Solvable groups and affine structures

Wolfgang Rump

In this talk, I will focus upon (affine structures of)groups, (metacommutation in) rings, in connectionwith the integrability principle for braided structures,the Yang-Baxter equation, according to the threemain topics of the conference.

0. Finite solvable groups.

By a 1937 theorem of Hall, a finite group G is solvableif and only if it admits a Sylow basis, that is, arepresentative system of pairwise commuting Sylowsubgroups. Moreover, he proved that the Sylow basesof G are conjugate.

Thus, to understand the structure of a finite solvablegroup G, a first step is to fix a Sylow basis S . ForP, Q ∈ S , we have P ∩ Q = 1 and PQ = QP .Hence, if π ∈ P and σ ∈ Q, there are unique elementsπσ ∈ P and πσ ∈ Q with πσ = (πσ)(πσ). The mapsπ 7→ πσ are bijective and provide an action of thegroup Q on the set P . So there is an inverse actionπ 7→ σ · π, which yields πσ = πσ · σ.

Rewriting the equation πσ = (πσ)(πσ), we obtain

(π · σ)π = (σ · π)σ

π1π2 · σ = π1 · (π2 · σ)

and 1 · σ = σ for π, π1, π2 ∈ P and σ ∈ Q. If theseequations hold for all π, π1, π2, σ ∈ G, we will speakof an affine structure of G, and G together withthe operation · will be called a brace.

Now we are ready to deal with our three topics:Groups, rings, and the Yang-Baxter equation.

1. Groups: Affine structures

An affine structure of an n-dimensional manifoldX is given by an atlas with affine transition maps.This allows parallel transport of tangent vectors. Forthe universal cover ˜

X of X , there is a developing

map, an immersion D : ˜X → R

n into Rn. If D is

diffeomorphic, X is said to be complete. Example:The n-torus.

Let G be a simply connected Lie group with a right

invariant affine structure, i. e. right translations Rg

are affine. Then G must be solvable (Auslander 1977).So there is an affine representation α : G → Affn(R)with

D ◦ Rg = α(g) ◦ D

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and α(g)(x) = xg+γ(g), with linear part x 7→ xg anda 1-cocycle γ : G → R

n which is bijective if and onlyif the affine structure of G is complete. Identifying G

with A := Rn via γ leads to a brace: An abelian

group A with a group structure G = (A; ◦) satisfyingthe 1-cocycle condition

a ◦ b = ab + b,

where a 7→ ab is a right action G → Aut(A)op.

Example. The Jacobson radical J of any ring is abrace with respect to adjoint group structure

a ◦ b := ab + a + b.

Therefore, the group A◦ := (A; ◦) of a brace A iscalled the adjoint group of A.

Definition 1. An affine structure of an arbitrarygroup G is a brace with adjoint group G.

2. Rings: Metacommutation

Let H = R ⊕ Ri ⊕ Rj ⊕ Rk be the skew-field ofquaternions. The reduced norm

N(α) = αα = t2 + x2 + y2 + z2

of α = t+xi+yj+zk ∈ H is a group homomorphismN : H× → R

×, where α := t − xi − yj − zk is the

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conjugate of α. The subring of elements α ∈ H withinteger coefficients is contained in the left and rightprincipal ideal domain H = Zρ + Zi + Zj + Zk ofHurwitz quaternions, where ρ := 1

2(1+ i+j +k).The unit group H× = {ε ∈ H | N(ε) = 1} is isomor-phic to the binary tetrahedral group SL2(F3). Everyα ∈ H is a product of primes, i. e. elements π ∈ H

with N(π) ∈ N prime. There are three obstructionsto unique factorization (Conway and Smith, 2003):

1. Migration of units. Hurwitz (1896) solvedthis problem: He found a natural set P of primes suchthat every prime α ∈ H with N(α) odd has a uniquerepresentation α = πε with π ∈ P and ε ∈ H×.In analogy with irreducible polynomials, we speak ofmonic primes π ∈ P .

