Commutativity in Double Interchange Semigroups Murray Bremner Joint - - PowerPoint PPT Presentation

Commutativity in Double Interchange Semigroups

Murray Bremner Joint work with Gary Au, Fatemeh Bagherzadeh, and Sara Madariaga

Department of Mathematics and Statistics, University of Saskatchewan

Workshop on Operads and Higher Structures in Algebraic Topology and Category Theory July 29 – August 2, 2019, University of Ottawa w x y z = w x y z = w x y z

Diagrammatic representation of the interchange law

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A very gentle introduction to higher-dimensional algebra

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Horizontal and vertical multiplications

Suppose that x and y are two indeterminates. When we compose x and y we usually write the result as xy. This composition may not be commutative: xy = yx in general. We write the products xy and yx horizontally — but why? Why don’t we write the products vertically, y x

• r x

y ? Paper (or blackboards, or computer screens, or . . . ) are 2-dimensional. Why don’t we use the third dimension (orthogonal to the screen)? We could write x in front of y, or y in front of x, using 3 dimensions. Is this just silly, or could it possibly have some important applications? Multiplying in different directions leads to higher-dimensional algebra.

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From Ronald Brown’s survey paper Out of Line

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A crossword puzzle: is this higher-dimensional algebra?

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The interchange law for two operations

Consider the 2 × 2 square product y z w x

• f four variables.

There are two different ways to write this using ◦ and •: (w ◦ x) • (y ◦ z)

• r

(w • y) ◦ (x • z). Note the transpositions of x and y and of the operations ◦ and •. We assume these two ways of writing y z w x always give the same result. This assumption is called the interchange law: for all w, x, y, z we have (w ◦ x) • (y ◦ z) = (w • y) ◦ (x • z). This relation between the two operations can be expressed as w x y z = w x y z = w x y z

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The surprising Eckmann-Hilton argument (1)

Assume that ◦ and • satisfy the interchange law: (w ◦ x) • (y ◦ z) = (w • y) ◦ (x • z). Assume that ◦ and • have identity elements 1◦ and 1• respectively. In the interchange law, take w = z = 1◦ and x = y = 1•: (∗) (1◦ ◦ 1•) • (1• ◦ 1◦) = (1◦ • 1•) ◦ (1• • 1◦). By definition of identity elements, 1◦ ◦ 1• = 1• ◦ 1◦ = 1•, 1◦ • 1• = 1• • 1◦ = 1◦. Using these to simplify equation (∗) gives 1• • 1• = 1◦ ◦ 1◦. By definition of identity elements, this implies that 1◦ = 1•. The two identity elements are equal.

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The surprising Eckmann-Hilton argument (2)

Write 1 for the common identity element: 1 = 1◦ = 1•. In the interchange law, take w = z = 1: (1 ◦ x) • (y ◦ 1) = (1 • y) ◦ (x • 1). By definition of identity element, this implies that x • y = y ◦ x. The black operation is the opposite of the white operation. Using this to simplify the interchange law gives (y ◦ z) ◦ (w ◦ x) = (y ◦ w) ◦ (z ◦ x). Taking x = y = 1 in the last equation gives (1 ◦ z) ◦ (w ◦ 1) = (1 ◦ w) ◦ (z ◦ 1). By definition of identity element, this implies that z ◦ w = w ◦ z. The white operation is commutative.

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The surprising Eckmann-Hilton argument (3)

But if x • y = y ◦ x, and y ◦ x = x ◦ y, then x • y = x ◦ y. The two operations are equal. (There is really only one operation.) Hence the interchange law can be simplified to (w ◦ x) ◦ (y ◦ z) = (w ◦ y) ◦ (x ◦ z). Taking x = 1 gives (w ◦ 1) ◦ (y ◦ z) = (w ◦ y) ◦ (1 ◦ z). By definition of identity element, this implies w ◦ (y ◦ z) = (w ◦ y) ◦ z. The operation is associative. We have proved the (in)famous Eckmann-Hilton Theorem.

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The Eckmann-Hilton Theorem

Theorem (Eckmann-Hilton) Let S be a set with two binary operations S × S → S denoted ◦ and •. Suppose that these two operations satisfy the interchange law. Suppose also that these two operations have identity elements 1◦ and 1•. Then the two identity elements are equal, the two operations are equal, and the single remaining operation is both commutative and associative. Beno Eckmann, Peter Hilton: Group-like structures in general categories, I: Multiplications and comultiplications. Mathematische Annalen 145 (1961/62) 227–255.

