of X as the generators ofthe presentation, and to each X -word in a pair ( q , q ′ ) ∈ R as a defining relation . If there is no real risk ofconfusion, we identify, as is customary, an X -word z with its equivalence class in Mon h X | R i .The left graph of a presentation Mon h X | R i is the undirected multigraph with vertex set X and anedge from y to z for each pair ( y ∗ y , z ∗ z ) ∈ R with y, z ∈ X and y , z ∈ F ( X ) ; note that this results ina loop when y = z , and in multiple (or parallel) edges between y and z if there are two or more definingrelations of the form ( y ∗ y , z ∗ z ) . The right graph of a presentation is defined analogously, using theright-most (instead of left-most) letters of each word from a defining relation.A monoid is Adian if it is isomorphic to a finite presentation whose left and right graphs are cycle-free ,that is, contain no cycles (including loops). Our interest for Adian monoids stems from the following:
Theorem 2.1.
Every Adian monoid embeds into a group (and hence is cancellative).The result is attributed to S.I. Adian [1, Theorem II.4], and therefore it is commonly referred to as
Adian’s Embedding Theorem : It will come in useful in Example 4.8.3.
Preorders and their Interplay with Monoids
In the present section, we aim to generalize fundamental aspects of the classical theory of factorizationby combining the language of monoids with that of preorders: This will prepare the ground for theabstract factorization theorem (Theorem 3.12) promised in the introduction. The section also includes avariety of examples that will help illustrate some key points: Certain of these examples are of independentinterest and we will return to them later, when discussing applications in Sect. 4.We start with the following definition (see, e.g., [17, Definition 2.1] and note that some authors preferthe terms “quasi-order” or “quasi-ordering” to the term “preorder”):
Definition 3.1.
Let X be a set. A preorder on X is a relation R on X such that x R x for all x ∈ X (i.e., R is reflexive ), and x R z whenever x R y and y R z (i.e., R is transitive ).In particular, we say a preorder R on X is total if, for all x, y ∈ X , x R y or y R x ; and is an order if x R y and y R x imply x = y (i.e., R is antisymmetric ).We will usually denote a preorder on a set X by either of the relational symbols ≤ and (cid:22) , with orwithout subscripts or superscripts. In particular, we shall reserve the symbol ≤ for the standard orderon R ∪ {±∞} and its subsets. With this in mind, we make the following: Definition 3.2.
Given a preorder (cid:22) on a set X , we write x ≺ y to signify that x (cid:22) y and y (cid:14) x .Accordingly, we say that a sequence ( x k ) k ≥ of elements of X is (cid:22) -non-increasing (resp., (cid:22) -decreasing ) if x k +1 (cid:22) x k (resp., x k +1 ≺ x k ) for every k ∈ N ; and is (cid:22) -non-decreasing (resp., (cid:22) -increasing ) if x k (cid:22) x k +1 (resp., x k ≺ x k +1 ) for every k ∈ N .It is perhaps worth remarking that, for a preorder (cid:22) , the condition “ x ≺ y ” is stronger than “ x (cid:22) y and x = y ”: The two conditions are, in fact, equivalent if and only if (cid:22) is an order. In addition, notethat “ (cid:22) -decreasing” means “strictly (cid:22) -decreasing” (and similarly for “ (cid:22) -increasing”). Salvatore Tringali
Examples 3.3.
All through this example, we let X be a fixed set.(1) An equivalence on X is a preorder R on X with the property that x R y if and only if y R x .(2) Let (cid:22) be a preorder (resp., an order) on X . The relation (cid:22) op on X defined by taking x (cid:22) op y ifand only if y (cid:22) x , is still a preorder (resp., an order) on X : We will refer to (cid:22) op as the dual preorder (resp., the dual order ) of (cid:22) , or simply as the dual of (cid:22) . It is common practice to denote the preorder (cid:22) op by the “dual” of the relational symbol (cid:22) (that is, by (cid:23) ). However, we will avoid this practice, except forthe dual of the standard order ≤ on R ∪ {±∞} , which, as usual, we denote by ≥ .(3) Let ⊆ X be the restriction to the power P ( X ) of X of the “global relation” of containment ⊆ . Then ⊆ X is an order on P ( X ) , whose dual (see item (2) on this list) is the restriction to P ( X ) of the “globalrelation” ⊇ . We will refer to ⊆ X as the inclusion order on X .(4) Given a function φ : X → Y and a preorder (cid:22) on Y , the relation (cid:22) φ on X defined for all x, y ∈ X by x (cid:22) φ y if and only if φ ( x ) (cid:22) φ ( y ) , is a preorder on X . We will refer to (cid:22) φ as the pullback preorder induced by (cid:22) through φ or, more simply, as the φ -pullback (cid:22) .(5) Let R be a family of (binary) relations on X . Then V R := T R ∈R R is still a relation on X ,with the understanding that V R = X × when R = ∅ . We shall refer to V R as the relational wedge of the family R . By construction, ( x, y ) ∈ V R if and only if x R y for each R ∈ R . In consequence, itis straightforward that, if every relation R ∈ R is a preorder, then so also is V R . In particular, if thefamily R is empty, then V R is the trivial preorder on X (that is, the relation X × X on X ).(6) Given a reflexive relation R on X , let (cid:22) R be the relational wedge (see item (5) on this list) of alltransitive relations on X containing R . It is easily seen that (cid:22) R is a preorder on X containing R , with x (cid:22) R y if and only if there exist z , z , . . . , z n ∈ X with z = x and z n = y such that z i R z i +1 for each i ∈ J , n − K . We will refer to (cid:22) R as the transitive closure of R .We are mainly interested in preorders that are, in a way, compatible with the operation of a monoid.The basic idea is nothing new (see, e.g., [9] or [14, Sect. 1.2]) and leads straight to the following: Definition 3.4.
We let a premonoid be a pair consisting of a monoid and a preorder on its underlyingset; and a preordered monoid is a premonoid ( H, (cid:22) ) such that, if x (cid:22) u and y (cid:22) v , then xy (cid:22) uv .In particular, a totally preordered monoid is a preordered monoid ( H, (cid:22) ) with the further property that (cid:22) is a total preorder; and a linearly preordered monoid is a totally preordered monoid such that ux ≺ uy and xu ≺ yu for all u, x, y ∈ H with x ≺ y .Ordered monoids, totally ordered monoids, and linearly ordered monoids are defined in a similarfashion, simply by replacing the word “preorder” with the word “order” everywhere in Definition 3.4. Examples 3.5.
Let H be a monoid. In principle, there are many preorders one can put on H , but their“compatibility” with the monoid operation is often depending on specific properties of H .(1) Given a preorder on H , it is fairly obvious that ( H, (cid:22) ) is a preordered monoid (resp., an orderedmonoid) if and only if so is ( H, (cid:22) op ) , where (cid:22) op is the dual of (cid:22) (see Example 3.3(2)).(2) In the notation of Example 3.3(3), define X ∩ := ( P ( X ) , ∩ X ) and X ∪ := ( P ( X ) , ∪ X ) , where ∩ X and ∪ X are, resp., the restrictions to P ( X ) of the “global operations” of union and intersection. It is easyto verify that ( X ∩ , ⊆ X ) and ( X ∪ , ⊆ X ) are ordered monoids. n Abstract Factorization Theorem and Some Applications | H on H defined for all x, y ∈ H by x | H y (read“ x divides y ”) if and only if y ∈ HxH , is a preorder on H . We will refer to | H as the divisibility preorder on H and write x ∤ H y (read “ x does not divide y ”) if y / ∈ HxH . In general, | H is not an order, as seen,e.g., by considering the case where H has at least two units. Moreover, ( H, | H ) need not be a preorderedmonoid: If H is, for instance, the free monoid over the -element set { a, b } , then a | H a and a | H b ∗ a ,but a ∗ a ∤ H a ∗ b ∗ a (see Sect. 2.3 for notation).(4) Let ⊢ H and ⊣ H be, resp., the relations on H defined for all x, y ∈ H by x ⊢ H y (read “ x divides y from the left”) if and only if y ∈ xH , and x ⊣ H y (read “ x divides y from the right”) if and only if y ∈ Hx . In addition, denote by ⊥ H the relational wedge of ⊢ H and ⊣ H (see Example 3.3(5)), in such away that x ⊥ H y if and only if y ∈ xH ∩ Hx .It is quickly checked that each of ⊢ H , ⊣ H , and ⊥ H is a preorder on H ; that these preorders are allequal to one another and to the divisibility preorder | H when H is normal (item (3) on this list); andthat none of ( H, ⊢ H ) , ( H, ⊣ H ) , or ( H, ⊥ H ) is, in general, a preordered monoid. In addition, it turns outthat | H is the transitive closure (see Example 3.3(6)) of the relation ⊢ H ∪ ⊣ H .Indeed, assume first that x | H y . Then y = uxv for some u, v ∈ H , and hence x ⊢ H xv and xv ⊣ H y .But this means that ( x, y ) is in the transitive closure of ⊢ H ∪ ⊣ H , implying that the transitive closure of ⊢ H ∪ ⊣ H contains | H . It remains to prove the opposite inclusion.To this end, suppose that ( x, y ) is in the transitive closure of ⊢ H ∪ ⊣ H . By definition, there exists afinite sequence z , z , . . . , z n of elements of H with z = x and z n = y such that z i ⊢ H z i +1 or z i ⊣ H z i +1 for each i ∈ J , n − K . It follows that z | H z , . . . , z n − | H z n ; and by the transitivity of | H , we concludethat x | H y . In consequence, the transitive closure of ⊢ H ∪ ⊣ H is contained in | H (as wished).(5) Assume that ( K, (cid:22) ) is a preordered monoid, and denote by (cid:22) φ the pullback preorder induced by (cid:22) through a monoid homomorphism φ from H to K (see Example 3.3(4)). If x (cid:22) φ y and u (cid:22) φ v , then φ ( x ) (cid:22) φ ( y ) and φ ( u ) (cid:22) φ ( v ) ; and by the assumptions made on φ and (cid:22) , we have φ ( xu ) = φ ( x ) φ ( u ) (cid:22) φ ( y ) φ ( v ) = φ ( yv ) , which is equivalent to xu (cid:22) φ yv . In consequence, ( H, (cid:22) φ ) is a preordered monoid.The preorders defined in items (3) and (4) of Example 3.5 are extensively studied by J.A. Green in hisseminal paper [22], whence they are sometimes called Green’s preorders : We will pay special attention tothem in Sect. 4.1.One of the key insights of this whole work is that every premonoid comes in with a natural generaliza-tion of the notion of unit, which, in turn, results in a natural generalization of the notions of atom andirreducible discussed in the introduction (see Example 3.8(4)). More precisely, we have the following:
Definition 3.6.