2. Recombination. For an odd prime p ∈ N,the p + 1 primes π ∈ P with N(π) = p form a pro-jective line Π(p) = P

1(Fp). The conjugate π is asso-ciated to a prime π∗ ∈ P with ππ∗ =

(−1p

)p. Thus,

any product ππ∗ is equal to σσ∗ whenever π and σ

belong to the same Π(p).

3. Metacommutation. For π ∈ Π(p) and σ ∈Π(q) with p 6= q, there are unique πσ ∈ Π(q) andπσ ∈ Π(p) with

πσ = (πσ)(πσ).

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Our notation suggests that π 7→ πσ is an action of σ

on Π(p). This action is bijective. Let π 7→ σ ·π be itsinverse. Then the bijective action σ 7→ πσ is given by

πσ = πσ · σ.

Using the left action π 7→ σ ·π, the metacommutationequation πσ = (πσ)(πσ) can be rewritten as

(π · σ)π = (σ · π)σ (1)

Thus, metacommutation is captured by Eq. (1) and

π1π2 · σ = π1 · (π2 · σ) (2)

We mention that Gauss’ reciprocity plays a role here:By a theorem of Cohn and Kumar (2015), the sign ofthe permutation π 7→ σ · π of Π(p) is the Legendresymbol

(qp

). The prime 2 is ramified: |Π(2)| = 1.

Now let us return to affine structures of a groupG. Recall that this amounts to a G-module structurewith a bijective 1-cocycle. We have:

Theorem 1. An affine structure of a group G is

equivalent to a binary operation · on G satisfying

Eqs. (1) and (2) - compare with Section 0!

The additive group is given by

a + b := (a · b)a, −a := a−1 · a−1.

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3. The Yang-Baxter equation

Let X be a set. For a permutation S of X × X , theYang-Baxter equation is a relation in X3:

(S×1X)(1X×S)(S×1X) = (1X×S)(S×1X)(1X×S).

Denote the components of S by

S(x, y) = (xy, xy).

Assume that S is non-degenerate, that is, the mapsx 7→ xy and y 7→ xy are bijective. Let x 7→ y · x bethe inverse of x 7→ xy. Define a second operation

x : y = y·xy.

The invertibility of S says that the maps y 7→ x : y

are bijective, too, and the Yang-Baxter equation saysthat X is a quantum cycle set, that is,

(x · y) · (x · z) = (y : x) · (y · z)

(x : y) : (x : z) = (y · x) : (y : z)

(x · y) : (x · z) = (y : x) · (y : z).

Definition 2. We define a q-affine structure ofa group G to be a quantum cycle set structure of G

such that for all a, b, c ∈ G,

ab · c = a · (b · c) a · bc =((c : a) · b

)(a · c)

ab : c = a : (b : c) a : bc =((c · a) : b

)(a : c).

Accordingly, we also speak of a q-brace.

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Theorem 2. A non-deg. q-cycle set X extends to

a unique q-affine structure of the free group F (X).

Etingof et al. (1999) introduced the structure

group GX of (the solution S corresponding to X)by assuming ab = (ab)(ab) (metacommutation), i. e.

(a : b)a = (b · a)b (3)

For the q-brace F (X), we only have

(a : b)a · c = (b · a)b · c

(a : b)a : c = (b · a)b : c.(4)

As this does not imply Eq. (3), the canonical mapX → GX need not be injective. A q-brace satisfiesEq. (3) if and only if it is a skew-brace in the senseof Guarnieri and Vendramin (2017). Thus, with itsq-affine structure, the structure group GX is a skew-brace. Define the socle of a q-brace A to be the ideal

Soc(A) := {a ∈ A | ∀ b ∈ A : a · b = a : b = b}.

The map X → GX extends uniquely to a q-bracemorphism F (X) ։ GX . By Eqs. (4), the kernel iscontained in the socle of F (X).

Skew-braces are important in Galois theory: Hopf-Galois extensions are K-linear versions of skew-braces(Byott (1996). Ordinary Galois extensions are Hopf-Galois with the group ring KG as Hopf algebra.

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Proposition 1. A skew-brace is equivalent to a

group (A; +) with a binary operation · satisfying

a · (b + c) = (a · b) + (a · c)

(a + b) · c = (a · b) · (a · c)

for all a, b, c ∈ A.