• Theorem 3.33 (page 236)
• The definition of H-structure (page 241)
• Theorem 4.17 (page 244)

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A brief digression on homotopy groups

The homotopy groups for n ≥ 1 of a topological space are defined in terms

• f continuous maps of the n-sphere into the space.

The first homotopy group (the fundamental group) is defined in terms of loops in the space, and is usually noncommutative. Corollary (of the Eckmann-Hilton Theorem) For n ≥ 2 the higher homotopy groups are always commutative. Ronald Brown: groupoids.org.uk (slightly edited quotation) “The nonabelian fundamental group gave more information than the first homology group. Topologists were seeking higher dimensional versions of the fundamental group. ˇ Cech submitted to the 1932 ICM a paper on higher homotopy groups. These were quickly proved to be abelian in dimensions > 1, and Cech was asked (by Alexandrov and Hopf) to withdraw his paper. Only a short paragraph appeared in the Proceedings.”

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Familiar examples of the interchange law (1)

Example (multiplication and division) Suppose that the underlying set is the nonzero real numbers R \ {0}. Let the operations ◦ and • be multiplication and division, respectively. Then the interchange law states that (w · x)/(y · z) = (w/y) · (x/z), which is simply the familiar rule for multiplying fractions: w · x y · z = w y · x z . (Why does the Eckmann-Hilton Theorem not imply that multiplication and division are the same commutative associative operation?)

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Familiar examples of the interchange law (2)

Example (composition of functions on Cartesian products) Suppose that ◦ denotes the operation of forming an ordered pair: f1 ◦ f2 = (f1, f2). Suppose that • denotes the operation of function composition: (f • g)(−) = f (g(−)). What does the interchange law state in this setting? (f1 ◦ f2) • (g1 ◦ g2) = (f1 • g1) ◦ (f2 • g2) ⇐ ⇒

• (f1, f2) • (g1, g2)
• (−, −)
• =
• (f1 • g1)(−), f2 • g2)(−)
• .

The interchange law shows how to define composition of Cartesian products of functions on Cartesian products of sets.

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Familiar examples of the interchange law (3)

Example (endomorphism PROP of a set) Let X be a nonempty set, and let X n be its n-th Cartesian power (n ≥ 1). Let Map(m, n) be the set of all functions f : X m → X n for m, n ≥ 1. Let Map(X) be the disjoint union of the sets Map(m, n): Map(X) =

• m,n≥1

Map(m, n) On Map(X) there are two natural binary operations: The horizontal product ◦: For f : X p → X q and g : X r → X s we define the operation ◦: Map(p, q) × Map(r, s) − → Map(p+r, q+s) as follows: f ◦ g : X p+r = X p × X r

(f ,g)

− − − − − − → X q × X s = X q+s. This operation is defined for all p, q, r, s ≥ 1.

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Example (continued) The vertical product •: For f : X q → X r and g : X p → X q we define the operation •: Map(q, r) × Map(p, q) − → Map(p, r) as follows: f • g : X p

f • g

− − − − − − → X r. This operation is defined on Map(q1, r) × Map(p, q2) only if q1 = q2. These two operations satisfy the interchange law: (f ◦ g) • (h ◦ k) = (f • h) ◦ (g • k). To verify this: compare the following maps, check the results are equal: (f ◦ g) • (h ◦ k): X p × X q

(h,k)

− − − − − − → X r × X s

(f ,g)

− − − − − − → X t × X u, (f • h) ◦ (g • k): X p × X q

( f •h, g•k )

− − − − − − − − − − → X t × X u.

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Familiar examples of the interchange law (4)

Definition A double category D is a pair of categories (D0, D1) with functors e : D0 → D1 (identity map), s, t : D1 → D0 (source and target objects). In D0 we denote objects by capital Latin letters A, . . . (the 0-cells of D) and morphisms by arrows labelled by lower-case italic letters u, . . . (the vertical 1-cells of D). In D1 the objects are arrows labelled by lower-case italic letters h, . . . (the horizontal 1-cells of D), and the morphisms are arrows labelled by lower-case Greek letters α, β, . . . (the 2-cells of D). If A is an object in D0 then e(A) is the (horizontal) identity arrow on A. (By Eckmann-Hilton, identity arrows may exist in only one direction.) The functors s, t are source and target: if h is a horizontal arrow in D1 then s(h) and t(h) are its domain and codomain, objects in D0. These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A.