Let ( H, (cid:22) ) be a premonoid. An element u ∈ H is a (cid:22) -unit (of H ) if u is (cid:22) -equivalentto H (i.e., u (cid:22) H (cid:22) u ); otherwise, u is a (cid:22) -non-unit . Accordingly, a (cid:22) -non-unit a ∈ H is • a (cid:22) -irreducible (of H ) if a = xy for all (cid:22) -non-units x, y ∈ H with x ≺ a and y ≺ a ; • a (cid:22) -atom if a = xy for all (cid:22) -non-units x, y ∈ H ; • a (cid:22) -quark if there exists no (cid:22) -non-unit b ∈ H with b ≺ a ; • a (cid:22) -prime if a (cid:22) xy , for some x, y ∈ H , implies a (cid:22) x or a (cid:22) y .We say that H is (cid:22) -factorable if each (cid:22) -non-unit factors as a (non-empty, finite) product of (cid:22) -irreducibles;and (cid:22) -atomic if each (cid:22) -non-unit factors as a product of (cid:22) -atoms. Salvatore Tringali
It is actually the notion of (cid:22) -irreducible as per the above definition that is crucial to Theorem 3.12: Thenotions of (cid:22) -quark and (cid:22) -atom are of independent interest, for understanding the interrelation between (cid:22) -irreducibles, (cid:22) -atoms, and (cid:22) -quarks in a specific scenario is often pivotal to a deeper comprehensionof various phenomena (see, e.g., Propositions 3.14 and 4.3 and Theorem 4.12).
Remark 3.7.
The rationale behind Definition 3.6 is (vaguely) reminiscent of certain ideas set forth in[5], where, among other things, N.R. Baeth and D. Smertnig axiomatize a notion of “divisibility relation”(ibid., Definition 5.1): Every divisibility relation corresponds to a notion of “prime-like element” (ibid.,Definition 5.3), similarly to how a preorder (cid:22) on a monoid H is associated with notions of (cid:22) -irreducible, (cid:22) -atom, and (cid:22) -quark. But while Baeth and Smertnig’s approach is firmly anchored to a classical paradigmof factorization (as seen, e.g., from the critical role that “ordinary units” keep playing in their framework),this is not the case with our approach. Moreover, Baeth and Smertnig’s notion of prime-like element isnot really a generalization of the classical notion of atom: It is rather a generalization of the Euclideannotion of “prime”, which, in turn, has a natural generalization in the notion of (cid:22) -prime (note that we willnot discuss (cid:22) -primes any further in this work). Examples 3.8. (1) In the notation of Example 3.5(1), it is immediate that an element u ∈ H is a (cid:22) -unitif and only if u is a (cid:22) op -unit, and this implies at once that an element a ∈ H is a (cid:22) -atom if and only if a is a (cid:22) op -atom. However, a (cid:22) -quark need not be a (cid:22) op -quark; and similarly, a (cid:22) -irreducible need notbe a (cid:22) op -irreducible (see the next item on this list).(2) In the notation of Example 3.5(2), assume X is a non-empty set. Obviously, the unique ⊆ X -unit ofthe monoid X ∪ is the empty set (i.e., the identity). In consequence, it is easily found that the ⊆ X -quarksof X ∪ are the -element subsets of X , while the only ⊆ op X -quark of X ∪ is X itself (recall from item (1) onthis list that the ⊆ X -units of X ∪ are the same as the ⊆ op X -units). On the other hand, every non-emptysubset A of X is a ⊆ op X -irreducible of X ∪ (if A = B ∪ C for certain sets B and C , then A cannot be properly contained in B ), while the ⊆ X -irreducibles of X ∪ are still the -element subsets of X (i.e., theonly non-empty subsets of X that are not a union of two proper subsets). So, X ∪ is ⊆ X -factorable if andonly if X is a finite set, while it is ⊆ op X -factorable regardless of whether X is finite or not.(3) If H is an idempotent monoid (meaning that x = x for each x ∈ H ), then the set of (cid:22) -atoms of H is empty for every preorder (cid:22) on H . The conclusion applies, in particular, to the monoid X ∪ consideredin item (2) on this list and to the monoid X ∩ of Example 3.5(2), as X ∪ and X ∩ are both idempotent.(4) Let H be a monoid. With the notation of items (3) and (4) of Example 3.5, a ∈ H is a | H -irreducibleif and only if a is a | H -non-unit and a = xy for all | H -non-units x, y ∈ H such that a ∤ H x and a ∤ H y . Inaddition, it is evident that u ∈ H is a ⊥ H -unit if and only if u is a unit (that is, uv = vu = 1 H for some v ∈ H ); whence a ∈ H is a ⊥ H -atom (that is, a ⊥ H -non-unit with a = xy for all ⊥ H -non-units x, y ∈ H )if and only if a is an atom (that is, a non-unit with a = xy for all non-units x, y ∈ H ).Assume, on the other hand, that H is a Dedekind-finite monoid (see Sect. 2.2). It is then easily checkedthat u ∈ H is a ⊢ H -unit (i.e., H ∈ uH ) if and only if u is a ⊣ H -unit (i.e., H ∈ Hu ), if and only if u is a | H -unit (i.e., H ∈ HuH ), if and only if u is a unit. Therefore, a ∈ H is a ⊢ H -atom (i.e., a ⊢ H -non-unitwith a = xy for all ⊢ H -non-units x, y ∈ H ) if and only if a is a ⊣ H -atom (i.e., a ⊣ H -non-unit with a = xy for all ⊣ H -non-units x, y ∈ H ), if and only if a is a | H -atom (i.e., a | H -non-unit with a = xy for all | H -non-units x, y ∈ H ), if and only if a is an atom. And in a similar way, a ∈ H is a | H -irreducible if andonly if a is irreducible in the sense of Anderson and Valdes-Leon (see Sect. 1), i.e., a is a non-unit such n Abstract Factorization Theorem and Some Applications a = xy for all non-units x, y ∈ H with HxH = HaH and
HyH = HaH .In particular, it follows from the above that the | H -atoms of a monoid H are ultimately a generalizationof the standard notion of atom, since it is not difficult to show that the set of atoms of H is non-emptyif and only if H is Dedekind-finite (see [16, Lemma 2.2(i)] for details).(5) Let ( H, (cid:22) ) be a premonoid. It is straightforward from Definition 3.6 that, if a ∈ H is a (cid:22) -atom ora (cid:22) -quark, then a is also a (cid:22) -irreducible; while, in general, the converse is not true.For instance, let H be the multiplicative monoid of a (commutative or non-commutative) domain R .In the notation of Example 3.5(3), the zero R of R is a | H -irreducible of H , because R = xy for all x, y ∈ R r { R } . However, R is not a | H -atom of H , for R is not a | H -unit and R = 0 R R (note that,by item (4) on this list, the | H -units are precisely the units of H , since H is a Dedekind-finite monoid).If, in addition, R is not a skew field, then R is not a (cid:22) -quark either: Just let x be a non-zero non-unitof R and observe that x | H R but R ∤ H x .It is a natural question to look for conditions under which the elements of a certain subset S of a monoid H can all be factored through the elements of another subset A : In essence, the key contributions of thepresent work provide a partial answer to this question in the case where, given a preorder (cid:22) on H , welet S be the set of (cid:22) -non-units (of H ) and A be either the set of (cid:22) -irreducibles, the set of (cid:22) -atoms, orthe set of (cid:22) -quarks. Most notably, we aim to obtain sufficient conditions for H to be (cid:22) -factorable thatextend the ideal-theoretic conditions reviewed in Sect. 1 (see Remark 3.11(4) for additional details). Definition 3.9.