The operation : is given by the metacommutationequation (a : b)a = (b · a)b. The group (A; +) iscommutative if and only if the operations · and : co-incide. So we have the following specializations:

q-braces

skew-braces

↓(a : b)a = (b · a)b

braces

↓a · b = a : b

The adjoint group (A; ◦) of a skew-brace is given by

a ◦ b := b + ab

where a 7→ ab is inverse to a 7→ b · a.

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q-Braces are more “non-commutative” than skew-braces. They can be described as pairs of skew-braces(A; +0, ·) and (A; +1, :) of skew-braces with the sameadjoint group such that (A; +op

0 , :) and (A; +op1 , ·) are

again skew-braces. The corresponding solution

(a, b) 7→ (ab, ab)

of the Yang-Baxter equation gives rise to an infinitudeof group structures (n ∈ Z):

a ◦n+1 b := ab ◦n ab

For a skew-brace, all these group structures coincide.In general, there are two sequences of skew-braces(A; +n, ·) and (A; +op

n+1, :) with adjoint group (A; ◦n).The passage from +n to +n+1 is interesting, too:

a +n+1 b = b +n (b : a)b

Namely, (a, b) 7→(b, (b : a)b

)is again a solution, a so-

called derived solution (Drinfeld 1990; Lu, Yan,Zhu, 2000). The operation a ⊲ b := (b : a)b is rightself-distributive: (a ⊲ b) ⊲ c = (a ⊲ c) ⊲ (b ⊲ c).

Right self-distributive systems (X ;⊲) with bijectivemaps x 7→ x⊲y are called racks. They are equivalentto quantum cycle sets X satisfying x : y = y for allx, y ∈ X . With the extra condition x · x = x, theyform a classifying invariant for unoriented knots andlinks (Joyce, Matveev, 1982).

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4. Finite solvable groups - again

For a brace A with adjoint group G, the additivegroup admits a primary decomposition A = ΠAp,and the Ap are subbraces. So the adjoint group ofeach Ap is a Sylow p-subgroup of G. Therefore, theadjoint groups A◦

p form a (unique) Sylow basis.

We have seen (Section 0) that any finite solvablegroup with Sylow basis S satisfies the equations

(π · σ)π = (σ · π)σ

π1π2 · σ = π1 · (π2 · σ)

for π, π1, π2 ∈ P and σ ∈ Q, and distinct P, Q ∈ S .In other words: Any finite solvable group carries apartial affine structure. To make it global, it has tobe completed within the Sylow subgroups P ∈ S . Innumber-theoretic terms, the P ∈ S are the “primes”,and elements of P are the “Hurwitz primes” above P .

If the P ∈ S are equipped with affine structures,the metacommutation rules imply that an affine struc-ture of G which induces these affine structures mustbe unique. Existence is characterized by the following

Theorem 3. Let G be a group with Sylow basis S .

Affine structures of the P ∈ S extend to an affine

structure of G if and only if they extend to affine

structures of each subgroup PQ with P, Q ∈ S .

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It is known that certain big p-groups (p > 23 andnilpotency class 9) have no affine structure (Bachiller,2016). If non-nilpotent counterexamples (Sylow sub-groups admitting affine structures) exist, Theorem 3tells us that they have to be found among groups oforder pmqn. It seems that this question is easier todecide than the p-group case. A good starting pointmay be m, n 6 3, using Bachillers classification ofbraces of order p3.

5. Dequantization - a tropical view

Let S be a non-degenerate solution of the Yang-Baxterequation with corresponding q-cycle set X and struc-ture group GX . Then

S−1 = S ⇐⇒ x : y = x·y in X ⇐⇒ (GX ; +) abelian.

Indeed, x + y = (x · y)x = (y : x)y gives the secondequivalence. For such involutive solutions, X is acycle set, that is, x 7→ y · x is bijective, and

(x · y) · (x · z) = (y · x) · (y · z).

Non-degeneracy is equivalent to the bijectivity of thesquare map D : X → X with D(x) := x · x.