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Definition If α, β are 2-cells then horizontal and vertical composition α ⊟ β, α ⊟ β are defined by the following diagrams (and satisfy the interchange law): A C E B D F

u

• v
• w
• h
• k
• ℓ

m α

• β
• ∼

= A E B F

u

• w
• k◦h
• m◦ℓ
• α

⊟

β

• A

B C D E F

u

• v
• w
• x
• h
• ℓ
• α
• β
• ∼

= A C D F

h

• ℓ
• v◦u
• x◦w
• α⊟β
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Introductory references on higher-dimensional algebra

• J. C. Baez: An introduction to n-categories. 7th Conf. on Category

Theory and Computer Science (1997), pages 1–33. Lecture Notes in Computer Science, 1290.

• J. C. Baez, M. Stay: Physics, topology, logic and computation: a

Rosetta Stone. New Structures for Physics (2011), pages 95–172. Lecture Notes in Physics, 813.

• M. A. Batanin: The Eckman-Hilton argument and higher operads.
• Adv. Math. 217 (2008) 334–385.
• R. Brown: Out of line. Royal Institution Proc. 64 (1992) 207–243.
• R. Brown, T. Porter: Intuitions of higher dimensional algebra for

the study of structured space. Revue de Synth´ ese 124 (2003) 173–203.

• R. Brown, T. Porter: Category theory, higher-dimensional algebra:

potential descriptive tools in neuroscience. Proc. International Conference

• n Theoretical Neurobiology, Delhi, 2003.

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Nonassociative algebra in

• ne and two dimensions

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Nonassociativity for one operation

Associativity can be applied in various ways to a given association type. In each arity, this gives a partially ordered set, the Tamari lattice. On the left, we have the Tamari lattice of binary trees for n = 4:

(((ab)c)d)e ((a(bc))d)e ((ab)(cd))e (a((bc)d))e (a(b(cd)))e ((ab)c)(de) (a(bc))(de) (ab)((cd)e) (ab)(c(de)) a(((bc)d)e) a((b(cd))e) a((bc)(de)) a(b((cd)e)) a(b(c(de)))

On the right, association types are represented by dyadic partitions of the unit interval [0, 1], where multiplication corresponds to bisection.

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Two nonassociative operations with interchange law (1)

How do we generalize association types to two dimensions? Thinking in terms of placements of parentheses, we could use two different types of brackets, (−−) to represent − ◦ −, and [−−] to represent − • −. Thinking in terms of binary trees, we could label each internal node (including the root) by one or the other of the operation symbols ◦, •. Alternatively, in one dimension, we’ve seen that association types correspond bijectively to dyadic partitions of the unit interval [0, 1]. This suggests that in two dimensions, we should be able to interpret association types as dyadic partitions of the unit square [0, 1] × [0, 1]. In this case, ◦ represents the horizontal operation (vertical bisection) and • represents the vertical operation (horizontal bisection). This geometric interpretation has the advantage that the two operations (both nonassociative) automatically satisfy the interchange law.

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Two nonassociative operations with interchange law (2)

Definition A 2-dimensional association type of arity n is a dyadic partition of the unit square [0, 1] × [0, 1]: that is, obtained by n−1 vertical or horizontal bisections of subrectangles, starting from the (empty) unit square.

n = 1: ∗ (∗ is the argument symbol: − takes up too much space) n = 2: ∗◦∗ ∗•∗ n = 3: (∗◦∗)◦∗ ∗◦(∗◦∗) (∗•∗)•∗ ∗•(∗•∗) (∗•∗)◦∗ ∗◦(∗•∗) (∗◦∗)•∗ ∗•(∗◦∗)

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Two-dimensional association types in arity n = 4 (part 1)

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Two-dimensional association types in arity n = 4 (part 2)

total 39

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Enumeration of two-dimensional association types (1)

The sequence 1, 2, 8, 39, . . . satisfies the following recurrence relation, which generalizes the familiar recurrence relation for the Catalan numbers: C(1) = 1, C(n) = 2

i,j C(i)C(j) − i,j,k,l C(i)C(j)C(k)C(l).