We say that a preorder (cid:22) on a set X is artinian or satisfies the descending chain condition (DCC) if, for every (cid:22) -non-increasing sequence ( x k ) k ≥ of elements of X , there exists k ′ ∈ N such that,for k ≥ k ′ , x k (cid:22) x k +1 (and hence x k is (cid:22) -equivalent to x k +1 ). Consequently, we say that (cid:22) is noetherian or satisfies the ascending chain condition (ACC) if the dual (cid:22) op of (cid:22) is artinian.In other terms, a preorder (cid:22) on a set X is artinian (resp., noetherian) if and only if there is no sequence ( x k ) k ≥ of elements of X with x k +1 ≺ x k (resp., x k ≺ x k +1 ) for all k ∈ N . (Cf. [17, Definition 2.2], wherethe term “noetherian” is used in a way that is dual to how we are using it here.) Definition 3.10.
We let an artinian (resp., noetherian ) premonoid be a premonoid ( H, (cid:22) ) with the prop-erty that the preorder (cid:22) is artinian (resp., noetherian).In the remainder, we will often use the word “artinianity” (resp., “noetherianity”) to refer to the prop-erty that a preorder or a premonoid is artinian (resp., noetherian).Artinian (and noetherian) premonoids lie at the heart of the approach to factorization set forth in thiswork. But before going into the details, a few remarks are in order (no pun intended). Remark 3.11. (1) Let (cid:22) be a preorder on a set X and assume that there is a function λ : X → N suchthat λ ( x ) < λ ( y ) whenever x ≺ y . Then (cid:22) is artinian, or else there would exist a sequence ( N k ) k ≥ ofnon-negative integers with N k +1 < N k for each k ∈ N (absurd). In particular, note that, if H is acyclic as per Definition 4.2 and (cid:22) is the divisibility preorder | H , then λ is a length function in the sense of [18,Definition 1.1.3.2] (the commutative case) and [16, Definition 2.26].(2) Every preorder (cid:22) on a finite set X is artinian. In fact, let λ be the function X → N that mapsan element x ∈ X to the largest k ∈ N for which there are x , . . . , x k ∈ X with x = x and x i +1 ≺ x i for each i ∈ J , k − K . Since ≺ is a transitive relation on X and x ≺ y implies x = y , the finiteness of X Salvatore Tringali guarantees the well-definiteness of λ (by the Pigeonhole Principle). It is then clear by construction that x ≺ y yields λ ( x ) < λ ( y ) . So, by item (1) on this list, (cid:22) is artinian (as wished).(3) Assume that (cid:22) is an artinian preorder on a set X , and let S be a non-empty subset of X . Thenit is well known that S has at least one (cid:22) -minimal element , meaning that there exists ¯ x ∈ S such that, if y (cid:22) ¯ x for some y ∈ S , then ¯ x (cid:22) y : We include the short proof here for the sake of completeness.To begin, choose an arbitrary x ∈ S (this is possible because S is not the empty set). Next, recursivelydefine an S -valued sequence ( x k ) k ≥ as follows: Start with x := x . If, for some k ∈ N , x k is not a (cid:22) -min-imal element of S , then pick y ∈ S such that y ≺ x k and set x k +1 := y ; otherwise, set x k +1 := x k . Since (cid:22) is assumed to be artinian, there exists k ∈ N such that x k +1 is (cid:22) -equivalent to x k for every k ≥ k ;and by construction of the sequence ( x k ) k ≥ , this means that x k is a (cid:22) -minimal element of S .(4) In the notations of items (3) and (4) of Example 3.5, H satisfies the ACCP (as formulated in thefirst paragraph of the introduction) if and only if the divisibility preorder | H is artinian, while H satisfiesthe ACCPR (resp., ACCPL) if and only if ⊢ H (resp., ⊣ H ) is artinian.At long last, we are finally ready to state the main (though probably the easiest) result of the paper. Theorem 3.12.
Let ( H, (cid:22) ) be an artinian premonoid. Then H is a (cid:22) -factorable monoid. Proof.
Let Ω be the set of (cid:22) -non-units of H that do not factor as a product of (cid:22) -irreducibles of H , andsuppose for a contradiction that Ω = ∅ . By Remark 3.11(3), Ω has a (cid:22) -minimal element ¯ x . In particular, ¯ x is a (cid:22) -non-unit, but not a (cid:22) -irreducible of H . So, ¯ x = yz for some (cid:22) -non-units y, z ∈ H with y ≺ ¯ x and z ≺ ¯ x . But this is only possible if y / ∈ Ω and z / ∈ Ω , since ¯ x is a (cid:22) -minimal element of Ω . Therefore,each of y and z factors as a product of (cid:22) -irreducibles; whence the same is also true for ¯ x (absurd). (cid:4) Theorem 3.12 has a fairly abstract formulation and a rather simple proof: Both of these features arepart of the reason why the result applies to a wide range of different situations (as we will see). But beforeturning to applications, we aim to show that, in addition to the mere existence of certain factorizations,one can say a little more about the “arithmetic of a premonoid” ( H, (cid:22) ) when H and (cid:22) are related by acondition that, while much stronger than artinianity, is often met in practice. Definition 3.13.
Given a premonoid ( H, (cid:22) ) and an element x ∈ H , we denote by ht H (cid:22) ( x ) the supremumof the set of all n ∈ N + for which there exist (cid:22) -non-units x , . . . , x n ∈ H with x = x and x i +1 ≺ x i foreach i ∈ J , n − K , where sup ∅ := 0 . We call ht H (cid:22) ( x ) the (cid:22) -height of x (relative to the monoid H ) andsay that ( H, (cid:22) ) is a strongly artinian premonoid if ht H (cid:22) ( y ) < ∞ for every y ∈ H . We will usually write ht( · ) instead of ht H (cid:22) ( · ) if no confusion can arise.Definition 3.13 is resonant with the notions of “ideal height” and “Krull dimension” in ring theory:This is no coincidence and we hope to discuss the details in future work. Proposition 3.14.
Let ( H, (cid:22) ) be a strongly artinian premonoid and suppose that, for each x ∈ H thatis neither a (cid:22) -unit nor a (cid:22) -quark, there are (cid:22) -non-units y, z ∈ H with y (cid:22) x and z (cid:22) x such that x = yz and ht( y ) + ht( z ) ≤ ht( x ) . The following hold:(i) The preorder (cid:22) is artinian, the monoid H is (cid:22) -factorable, and every (cid:22) -irreducible is a (cid:22) -quark.(ii) Every (cid:22) -non-unit x ∈ H factors into a non-empty product of ht( x ) or fewer (cid:22) -quarks. n Abstract Factorization Theorem and Some Applications Proof. (i) As the (cid:22) -height of each element of H is finite (by hypothesis), the function λ : H → N : x ht( x ) is well defined. In particular, it is obvious from Definition 3.13 that λ ( u ) = ht( u ) < ht( v ) = λ ( v ) for all u, v ∈ H with u ≺ v . Therefore, we get from Remark 3.11(1) that (cid:22) is an artinian preorder; andby Theorem 3.12 this yields that H is a (cid:22) -factorable monoid.We are left to check that every (cid:22) -irreducible of H is also a (cid:22) -quark. Let x ∈ H be neither a (cid:22) -unit nora (cid:22) -quark. It is then guaranteed by our assumptions that there exist (cid:22) -non-units y, z ∈ H with y (cid:22) x and z (cid:22) x such that x = yz and ht( y ) + ht( z ) ≤ ht( x ) . It follows y ≺ x ; otherwise, y is (cid:22) -equivalent to x and, hence, ht( x ) = ht( y ) and ht( z ) = 0 , which is impossible because the only elements u ∈ H with ht( u ) = 0 are the (cid:22) -units. Likewise, we see that z ≺ x . In consequence, x is not a (cid:22) -irreducible.(ii) Let x be a (cid:22) -non-unit of H and set n := ht( x ) . If n = 1 , then x is a (cid:22) -quark and the conclusionis trivial. Hence assume n ≥ , and suppose inductively that every (cid:22) -non-unit of H of (cid:22) -height h < n factors into a product of h or fewer (cid:22) -quarks. Since x is neither a (cid:22) -unit nor a (cid:22) -quark, we have by theassumptions made in the statement that x = yz for some (cid:22) -non-units y, z ∈ H with ht( y ) + ht( z ) ≤ n .As in the proof of part (i), it follows ≤ ht( y ) < n and ≤ ht( z ) < n . So, by the inductive hypothesis,there exist k ∈ J , ht( y ) K and ℓ ∈ J , ht( z ) K such that y = a · · · a k and z = b · · · b ℓ for certain (cid:22) -quarks a , . . . , a k , b , . . . , b ℓ ∈ H . Then x = yz = a · · · a k b · · · b ℓ ; and this is enough to finish the proof (byinduction on n ), upon considering that k + ℓ ≤ ht( x ) . (cid:4) As a first test bench for the ideas heretofore set forth, we are going to apply Proposition 3.14 to aclassical problem in group theory (further applications will be discussed in Sect. 4):
Example 3.15.