Although · and : coincide, the operation · still hasa hidden dual. If x 7→ y ∗ x denotes the inverse ofx 7→ yx, then (X ; ∗) is a cycle set, the dual of X .

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The connection is given by

(x · y) ∗ (y · x) = x = (x ∗ y) · (y ∗ x).

The additive group of GX is the free abelian groupZ

(X). Its partial order with negative cone G−X =

−N(X) coincides with the algebraic order of GX :

a 6 b ⇐⇒ ab−1 ∈ G−X.

It is right invariant:

a 6 b =⇒ ac 6 bc

and makes GX into a lattice: a right ℓ-group. Thebinary operation (on G−

X)

a → b := ba−1 ∧ 1

satisfiesab → c = a → (b → c)

and(a → b)a = (b → a)b (5)

Another instance of metacommutation! However, theexpression in Eq. (5) is an idempotent operation

(a → b)a = a ∧ b

in the lattice GX , instead of

(a · b)a = a + b

in the brace GX .

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What happens here, is a case of dequantization,a process in tropical mathematics: The affinestructure (GX ; ·) of the structure group GX has been“melt” into a tropical affine structure (GX ; →), the+ has become a logical conjunction ∧.

Indeed, the right ℓ-group GX is a piece of pure logic:The above equations, e. g.,

ab → c = a → (b → c)

can be interpreted as logical equivalences, a, b, c arepropositions, and → stands for the logical implication.(In classical logic, non-commutative conjunctionab is replaced by a ∧ b.)

6. L-algebras

What is the connection between the two essentiallydifferent versions of affine structure? For x 6= y in X ,

x → y = x · y,

while x · x = D(x) ∈ X but x → x = 1 (=“true”).In logic, the truth-value 1 represents a logical unit,an element which satisfies

1 → x = x, x → x = x → 1 = 1.

An element 1 must be unique. The anti-symmetry

of the implication → is given by

x → y = y → x = 1 =⇒ x = y

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Definition 3. An L-algebra is a set X with ananti-symmetric operation → and a logical unit, so that

(x → y) → (x → z) = (y → x) → (y → z)

Any L-algebra X has a partial order (entailment)

x 6 y :⇐⇒ x → y = 1

with greatest element 1. There is a structure group

G(X) with a canonical map X → G(X).

Examples. 1. Let X be a non-degenerate cycleset. If we adjoin a logical unit, then ˜

X := X ⊔ {1}is an L-algebra. This L-algebra is discrete, i. e.x < y ⇒ y = 1. Furthermore, ˜

X has a duality, abijection D : ˜

X → ˜X with D(1) = 1 and

D(x → y) = (y → x) → D(y)

for distinct x, y ∈ ˜X . And ˜

X is non-degenerate,which means that the map (x, y) 7→ (x → y, y → x)is bijective off the diagonal.

Theorem 4. X 7→ ˜X is a bijection between non-

degenerate cycle sets and non-degenerate discrete

L-algebras with duality.

Proof. The case x = y where x → y = x · y failshas to be replaced by the square map D(x) = x · x.

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So the L-algebras of Theorem 4 can be regarded as“tropical” (non-degenerate) cycle sets.

2. If G is a right ℓ-group, G− is an L-algebra with

a → b := ba−1 ∧ 1.

3. The propositions of classical logic, and the logicof quantum mechanics (Birkhoff, von Neumann, 1936),form an L-algebra.

4. The Hurwitz primes (Section 2) form a discreteL∗-algebra (an L-algebra with a special involution).Its structure group is a subgroup of H×.

5. For vectors x, y with x 6= 0 of the Euclideann-space,

x → y := y −(y, x)

(x, x)x.

x y

x → y

gives a well-defined operation on the one-dimensionalsubspaces, which makes the projective space Pn−1 intoa discrete L-algebra. The fact that there is a uniqueextension to an L-algebra of all subspaces of R

n isthe essence of the Gram-Schmidt orthogonalizationprocess. Keep this in mind when you teach your nextcourse in

L(inear)-Algebra.