First sum is over all 2-compositions of n (i+j = n) into positive integers; second sum is over all 4-compositions of n (i+j+k+l = n). The generating function for this sequence, G(x) =

n≥1 C(n)xn,

satisfies a quartic polynomial equation, which generalizes the familiar quadratic polynomial equation for the Catalan numbers: G(x)4 − 2G(x)2 + G(x) − x = 0. We’ve generalized this to all dimensions (proof by homological algebra): Murray Bremner, Vladimir Dotsenko: Boardman-Vogt tensor products of absolutely free operads. Proceedings A of the Royal Society of Edinburgh (to appear).

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Enumeration of two-dimensional association types (2)

The sequence counting 2-dimensional association types starts like this: 1, 2, 8, 39, 212, 1232, 7492, 47082, 303336, 1992826, 13299624, 89912992, 614474252, 4238138216, 29463047072, 206234876287, 1452319244772, . . . . . . (oeis.org/A236339). The number of dyadic partitions of the unit square into n rectangles is C2,2(n) = 1 n

⌊ n−1

3 ⌋

• i=0
• 2(n−1−i)

n−1, n−1−3i, i

• (−1)i 2n−1−3i.

Yu Hin (Gary) Au, Fatemeh Bagherzadeh, Murray Bremner: Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube arXiv:1903.00813v1[math.CO]

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A related application: VLSI design floorplanning

alumni.soe.ucsc.edu/~slogan/research.html

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A related application: rectangular cartograms

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A related application: “squaring the square”

www.squaring.net/history_theory/duijvestijn.html

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A related application: “guillotine partitions” in 3D

• A. Asinowski, G. Barequet, T. Mansour, R. Pinter:

Cut equivalence of d-dimensional guillotine partitions. Discrete Mathematics Volume 331, 28 September 2014, Pages 165–174.

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A related application: the “little n-cubes” operad

www.math3ma.com/mathema/2017/10/30/what-is-an-operad-part-2

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Double interchange semigroups: two associative operations satisfying interchange (a toy model of double categories)

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What about two associative operations related by the interchange law? An unexpected commutativity relation in arity 16 was discovered by: Joachim Kock: Commutativity in double semigroups and two-fold monoidal categories.

• J. Homotopy and Related Structures 2 (2007), no. 2, 217–228.

A former postdoctoral fellow and I used computer algebra (Maple) to show that arity 9 is the lowest in which such commutativity relations occur: Murray Bremner, Sara Madariaga: Permutation of elements in double semigroups. Semigroup Forum 92 (2016), no. 2, 335–360. Further relations in arity 10 were discovered by my current postdoc: Fatemeh Bagherzadeh, Murray Bremner: Commutativity in double interchange semigroups. Applied Categorical Structures 26 (2018), no. 6, 1185–1210.

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Kock’s surprising observation

Relation of arity 16: associativity and the interchange law combine to imply a commutativity relation, the equality of two monomials with: − same skeleton (placement of parentheses and operation symbols), − different permutations of arguments (transposition of f , g).

(a b c d) (e f g h) (i j k ℓ) (m n p q) ≡ (a b c d) (e g f h) (i j k ℓ) (m n p q) a b c d e f g h i j k ℓ m n p q ≡ a b c d e g f h i j k ℓ m n p q

The symbol ≡ indicates that the equation holds for all arguments.

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Nine is the least arity for a commutativity relation

((a b) c) (((d (e f )) (g h)) i) ≡ ((a b) c) (((d (g f )) (e h)) i) a b c d e f g h i ≡ a b c d e f g h i

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Geometric proof of commutativity in arity 10

Watch what happens to the arguments d and g as we repeatedly apply associativity horizontal and vertically, together with the interchange law. At the end, we have exactly the same dyadic partition of the square, but the arguments d and g have changed places: a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j

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a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b f g h c d e i j a b c d e f g h i j a b f g h c d e i j a b f g h c d e i j a b f d h c g e i j a b f d h c g e i j a b f d h c g e i j

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Four nonassociative operads: Free, Inter, BP, DBP

Definition Free: free symmetric operad, two binary operations with no symmetry, denoted △ (horizontal) and (vertical). Definition Inter: quotient of Free by ideal I = ⊞ generated by interchange law: ⊞: (a △ b) (c △ d) − (a c) △ (b d) ≡ 0 Definition BP: set operad of block partitions of open unit square I 2, I = (0, 1). Horizontal composition x → y (vertical composition x ↑ y):

• translate y one unit east (north) to get y + ei (i = 1, 2)
• form x ∪ (y + ei) to get a partition of width (height) two
• scale horizontally (vertically) by one-half to get a partition of I 2

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Algorithm In dimension d, to get a dyadic block partition of I d (unit d-cube): Set P1 ← {I d}. Do these steps for i = 1, . . . , k−1 (k parts): Choose an empty block B ∈ Pi and an axis j ∈ {1, . . . , d}. If (aj, bj) is projection of B onto axis j then set c ← 1

2(aj+bj).