Let X be a finite k -element set and S ( X ) be the symmetric group of X , that is, the set ofall permutations of X endowed with the operation of ( functional ) composition ◦ : S ( X ) × S ( X ) → S ( X ) that maps a pair ( f, g ) of permutations of X to the permutation f ◦ g : X → X : x f ( g ( x )) . We willdenote the identity of S ( X ) (that is, the identity function on X ) by id X .It is well known (see the introduction) that every f ∈ S ( X ) factors as a composition of transpositions ,i.e., permutations of X that exchange two elements and keep all others fixed. Below we give a new proofof this result, by showing that, for k ≥ , every f ∈ S ( X ) r { id X } is a composition of k − − | Fix( f ) | or fewer transpositions, where Fix( f ) := { x ∈ X : f ( x ) = x } is the set of fixed points of f : The bound k − − | Fix( f ) | is sharp but not best possible (see, e.g., [26] and references therein); this, however, is notthe point here. (For k = 0 or k = 1 , S ( X ) is a one-element group and there is nothing to prove.)To begin, assume k ≥ and let (cid:22) be the dual of the pullback of the standard order ≤ on N through thefunction φ : S ( X ) → N : f
7→ |
Fix( f ) | (see items (2) and (4) of Example 3.3 for the terminology); to wit,we have that f (cid:22) g , for some f, g ∈ S ( X ) , if and only if f has at least as many fixed points as g . Clearly,a permutation f of X has k − or more fixed points if and only if f = id X . It follows that ( S ( X ) , (cid:22) ) isa strongly artinian premonoid with ht( f ) = k − − | Fix( f ) | ≤ k − for each f ∈ S ( X ) r { id X } ; whencethe (cid:22) -quarks of S ( X ) are the permutations with exactly k − fixed points, viz., the transpositions.Now, suppose that f ∈ S ( X ) is neither the identity id X nor a transposition, so that φ ( f ) = | Fix( f ) | ≤ k − . Accordingly, pick an element ¯ x ∈ X that is not a fixed point of f , and let τ be the transpositionthat exchanges ¯ x and f (¯ x ) . On the one hand, f = τ ◦ ( τ ◦ f ) = ( τ ◦ τ ) ◦ f = id X ◦ f = f . On the other,it is readily checked that Fix( f ) ∪ { ¯ x } ⊆ Fix( τ ◦ f ) ; note, in particular, that f (¯ x ) is not a fixed point of f , or else we would have f ( f (¯ x )) = f (¯ x ) , contradicting that f is injective. Consequently, τ ◦ f has morefixed points than f and, hence, ht( τ ) + ht( τ ◦ f ) = 1 + ht( τ ◦ f ) ≤ ht( f ) .2 Salvatore Tringali
So, putting it all together, we can conclude from Proposition 3.14 that every f ∈ S ( X ) r { id X } factorsas a composition of k − − | Fix( f ) | or fewer transpositions (as wished).We close the section by remarking that the artinianity of a premonoid ( H, (cid:22) ) , while sufficient for each (cid:22) -non-unit of H to factor as a product of (cid:22) -irreducibles (Theorem 3.12), is not necessary: Just let H bethe multiplicative monoid of the non-zero elements of the integral domain constructed by Grams in [21,Sect. 1] and (cid:22) be the divisibility preorder | H on H ; and consider that the | H -irreducibles of a cancellative,commutative monoid H are precisely the atoms of H (Corollary 4.4). A different construction will bepresented in Example 4.8, where, among other things, we show that it is even possible for a monoid H to be reduced, finitely generated, cancellative, and | H -atomic, and yet not satisfy the ACCP (note that,by [18, Proposition 2.7.4.2], this can only happen if H is non-commutative).4. Applications
Below, we discuss some applications of Theorem 3.12. We organize the section into three subsections.Sects. 4.1 and 4.2 are devoted to the classical theory of factorization: The former is mainly about“nearly cancellative” monoids; and the latter is about power monoids , a “highly non-cancellative” class ofmonoids first introduced in [16] and further studied in [3]. In Sect. 4.3, we obtain an “object decompositiontheorem” (Corollary 4.13) for certain categories with finite products which, among other things, leadsto a monoid-theoretic and, in a way, more conceptual proof that every R -module M of finite uniformdimension over a ring R is isomorphic to a direct product of finitely many indecomposable R -modules.4.1. Classical factorizations.
Many fundamental aspects of the classical theory of factorization comedown to the study of various phenomena that are related to the possibility or impossibility of factoring the (cid:22) -non-units of a monoid H into (cid:22) -atoms or (cid:22) -irreducibles, where (cid:22) is either the divisibility preorder | H ,or the “divides from the left” preorder ⊢ H , or the “divides from the right” preorder ⊣ H (see, in particular,items (3) and (4) of Example 3.5). The following corollary will help us to substantiate our claims. Corollary 4.1. If H is a Dedekind-finite monoid and the divisibility preorder | H is artinian, then everynon-unit of H factors as a (non-empty, finite) product of | H -irreducibles. Proof.
It is enough to apply Theorem 3.12, after recalling from Example 3.8(4) that, if H is Dedekind-finite, then the | H -units of H are precisely the units of H . (cid:4) As simple as it may be, Corollary 4.1 is a non-commutative generalization of Theorem 3.2 in Andersonand Valdes-Leon’s seminal paper [2] on irreducible factorizations in commutative rings: Every commuta-tive monoid is Dedekind-finite and, by Example 3.8(4), the | H -irreducibles of a Dedekind-finite monoid H are precisely the irreducibles of H as per Anderson and Valdes-Leon’s work.Actually, we will show in the remainder of this section that Corollary 4.1 is also a generalization of[10, Proposition 0.9.3], i.e., Cohn’s classical result on atomic factorizations in cancellative monoids. Definition 4.2.
A monoid H is unit-cancellative if xy = x = yx for all x, y ∈ H with y / ∈ H × ; and is acyclic if uxv = x for all u, v, x ∈ H with u / ∈ H × or v / ∈ H × .Unit-cancellative monoids were recently introduced in [15, 16], as part of a broader program aimedto extend various aspects of the classical theory of factorization to the non-cancellative setting (every n Abstract Factorization Theorem and Some Applications Proposition 4.3.
Let H be a unit-cancellative monoid. The following hold:(i) H is Dedekind-finite and an element u ∈ H is a ⊢ H -unit (resp., a ⊣ H -unit), if and only if u is a | H -unit, if and only if u is a unit.(ii) An element a ∈ H is a ⊢ H -quark (resp., a ⊣ H -quark), if and only if a is a ⊢ H -atom (resp., a ⊣ H -atom), if and only if a is a | H -atom, if and only if a is an atom.(iii) Every | H -atom of H is a | H -quark. Proof. (i) By Remark 3.8(4), it suffices to check that H is Dedekind-finite, and this is straightforward: If xy = 1 H for some x, y ∈ H , then xyx = x and hence yx ∈ H × (by the fact that H is unit-cancellative).(ii) Let a ∈ H . We will only show that a is a ⊢ H -quark if and only if a is an atom, since we get fromRemark 3.8(4) and part (i) that a is a ⊢ H -atom if and only if a is a | H -atom, if and only if a is an atom:The analogous statement for ⊣ H can be proved in a similar way.Assume first that a is an atom but not a ⊢ H -quark. Then aH ( xH for some x ∈ H r H × , as every ⊢ H -unit is, by part (i), a unit. In consequence, a = xu for some u ∈ H , which can only happen if u ∈ H × ,because a is an atom and x is not a unit. It follows that a ⊢ H x and, hence, aH ( xH ⊆ aH (absurd).Now, suppose by way of contradiction that a is a ⊢ H -quark but not an atom. Then a = xy for some x, y ∈ H r H × ; and from here we get that a ⊢ H x , since x divides a from the left but is not a ⊢ H -unit(again by part (i)). Thus a = xy = auy for some u ∈ H , which is a contradiction, because it implies (bythe unit-cancellativity of H ) that y is a unit.(iii) Assume to the contrary that there is a | H -atom a ∈ H that is not a | H -quark. There then exist a | H -non-unit b ∈ H such that b | H a and a ∤ H b . In particular, a = ubv for some u, v ∈ H such that u or v is not a unit. This, however, is only possible if ub or bv is a unit, because a is a | H -atom and, by part(ii), every | H -atom is an atom. So, using that, by part (i), H is Dedekind-finite, we see that b is a unit;whence we get a contradiction, because, again by part (i), every unit is a | H -unit. (cid:4) Corollary 4.4.
Let H be an acyclic monoid, and let a ∈ H . The following are equivalent:(a) a is a ⊢ H -quark (resp., a ⊣ H -quark).(b) a is a ⊢ H -atom (resp., a ⊣ H -atom).(c) a is a | H -quark.(d) a is a | H -atom.4 Salvatore Tringali (e) a is an atom.(f) a is a | H -irreducible. Proof.