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7. OML-algebras and Garside groups

Now we exhibit a case where the map X → G(X) intothe structure group of an L-algebra X is injective andG(X) is even a classifying invariant of X . The lastexample is a very special case.

Quantum physics deals with the interaction betweenstates and observables of a quantum mechanicalsystem. In the simplest case, states are unit vectorsof a Hilbert space H . The observables are self-adjointoperators on H . Applied to a state, an operatorchanges the state. If it happens to be an eigenvector,the state is retained, and the output is a real number.

More generally, let A be a von Neumann alge-

bra, a weakly closed subalgebra of the algebra B(H)of bounded operators on H . Then the A-invariantclosed subspaces of H correspond to the self-adjointP = P 2 ∈ A, the projections of A. They form adense subset P (A) of A. The partial ordering of theimages of the P ∈ P (A) induces a lattice structureof P (A), with ortho-complements P ′ : = 1 − P ,and

P 6 Q =⇒ P ∨ (P ′ ∧ Q) = Q.

Such a lattice is called ortho-modular (OML). Mostof the theory of von Neumann algebras takes place inthe orthomodular lattice P (A).

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A theorem of Dye (1955) states that if A, B arevon Neumann algebras without factors ∼= M2(C), eachOML-isomorphism P (A) → P (B) lifts to a bijection

Af

→B

P (A)∪

↑

→P (B)∪

↑

f = f+ ⊕ f− with an isomorphism f+ and an anti-isomorphism f− of von Neumann algebras. So theorthomodular lattice P (A) is an almost complete in-variant of A. Now L-algebras appear on the scene:

Definition 4. We say that an L-algebra X is ortho-

modular if X has a smallest element 0 such thatx′ := x → 0 (negation) is involutive and satisfies

x′6 y =⇒ y → x = x.

Theorem 5. Every OML is an orthomodular L-

algebra with

x → y := (x ∧ y) ∨ x′

and vice versa.

The L-algebra P (A) of a von Neumann algebra A

represents the “logic” of the corresponding quantummechanical system. Dye’s theorem shows that the L-algebra P (A) captures the essential part of QM.

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Definition 5. Let X be an OML, and let G be agroup. Define a G-valued measure on X to be amap µ : X → G which satisfies

µ(x ∨ y) = µ(x)µ(y)

if x and y are orthogonal (x ⊥ y), i. e. x 6 y′. IfG = R and µ(1) = 1, then µ is called a state.

Now we turn our attention to the structure groupG(X) of X . Instead of explaining how G(X) is madefrom X , we characterize it by a universal property:

Theorem 6. Let X be an OML. There is a group-

valued measure µ : X → G(X) into the structure

group G(X) of X such that any G-valued measure

X → G factors uniquely through µ:

Xµ→G(X)

Gg

.

.

.

.

.

.∃ !

→

Definition 6. Let G be a right ℓ-group. An elements ∈ G is said to be normal if sG+s−1 = G+. Ifs ∈ G+ is normal and any a ∈ G+ is majorized bysome sn, then s is said to be a strong order unit.We call s ∈ G+ singular if for all x, y ∈ G−,

s−16 xy =⇒ xy = yx = x ∧ y.

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Example. In the (additive) abelian ℓ-group C(X)of continuous functions on a compact space X , thepositive constants are strong order units because everyf ∈ C(X) is bounded.

Proposition 2. Let G be a right ℓ-group with a

strong order unit u. If s ∈ G+ is normal and

singular, then s 6 u.

Corollary. A singular strong order unit of a right

ℓ-group must be unique.

In the above example, minimal strong order unitscannot exist, just as atoms have no place in classicalmechanics. In the quantum world, they abound:

Theorem 7. Let s be a singular strong order unit

of a right ℓ-group G. Then the interval [s−1, 1] is

an OML with ortho-complementation x′ := s−1x−1

and structure group G. Conversely, every OML

arises in this way. The universal group-valued

measure µ : X → G(X) is given by µ(x) = x−1.

Thus, OMLs are equivalent to a special class of rightℓ-groups. In particular, a von Neumann algebra isalmost completely determined by the structure groupof its projection lattice.

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Definition 7. A Garside group is a right ℓ-groupG with a strong order unit s such that the interval[1, s] is finite.

Prominent examples are the Artin groups of finiteCoxeter type, including Artin’s braid group.

Corollary. Let X be an OML. The structure group

G(X) is a Garside group if and only if X is finite.

Thus, finite OMLs give a big class of Garside groups.

8. Non-commutative prime factorization

One dimensional regular rings (=Dedekind domains)are characterized by unique prime ideal factorization.The non-commutative analogue of a Dedekind domainR is a hereditary order, an R-algebra Λ of globaldimension 1 which is an R-lattice, a finitely gen-erated torsion-free R-module. For example, the ringof Hurwitz quaternions is a hereditary order. Thereis a semisimple algebra A = KΛ over the quotientfield K of R. Fractional ideals of Λ are the R-latticesI = ΛIΛ with KI = A. They form a monoid and alattice (with + and ∩), the arithmetic A(Λ). Whatremains from the fundamental theorem of arithmeticis that any I ∈ A(Λ) factors uniquely into commutingideals Ip containing a power of a prime ideal p of R.

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So we can restrict ourselves to the case where R islocal and A is a simple K-algebra. The arithmeticA(Λ) then merely depends on the rank r(Λ), thenumber n of maximal ideals of Λ. If n = 1, thenA(Λ) consists of the powers of a single prime ideal,like in the commutative case. For n = 3, it looks asfollows (only the πn, n ∈ Z, are invertible):

...

...

u

π

π−1

p0 p1 p2

e1 e0 e2

p2π−1

p1p0 p2p1

p1p2 = e1p0p1

p0 = p1 = π−1p2p1

p0p2p1p0

p2p1

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Does non-commutative prime factorization over alocal ground ring R make any sense? If yes, eachelement of the monoid lattice A(Λ) stands for the se-quence of exponents of the n prime ideals P1, . . . , Pn

of Λ, lying over the prime ideal Rad R of the local ringR. Indeed, there are several analogies.

Let Hn denote the set of non-decreasing functionse : Z → Z which satisfy

e(k + n) = e(k) + n

for all k ∈ Z. Thus e is determined by its values on{1, . . . , n}. Then Hn is a lattice, and a monoid withrespect to composition.

Theorem 8. For n := r(Λ), the arithmetic A(Λ)is isomorphic to Hn.

In other words, the exponents of P1, . . . , Pn are partof a non-decreasing “n-periodic” function, multiplica-tion of ideals being the composition of these functions.

The distributive lattice Hn embeds canonically intoDiv(Λ) := Z

n, the additive group of divisors (on thecorresponding non-commutative curve). Every ideala ∈ A(Λ) represents a unique divisor ∂(a) ∈ Div(Λ),but not vice versa, in contrast to the commutativecase. The sequence of exponents of the P1, . . . , Pn isnothing else than the divisor ∂(a) of a.

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Theorem 9. The multiplication of ideals extends

to a multiplication of divisors, which makes Div(Λ)into a monoid Div(Λ)◦, acting on the additive group

of Div(Λ). The identity map Div(Λ)◦ → Div(Λ) is

a 1-cocycle.

So the divisors of Λ form a “brace” with an adjoint

monoid Div(Λ)◦ instead of a group. On the otherhand, the lattice structure makes Div(Λ) into a rightℓ-monoid - a tropical right ℓ-group!

For a right ℓ-group G, the normal elements forma (two-sided) ℓ-group

N(G) = {a ∈ G | ∀ b, c ∈ G : b 6 c ⇒ ab 6 ac}.

The same definition applies to Div(Λ). Here comesthe upshot:

Theorem 10. The arithmetic A(Λ) of an arbi-

trary hereditary order Λ can be recovered from the

“brace” of divisors:

N(Div(Λ)) = A(Λ)

The classical fundamental theorem of arithmetic fitsnicely into this broader context: The group of divisorsof a Dedekind domain is really a brace and coincideswith the group of fractional ideals. The adjoint groupcoincides with the additive group: A trivial brace.

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