Every acyclic monoid is also unit-cancellative. So, by Remark 3.8(5) and Proposition 4.3, H isDedekind-finite, units and | H -units are one thing, and we only need check that (f) ⇒ (d).To this end, let a ∈ H be a | H -irreducible and suppose for a contradiction that a is not a | H -atom.Since every | H -unit is a unit, it follows that a = xy for some x, y ∈ H r H × . Thus, x | H a and y | H a ;and since a is a | H -irreducible, this is only possible if a | H x or a | H y . To wit, x = uav or y = uav forsome u, v ∈ H . In consequence, a = uavy or a = xuav ; and this implies, by the acyclicity of H , that xu ∈ H × or vy ∈ H × . So, using that H is Dedekind-finite, we get x ∈ H × or y ∈ H × (absurd). (cid:4) Based on Remark 3.8, Proposition 4.3, and Corollary 4.4, the two diagrams below provide a succinctoverview of the logical relations between (cid:22) -quarks, (cid:22) -atoms, and (cid:22) -irreducibles for a unit-cancellative oracyclic monoid H , when (cid:22) is either the “divides from the left” preorder ⊢ H , the “divides from the right”preorder ⊣ H , or the divisibility preorder | H (by the way, we will find later on, in Example 4.8, that the | H -irreducibles of a cancellative monoid need not be | H -atoms or | H -quarks). ⊢ H -quark ⊢ H -atom ⊢ H -irreducible | H -quarkatom | H -atom | H -irreducible ⊣ H -quark ⊣ H -atom ⊣ H -irreducible(a). The unit-cancellative case. ⊢ H -quark ⊢ H -atom ⊢ H -irreducible | H -quarkatom | H -atom | H -irreducible ⊣ H -quark ⊣ H -atom ⊣ H -irreducible(b). The acyclic case. With this done, we are ready for the next theorem and its corollary: The latter subsumes [16, The-orem 2.28(i)], that is, Fan and the author’s generalization to unit-cancellative monoids of Cohn’s [10,Proposition 0.9.3] (see also the comments under Corollary 4.1).
Theorem 4.5.
The following conditions are equivalent for a monoid H :(a) H is unit-cancellative and the preorders ⊢ H and ⊣ H are both artinian.(b) H is acyclic and the divisibility preorder | H is artinian.Moreover, each of these conditions implies that every non-unit of H factors as a product of atoms. Proof. (a) ⇒ (b): We prove first that H is acyclic ( Part 1 ) and next that | H is artinian ( Part 2 ). Part 1: H is acyclic . Assume by way of contradiction that H is not acyclic. Since the statement tobe proved is “left-right symmetric”, it follows (without loss of generality) that the set Λ := { x ∈ H : x = uxv for some u, v ∈ H with u / ∈ H × } is non-empty; and since ⊣ H is artinian, we get from Remark 3.11(3) that Λ has at least one ⊣ H -minimalelement ¯ x . In particular, ¯ x is an element of Λ and, hence, ¯ x = u ¯ xv for some u, v ∈ H with u / ∈ H × . Set y := ¯ xv . Since uyv = ( u ¯ xv ) v = ¯ xv = y and u / ∈ H × , we see that y is in Λ . On the other hand, we have n Abstract Factorization Theorem and Some Applications ¯ x = u ¯ xv = uy and hence y ⊣ H ¯ x . Recalling that ¯ x is a ⊣ H -minimal element of Λ , it follows that ¯ x is ⊣ H -equivalent to y , that is, H ¯ x = Hy . This, however, means that ¯ xv = y = w ¯ x for some w ∈ H , withthe result that ¯ x = u ¯ xv = uw ¯ x . So, using that H is unit-cancellative and hence Dedekind-finite (by item(i) of Proposition 4.3), we conclude that u is a unit (absurd). Part 2: | H is artinian . Let ( x k ) k ≥ be a | H -non-increasing sequence. We need find that x k | H x k +1 for all large k , and we will actually show the stronger statement that x k ∈ H × x k +1 H × from some k on.To start with, we are given that, for each k ∈ N + , there exist u k , v k ∈ H such that x k − = u k x k v k .We claim that all but finitely many terms of the sequence v , v , . . . are units; mutatis mutandis, thesame argument also applies to the sequence u , u , . . . , and this will be enough to conclude.Fix k ∈ N + and set q k := u · · · u k . Since u k x k v k = x k − , it is evident that q k x k v k = q k − u k x k v k = q k − x k − and hence q k x k ⊢ H q k − x k − ; i.e., the sequence q x , q x , . . . is ⊢ H -non-increasing. Since ⊢ H isartinian (by hypothesis), it follows that there exists k ′ ∈ N + such that, for k ≥ k ′ , q k − x k − ⊢ H q k x k and,hence, q k x k = q k − x k − r k − for some r k − ∈ H . In consequence, we get from the above that, for k ≥ k ′ , q k − x k − = q k x k v k = q k − x k − r k − v k . So, recalling that H is unit-cancellative (and Dedekind-finite),we see that, for all large k ∈ N + , r k − v k is a unit and hence the same is true of v k (as wished).(b) ⇒ (a): Since every acyclic monoid is unit-cancellative (see the comments under Definition 4.2), itis sufficient to show that the preorders ⊢ H and ⊣ H are both artinian: We will work out the details for ⊢ H only, as the other case is essentially the same.Let ( x k ) k ≥ be a ⊢ H -non-increasing sequence, so that, for every k ∈ N , there is an element y k ∈ H such that x k = x k +1 y k . We need to show that x k ⊢ H x k +1 for all but finitely many k , and we will actuallyprove the stronger statement that y k is a unit for all large k .Indeed, it is clear from the standing assumptions that ( x k ) k ≥ is a | H -non-increasing sequence. Since | H is artinian, it follows that there exists k ′ ∈ N such that, for every k ≥ k ′ , x k +1 = u k x k v k = u k x k +1 y k v k for some u k , v k ∈ H . This yields, by the acyclicity of H , that y k is a unit for all large k (as wished). (cid:4) Corollary 4.6.
The following conditions are equivalent for a monoid H :(a) H is unit-cancellative and satisfies the ACCPR and the ACCPL.(b) H is acyclic and satisfies the ACCP.Moreover, each of these conditions implies that every non-unit of H factors as a product of atoms. Proof.
This is simply a reformulation of Theorem 4.5 based on Remark 3.11(4). (cid:4)
Corollary 4.6 is, in a way, best possible, as suggested by the next two examples (see Sect. 5 for a coupleof open questions that could further clarify the picture): The first shows that an acyclic, cancellativemonoid satisfying the ACCPL need not satisfy the ACCP; the second shows, among other things, that areduced, finitely generated, cancellative, non-commutative monoid H can be | H -atomic or | H -factorablewithout satisfying the ACCP (cf. the comments at the end of Sect. 3). Example 4.7.
In [27, Example 2.6], R. Mazurek and M. Ziembowski construct a linearly ordered (andhence cancellative) monoid ( H, (cid:22) ) that satisfies the ACCPL but not the ACCPR (we recall from Definition3.4 that “linearly ordered” means that ux ≺ uy and xu ≺ yu for all u, x, y ∈ H with x ≺ y ); moreover, H is positive , i.e., H (cid:22) x for every x ∈ H . It follows that H is acyclic, as it is straightforward to checkthat x ≺ uxv for all u, v, x ∈ H with u = 1 H or v = 1 H . Consequently, we see from Corollary 4.6 that H does not satisfy the ACCP, or else it would also satisfy the ACCPR (a contradiction).6 Salvatore Tringali
Example 4.8.
Fix n ∈ N + , and let H be the presentation Mon h A | R i , where A is the -element set { a, b } and R is the relation { ( b ∗ n , a ∗ b ∗ n ∗ a ) } on the free monoid F ( A ) (see Sect. 2.3 for terminology andnotation). By Theorem 2.1, H is a cancellative monoid, for it is defined by a finite presentation whose leftand right graphs are cycle-free (each is a path graph on two vertices, i.e., a single edge). In addition, itis clear from the nature of the defining relations in R that, if two A -words u and v are congruent modulo R ♯ , then v Hb ( u ) = v Hb ( v ) = nk for some k ∈ N + ; whence the only unit of H is the identity H , i.e., thecongruence class of ε A (the empty A -word) modulo R ♯ .Thus, we obtain from Proposition 4.3(i) that H is also the only | H -unit of H . It follows, by Example3.8(4) and the same “ b -adic argument” used in the above, that a is a | H -atom; and so is b for n ≥ . Onthe other hand, every | H -atom is a | H -irreducible (Example 3.8(5)); and we aim to show that • if n = 1 , then b is a | H -irreducible ( Part 1 ) but neither a | H -atom nor a | H -quark ( Part 2 ); • none of the preorders ⊢ H , ⊣ H , and | H is artinian ( Part 3 ).Overall, this will mean that H is | H -factorable for all values of n , and is | H -atomic if and only if n ≥ ;however, these conclusions cannot be drawn from Theorem 4.5 (note, incidentally, that H is not acyclic,because b ∗ n ≡ a ∗ b ∗ n ∗ a mod R ♯ and a / ∈ H ). Part 1: b is | H -irreducible . If not, then we see from Example 3.8(4) that b ≡ u ∗ v mod R ♯ for some u , v ∈ F ( A ) such that b ∤ H u and b ∤ H v (recall that the only | H -unit of H is H ). So, u and v are powersof a in F ( A ) ; whence b ≡ a ∗ k mod R ♯ for some k ∈ N . But this is a contradiction, as we know from theabove that two A -words are congruent modulo R ♯ only if they contain an equal number of b ’s. Part 2: If n = 1 , then b is neither a | H -atom nor a | H -quark . Suppose n = 1 . Then b ≡ a ∗ b ∗ a mod R ♯ (by the definition of H ), but neither a nor b ∗ a is a | H -unit. Therefore, b is a | H -non-unit and factorsin H as a product of two | H -non-units; viz., b is not a | H -atom. Moreover, it is clear that b ∤ H a , or else v Hb ( a ) = v Hb ( b ) = 1 (absurd); so, b is not a | H -quark, because a ∤ H b . Part 3:
None of ⊢ H , ⊣ H , and | H is artinian . Let ( z k ) k ≥ be the H -valued sequence whose k th term z k is the A -word b ∗ n ∗ a ∗ k ∗ b ∗ n (taken modulo R ♯ ). For every k ∈ N , we have that z k +1 ∗ a ≡ b ∗ n ∗ a ∗ k ∗ a ∗ b ∗ n ∗ a ≡ b ∗ n ∗ a ∗ k ∗ b ∗ n ≡ z k mod R ♯ ; and in a similar way, a ∗ z k +1 ≡ z k mod R ♯ . To wit, the sequence ( z k ) k ≥ is ⊢ H - and ⊣ H -non-increasing;therefore, it is also | H -non-increasing (by the fact that ⊢ H is a subrelation of | H ).Suppose for a contradiction that there exists k ∈ N such that z i ⊢ H z k +1 , z i ⊣ H z k +1 , or z i | H z k +1 for some i ∈ J , k K ; in fact, we may assume that i is the smallest integer between and k (inclusive) forwhich this holds. Accordingly, we can find u , v ∈ F ( A ) such that u ∗ z i ∗ v ≡ z k +1 mod R ♯ ; and since v Hb ( z i ) = v Hb ( z k +1 ) = 2 n , u and v are necessarily powers of a in F ( A ) , i.e., u = a ⋆r and v = a ⋆s for some r, s ∈ N (recall that two A -word are congruent modulo R ♯ only if they have the same b -adic valuation).We claim that neither r nor s can be zero. In fact, assume to the contrary that r = 0 (the other caseis symmetric). It then follows from the above that b ∗ n ∗ a ∗ i ∗ b ∗ n ∗ a ∗ s ≡ b ∗ n ∗ a ∗ ( k +1) ∗ b ∗ n mod R ♯ ; and by the cancellativity of H , we get b ∗ n ∗ a ∗ s ≡ a ∗ ( k +1 − i ) ∗ b ∗ n mod R ♯ . (1) n Abstract Factorization Theorem and Some Applications a ∗ j ∗ b n ∗ a ∗ j ≡ b n mod R ♯ , for every j ∈ N . (2)So, multiplying both sides of the congruence in Equ. (1) by a ∗ s , we obtain that a ∗ ( s + k +1 − i ) ∗ b ∗ n ≡ a ∗ s ∗ b n ∗ a ∗ s (2) ≡ b n mod R ♯ ; whence a ∗ ( s + k +1 − i ) ≡ ε A mod R ♯ (recall that H is cancellative). This, however, is impossible, since itgives i = s + k + 1 ≥ k + 1 . In consequence, r and s must be non-zero (as wished).To sum it up, we have established that a ∗ r ∗ z i ∗ a ∗ s ≡ z k +1 mod R ♯ for some r, s ∈ N + ; and we shallsee from here that i = 0 . In fact, assume that i is non-zero (i.e., a positive integer). Then z k +1 ≡ a ∗ r ∗ b ∗ n ∗ a ∗ i ∗ b n ∗ a ∗ s ≡ a ∗ ( r − ∗ b ∗ n ∗ a ∗ ( i − ∗ b n ∗ a ∗ s ≡ a ∗ ( r − ∗ z i − ∗ a ∗ s mod R ♯ ; and similarly, z k +1 ≡ a ∗ r ∗ z i − ∗ a ∗ ( s − mod R ♯ . Thus z i − ⊢ H z k +1 and z i − ⊣ H z k +1 , contradicting theminimality of i and yielding i = 0 (as wished). It follows that a ∗ r ∗ b ∗ n ∗ a ∗ ∗ b ∗ n | {z } z ∗ a ∗ s ≡ b ∗ n ∗ a ∗ ( k +1) ∗ b ∗ n | {z } z k +1 (2) ≡ a ∗ r ∗ b ∗ n ∗ a ∗ ( r + s + k +1) ∗ b ∗ n ∗ a ∗ s mod R ♯ ; which, by cancellativity, implies a ∗ ( r + s + k +1) ≡ ε A mod R ♯ . But this can only happen if r + s + k + 1 = 0 (absurd), because there is no non-empty A -word congruent to ε A modulo R ♯ (recall that H is reduced).So, putting it all together, we conclude that none of the preorders ⊢ H , ⊣ H , and | H is artinian, sincewe have shown that the sequence ( z k ) k ≥ is (strictly) decreasing with respect to each of them.We finish the subsection with a few remarks on Dedekind-fineteness, motivated by the critical role thiscondition plays in Corollary 4.1 and Theorem 4.5 (see also Proposition 4.11 and Theorem 4.12). Remarks 4.9. (1) Let H be a monoid. We will prove that, if the “divides from the left” preorder ⊢ H orthe “divides from the right” preorder ⊣ H is noetherian, then H is Dedekind-finite.In fact, pick x, y ∈ H such that xy = 1 H and assume ⊢ H is noetherian (the other case is symmetric);it suffices to prove that y is a unit. Since H ⊢ H y ⊢ H y ⊢ H · · · and, by hypothesis, no infinite sequenceof elements of H can be (strictly) ⊢ H -increasing, we have that y k +1 ⊢ H y k for some k ∈ N . It followsthat there exists u ∈ H such that x k y k +1 u = x k y k ; and this, in turn, gives that xy = 1 H = yu , for it isimmediate (by induction on k ) that x k y k = 1 H . So, y is a unit and we are done.(2) As a complement to the conclusions made in item (1), we will show that neither the artinianity northe noetherianity of the divisibility preorder | H is a sufficient condition for a monoid H to be Dedekind-fi-nite; nor is the artinianity of ⊢ H or ⊣ H .Indeed, let M be a monoid which is not Dedekind-finite (see, e.g., [3, Example 3.6]). Accordingly, pick x, y ∈ M such that xy = 1 M = yx , and let H be the submonoid of M generated by { x, y } . Of course, H is not Dedekind-finite. However, H = HzH for all z ∈ H , and hence | H is artinian and noetherian.In fact, fix z ∈ H . Then z ∈ { x, y } n for some n ∈ N + ; and since x k y k = 1 H for all k ∈ N (by inductionon k ), it is readily found (by induction on n ) that there exist a, b ∈ N such that z = y a x b . It follows that x a zy b = 1 H and hence H ⊇ HzH ⊇ Hx a zy b H = H . To wit, H = HzH (as wished).8
Salvatore Tringali
It remains to see that ⊢ H is artinian (the other case is symmetric). To start with, it is easily checkedthat, if y p x q u = y r x s for some p, q, r, s ∈ N and u ∈ H , then p ≤ r : Otherwise, we would have H = x r y r x s y s = x r y p x q u y s = y p − r x q u y s = yv, and hence y ∈ H × (absurd), where v := y p − r − x q u y s ∈ H (recall that x k y k = 1 H for all k ∈ N ). Itfollows that, if ( z k ) k ≥ is a ⊢ H -non-increasing sequence of elements of H , then there exist α, b , b , . . . ∈ N such that z k = y α x b k for every large k ∈ N (we proved above that each z ∈ H has the form y a x b forsome a, b ∈ N ). So H is ⊢ H -artinian, because y q x r y r x s = y q x s (i.e., y q x r ⊢ H y q x s ) for all q, r, s ∈ N .(3) Let H be a periodic monoid, meaning that, for each z ∈ H , the subsemigroup of H generated by z is finite. We claim that H is Dedekind-finite. Indeed, suppose xy = 1 H for some x, y ∈ H ; it suffices toshow that y is a unit. To this end, note that, since the set { y, y , . . . } is finite, we are guaranteed (by thePigeonhole Principle) that y m = y m + n for some m, n ∈ N + . Thus, we find H = x m y m = x m y m + n = y n ,because x k y k = 1 H for all k ∈ N (cf. item (1)). Therefore, y is a unit (as wished).4.2. Power monoids.
Let H be a monoid. Following [3], we let the reduced power monoid of H , hereafterdenoted by P fin , ( H ) , be the monoid obtained by endowing the set of all finite subsets of H containing theidentity H with the operation of setwise multiplication induced by H , so that XY = { xy : x ∈ X, y ∈ Y } for all X, Y ∈ P fin , ( H ) . Note that the identity of P fin , ( H ) is the singleton { H } .The arithmetic of P fin , ( H ) is rich and, in a way, rather intricate, even in the fundamental case where H is a cyclic group: Part of the reason lies in the “highly non-cancellative” nature of the operation ofsetwise multiplication, which results in a variety of algebraic and arithmetical phenomena not observablein the “nearly cancellative” scenarios discussed in Sect. 4.1 (see [16, 3] for further details).Below we add to this line of research by showing that P fin , ( H ) is | P fin , ( H ) -factorable, and by character-izing the monoids H for which every X ∈ P fin , ( H ) factors as a product of “ordinary atoms” (Proposition4.11(iii) and Theorem 4.12). In the proofs, we will repeatedly use that | XY | ≥ max( | X | , | Y | ) for all X, Y ∈ P fin , ( H ) , and X ⊆ Y whenever X | P fin , ( H ) Y (by the fact that each set in P fin , ( H ) contains H ).We start with a slight refinement of [3, Lemma 3.8], for which we recall that an element x in a monoidis an idempotent if x = x ; and is a non-trivial idempotent if x is an idempotent but not the identity. Lemma 4.10.
Let H be a monoid with no non-trivial idempotent. Then H is Dedekind-finite; and every x ∈ H which generates a finite subsemigroup, is a unit. Proof.
First, suppose that yz = 1 H for some y, z ∈ H . Then ( zy ) = z ( yz ) y = zy ; and since H has nonon-trivial idempotents, we conclude that zy = 1 H . Consequently, H is Dedekind-finite.Next, assume that Sgrp h x i H = { x, x , . . . } is a finite subsemigroup of H for some x ∈ H . There thenexist n, k ∈ N + such that x n = x n + k (by the Pigeonhole Principle); and this implies (by a routine induc-tion) that x n = x n + hk for all h ∈ N . So, we find that ( x nk ) = x nk = x ( k +1) n x ( k − n = x n x ( k − n = x nk .Since H has no non-trivial idempotents, it follows that x nk = 1 H . To wit, x is a unit. (cid:4) Proposition 4.11.
Let H be a monoid. The following hold:(i) P fin , ( H ) is a reduced and Dedekind-finite monoid, and the preorder | P fin , ( H ) is artinian.(ii) Each unit is a | P fin , ( H ) -unit, and vice versa; and each atom is a | P fin , ( H ) -atom, and vice versa.(iii) Every X ∈ P fin , ( H ) factors as a product of | P fin , ( H ) -irreducibles.(iv) A set A ∈ P fin , ( H ) is a | P fin , ( H ) -irreducible if and only if it is a | P fin , ( H ) -quark. n Abstract Factorization Theorem and Some Applications Proof.
Part (ii) is immediate from (i) and Example 3.8(5), and (iii) is straightforward from (i) and Cor-ollary 4.1 (in particular, note that the identity of P fin , ( H ) is an empty product of | P fin , ( H ) -irreducibles).So, we will focus our attention on parts (i) and (iv).(i) Since | XY | ≥ max( | X | , | Y | ) for all X, Y ∈ P fin , ( H ) , it is clear that XY = { H } if and only if X and Y are singletons, if and only if X = Y = { H } . Thus P fin , ( H ) is reduced and Dedekind-finite.On the other hand, since X is contained in Y whenever X | P fin , ( H ) Y , it is clear that a | -non-increasingsequence ( X k ) k ≥ of sets in P fin , ( H ) is also non-increasing with respect to inclusion; and this, in turn,can only happen if X k +1 = X k for all large k ∈ N , because the elements of P fin , ( H ) are finite sets. Inconsequence, | P fin , ( H ) is an artinian preorder.(iv) In view of Example 3.8(5), it is enough to prove the “only if” direction. Suppose for a contradictionthat there is a | P fin , ( H ) -irreducible A of P fin , ( H ) which is not a | P fin , ( H ) -quark. Since X ⊆ Y whenever X | P fin , ( H ) Y and, by part (i), P fin , ( H ) is reduced and Dedekind-finite, there then exists B ∈ P fin , ( H ) such that B | P fin , ( H ) A and { H } ( B ( A . Thus, A = U BV for some
U, V ∈ P fin , ( H ) with U = { H } or V = { H } . This, however, is only possible if A = U B or A = BV : Otherwise, both U and V are proper subsets of A , because U ⊆ U B ⊆ A and V ⊆ BV ⊆ A ; hence, A = XY for some | P fin , ( H ) -non-units X, Y ∈ P fin , ( H ) with A ∤ P fin , ( H ) X and A ∤ P fin , ( H ) Y , contradicting that A is a | P fin , ( H ) -irreducible (justtake X := U and Y := BV if U = { H } ; and X := U B and Y := V if V = { H } ).So, assume A = U B (the other case is similar). Then U = { H } , because B is properly contained in A .It follows that U = A ; or else U B is a factorization of A into two | P fin , ( H ) -non-units with A ∤ P fin , ( H ) U and A ∤ P fin , ( H ) B , again in contradiction to the fact that A is a | P fin , ( H ) -irreducible. As a result, we have { H } ( B ( AB = A . Accordingly, pick b ∈ B r { H } ⊆ A and set A b := A r { b } . Then ≤ | A b | < | A | (note that { H , b } ⊆ B ( A ); and since H ∈ A b ∩ B , it is readily seen that A b B ⊆ AB = A = A b ∪ { b } ⊆ A b B ∪ { b } ⊆ A b B ∪ B = A b B, i.e., A = A b B . But this is absurd, for it means similarly as above that A is not a | P fin , ( H ) -irreducible. (cid:4) The next result is a sensible refinement of [3, Theorem 3.9], where it is proved that, for a monoid H ,every set in P fin , ( H ) factors as a product of atoms if and only if H = x = x for all x ∈ H r { H } : Thekey difference is that, by Proposition 4.11, we here already know that every set in P fin , ( H ) factors as aproduct of | P fin , ( H ) -quarks (regardless of any condition on H ). Theorem 4.12.
The following are equivalent for a monoid H :(a) H = x = x for each x ∈ H r { H } .(b) Each | P fin , ( H ) -quark of P fin , ( H ) is an atom, and vice versa.(c) Every X ∈ P fin , ( H ) factors as a product of atoms. Proof.
The implication (b) ⇒ (c) is a trivial consequence of parts (iii) and (iv) of Proposition 4.11(iii).So we will concentrate on proving that (a) ⇒ (b) and (c) ⇒ (a).(a) ⇒ (b): Suppose for a contradiction that there is a | P fin , ( H ) -quark A ∈ P fin , ( H ) which is not anatom. Then A = XY for some non-units X, Y ∈ P fin , ( H ) , which implies by Proposition 4.11(i) that eachof X and Y is a | P fin , ( H ) -non-unit dividing A . But this can only happen if A , in turn, divides each of X and Y , because A is a | P fin , ( H ) -quark. So, using that, in P fin , ( H ) , “to divide” means “to be contained”,0 Salvatore Tringali we conclude that X = Y = A and hence A = XY = A . It follows (by a routine induction) that A = A n for every n ∈ N + . In consequence, it is clear that A = S n ≥ A n .Now, pick a ∈ A r { H } . The subsemigroup of H generated by a is finite, as we have that | A | < ∞ and { a, a , . . . } ⊆ S n ≥ A n = A . Since H = x = x for each x ∈ H r { H } (by hypothesis), we are thusguaranteed by Lemma 4.10 that a is a unit of H and a smallest n ∈ N + exists such that a n +1 = 1 H ; inparticular, n ≥ . So, setting B := A r { a n } and considering that A = A n +1 , we obtain { H , a } ⊆ B ( A ⊆ AB ⊆ A = AB ∪ Aa n ⊆ AB ∪ A n +1 = AB ∪ A = AB.
Then A = A = AB and hence B | P fin , ( H ) A . But this contradicts that A is a | P fin , ( H ) -quark, because { H } ( B ( A and hence A ∤ P fin , ( H ) B (recall that the only | P fin , ( H ) -unit of P fin , ( H ) is the identity).Every | P fin , ( H ) -quark is therefore an atom; and on the other hand, we have from Example 3.8(4) andparts (i) and (iv) of Proposition 4.11 that every atom is a | P fin , ( H ) -quark. So we are done.(c) ⇒ (a): Assume to the contrary that there exists an element x ∈ H r { H } with x = 1 H or x = x .Since P fin , ( H ) is a reduced monoid, { H , x } is by hypothesis a (non-empty) product A · · · A n of atoms A , . . . , A n ∈ P fin , ( H ) . It follows that A i = { H , x } for each i ∈ J , n K , because { H } ( A i ⊆ { H , x } .This is however a contradiction, by the fact that { H , x } = { H , x, x } = { H , x } . (cid:4) It is perhaps worth remarking that there is no obvious way to derive Proposition 4.11(iv) from Corollary4.4, or Theorem 4.12 from Proposition 4.3 and Theorem 4.5: The reason is that, in general, the reducedpower monoid P fin , ( H ) is far from being unit-cancellative (e.g., it is obvious that P fin , ( H ) is not unit-cancellative when ≤ | H | < ∞ ).4.3. Categories and “object decompositions”.
We begin with a quick review of some basic aspectsof category theory we will need below: We refer the reader to [25] for all terms used herein withoutdefinition, and we recall from Sect. 2.1 that we choose NBG set theory as foundations.Let C be a category. We denote by Ob( C ) and Arr( C ) , resp., the class of objects and the class of arrows(or morphisms) of C ; and given A, B ∈ Ob( C ) , we use Arr C ( A, B ) for the class of all arrows f ∈ Arr( C ) with domain A and codomain B . As usual, an object T ∈ Ob( C ) is terminal if Arr C ( A, T ) is a singletonfor each A ∈ Ob( C ) ; and an object P ∈ Ob( C ) is a product of an indexed set ( A i ) i ∈ I of objects of C if,for each i ∈ I , there is an arrow p i ∈ Arr C ( P, A i ) for which the following universal property holds:However we choose an object Q ∈ Ob( C ) and an indexed family ( p i : X → A i ) i ∈ I of arrowsof C , there is a unique u ∈ Arr C ( P, Q ) such that q i = p i ◦ C u for each i ∈ I , where we write g ◦ C f for the composite of a pair ( f, g ) ∈ Arr( C ) × Arr( C ) such that the codomain of f isthe same as the domain of g .It is an elementary fact that a product, when it exists, is unique up to isomorphism; and that an emptyproduct is nothing else than a terminal object (see [25, Sect. III.4] for further details).Suppose now that C is a category with finite products , meaning that every set of objects of C indexedby a finite set has a product in C : By [25, Sect. III.5, Proposition 1], this is equivalent to requiring that C has a terminal object and each pair ( A, B ) of objects of C has a product in C . We denote by V ( C ) the quotient of Ob( C ) by the equivalence that identifies two objects A and B of C if and only if thereis an isomorphism u ∈ Arr C ( A, B ) ; and we call an equivalence class in V ( C ) an isomorphism class of C .Accordingly, we can construct a monoid out of the objects of C by endowing the quotient V ( C ) with the(binary) operation ⊗ C that maps a pair ( a , b ) of isomorphism classes of C to the isomorphism class a ⊗ C b n Abstract Factorization Theorem and Some Applications A Π B ∈ Ob( C ) of an object A ∈ a by an object B ∈ b : The operation is well defined bythe universal property of products and makes the class V ( C ) into a reduced, commutative monoid (see,e.g., [14, Lemma 1.17]), herein referred to as the direct monoid of isomorphism classes of C and, by abuseof notation, identified with V ( C ) . This leads to the following: Corollary 4.13.
Let C be a category with finite products and assume there exists a function λ : Ob( C ) → N such that, for all A, B ∈ Ob( C ) , the following hold:(1) λ ( A ) = 0 if and only if A is a terminal object;(2) λ ( A ) + λ ( B ) ≤ λ ( A Π B ) for every product A Π B ∈ Ob( C ) of A by B .Then every X ∈ Ob( C ) is isomorphic to a finite product of non-terminal objects of C each of which isnon-isomorphic to a product of two non-terminal objects. Proof.
As noted in the comments above, the direct monoid V ( C ) of isomorphism classes of C is reduced andcommutative, and its identity is the isomorphism class of the terminal objects of C . On the other hand,we have from conditions (1) and (2) that λ ( B ) ≤ λ ( A Π B ) for all A, B ∈ Ob( C ) and every representative A Π B ∈ Ob( C ) of the product of A by B , with equality if and only if A is terminal. So, it is clear that V ( C ) is unit-cancellative and, by Remark 3.11(1), the divisibility preorder on V ( C ) is artinian. Therefore,we get from Corollaries 4.1 and 4.4 that every isomorphism class of C factors as a (finite) product ofatoms of V ( C ) . This finishes the proof, because an atom of V ( C ) is, obviously, the isomorphism class of anon-terminal object of C which is in turn non-isomorphic to a product of two non-terminal objects. (cid:4) Corollary 4.13 has many “concrete realizations”. Below, we discuss one of them in detail: The focuswill be on modules, but the same argument can be adapted to a whole variety of other objects for whicha “well-behaved” notion of “dimension”, “rank”, etc., is available.To begin, fix a (commutative or non-commutative) ring R . Following [24, Definition (6.2) and Corollary(6.6)], we let the uniform dimension u . dim R ( M ) of a (right) R -module M be the supremum of the set { k ∈ N + : N ⊕ R · · · ⊕ R N k ⊆ M, for some non-zero submodules N , . . . , N k of M } ⊆ R + ∪ {∞} , where ⊕ R denotes a direct sum of R -modules and we take sup ∅ := 0 . It is a basic fact that the uniformdimension is additive , in the sense that u . dim R ( M ⊕ R N ) = u . dim R ( M ) + u . dim( N ) , for all R -modules M and N, (3)see Part (1) of [24, Corollary (6.10)]. Together with Corollary 4.13, this leads straight to the following: Corollary 4.14.
Let R be a ring. Every R -module of finite uniform dimension is isomorphic to a directsum of finitely many indecomposable R -modules. Proof.
Let C be the full subcategory of the ordinary category Mod R of R -modules and module homomor-phisms whose objects are the R -modules with finite uniform dimension. It is a basic fact that Mod R is acategory with finite products: In particular, the terminal objects of Mod R are the zero R -modules, anda canonical representative of the product of two R -modules A and B is their direct sum A ⊕ R B . Sincethe inclusion functor of C in Mod R is fully faithful and, by [8, Proposition 2.9.9], fully faithful functorsreflect limits, it follows by Equ. (3) that C , too, is a category with finite products.On the other hand, if λ is the function Ob( C ) → N that maps an R -module to its uniform dimension,then we also get from Equ. (3) that λ ( A )+ λ ( B ) ≤ λ ( A ⊕ B ) for all A, B ∈ Ob( C ) ; in addition, λ ( A ) = 0 if2 Salvatore Tringali and only if A is a zero R -module (i.e., a terminal object of C ). Since λ ( A ) = λ ( B ) when the R -modules A and B are isomorphic, we thus conclude from Corollary 4.13 (and the very definition of an indecomposablemodule) that every R -module is isomorphic to a direct sum of indecomposable R -modules. (cid:4) By Part (1) of [24, Corollary (6.7)], Corollary 4.14 is seen to generalize the classical result that everyartinian or noetherian module over a ring R is isomorphic to a direct sum of indecomposable R -modules.In fact, it is perhaps worth stressing that Corollary 4.14 has a “direct and simple” proof all along thelines of the standard proof of the classical case. However, the point here is rather that we obtained theresult as a special case of an abstract and, in a way, elementary “object decomposition theorem” (thatis, Corollary 4.13), in which we get to characterize the indecomposable R -modules as the atoms of acertain (reduced, unit-cancellative, commutative) monoid where the divisibility preorder is artinian; andby the same “mechanical approach” analogous conclusions can be made for other objects with properties“similar” to those of modules with finite uniform dimension.5. Closing remarks and open questions
What we hope to have shown is that Theorem 3.12 and some of its descendants work as a sort ofblack box for a broad variety of problems: The inputs of the black box are a monoid H and an artinianpreorder (cid:22) on H ; the output is the existence of certain factorizations for every “large element” of H ,where an element is “large” if it is not (cid:22) -equivalent to the identity of H . In practice, if one’s goal is toprove some kind of factorization theorem (as in the “concrete cases” discussed in the previous sections),the recipe is always the same and consists of a sequence of four steps:(1) build up “the right monoid” H ;(2) find a “good candidate” for the preorder (cid:22) ;(3) prove that (cid:22) is artinian;(4) characterize the (cid:22) -irreducibles of H .One pro of this approach is that, as with other top-down approaches, one is able to bring apparentlydistant and independent questions under the unifying umbrella of a (natural) “grand theory of factor-ization”. Another pro is that Theorem 3.12 turns the task of proving the “global existence” of certainfactorizations into a “routine”, by revealing that the heart of the problem is not really in the existence ofa factorization, but rather in the four steps of the above recipe (some of which are often trivial).Be it as it may, there are many questions arising from this work that we were not able to answer.For instance, we do not know whether a monoid satisfies the ACCP if, or only if, it satisfies both theACCPR and the ACCPL (cf. Corollary 4.6 and Example 4.7). On a related note, is it true that everyunit-cancellative monoid satisfying the ACCP is acyclic (cf. Theorem 4.5)? Also, does an acyclic monoid H defined by a presentation Mon h X | R i with | X | < ∞ satisfy the ACCP? If not, what about the casewhen X and R are both finite? Cf. Example 4.8, in which we proved that the answer to the last questionis negative if we drop the requirement that H is acyclic and we ask in return that the monoid is reduced,cancellative, and | H -atomic. Acknowledgments
I, the author, am indebted to Laura Cossu, Alfred Geroldinger, and Daniel Smertnig for many fruitfulcomments. Part of the paper was written in summer 2020, while I was on a research stay at the University n Abstract Factorization Theorem and Some Applications
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