Set {B′, B′′} ← B \ { x ∈ B | xj = c } (hyperplane bisection). Set Pi+1 ← ( Pi \ {B} ) ⊔ {B′, B′′} (replace B by B′, B′′). Definition

• DBP: unital suboperad of BP generated by

⊟ and ⊟

• Unital: include unary operation I 2 (block partition with one empty block)
• DBP consists of dyadic block partitions:

every P ∈ DBP with n+1 parts is obtained from some Q ∈ DBP with n parts by bisection of a part of Q horizontally or vertically.

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Three associative operads: AssocB, AssocNB, DIA

Definition AssocB: quotient of Free by ideal A = A△ , A generated by A△ (a, b, c) = ( a △ b ) △ c − a △ ( b △ c ) (horizontal associativity) A (a, b, c) = ( a b ) c − a ( b c ) (vertical associativity) AssocNB: isomorphic copy of AssocB with following change of basis. ρ: AssocB → AssocNB represents rewriting a coset representative (binary tree) as a nonbinary (= not necessarily binary) tree. new basis consists of disjoint union {x1} ⊔ T△ ⊔ T isolated leaf x1 and two copies of T T = all labelled rooted plane trees with at least one internal node T△ : root r of every tree has label △ , labels alternate by level T : labels of internal nodes (including root) are reversed

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Algorithm for converting binary tree to nonbinary tree

We write Assoc if convenient for AssocB ∼ = AssocNB: △ △ △ T1T2T3T4

α

− − − → △ T1T2T3T4 △ △

• T1T2T3T4

α

− − − → △

• T1T2

T3T4 △

• △

T1T2T3T4

α

− − − → △

• T1T2

T3T4 △

• T1T2T3T4

α

− − − − − − →

no change

△

• T1T2T3T4

Switching △ , throughout defines α for subtrees with roots labelled .

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Associativity = ⇒ interchange applies almost everywhere

After converting a binary tree to a nonbinary (not necessarily binary) tree, if the root is white (horizontal) then all of its children are black (vertical), all of its grandchildren are white, all of its great-grandchildren are black,

• etc. . . . , alternating white and black according to the level:

△

• · · ·
• · · ·
• △ · · · △ · · · △

△ · · · △ · · · △ △ · · · △ · · · △ · · · · · · · · · · · · · · · · · · · · · · · · · · · △· · ·△ △· · ·△ △· · ·△ △· · ·△ △· · ·△ △· · ·△ △· · ·△ △· · ·△ △· · ·△ If the root is black then we simply transpose white and black throughout.

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Definition DIA: quotient of Free by ideal A△ , A , ⊞. This is the algebraic operad governing double interchange algebras, which possess two associative operations satisfying the interchange law.

• Inter, AssocB, AssocNB, DIA are defined by relations v1 − v2 ≡ 0

(equivalently v1 ≡ v2) where v1, v2 are cosets of monomials in Free.

• We could work with set operads (we never need linear combinations).
• Vector spaces and sets are connected by a pair of adjoint functors:

the forgetful functor sending a vector space V to its underlying set, the left adjoint sending a set S to the vector space with basis S.

• Corresponding relation between Gr¨
• bner bases and rewrite systems:

if we compute a syzygy for two tree polynomials v1 − v2 and w1 − w2, then the common multiple of the leading terms cancels, and we obtain another difference of tree monomials; similarly, from a critical pair of rewrite rules v1 → v2 and w1 → w2, we obtain another rewrite rule.

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Morphisms between operads

• Our goal is to understand the operad DIA.
• We have no convenient normal form for the basis monomials of DIA.
• There is a normal form if we factor out associativity but not interchange.
• There is a normal form if we factor out interchange but not associativity.
• We use the monomial basis of the operad Free.
• We apply rewrite rules which express associativity of each operation

(right to left, or reverse) and interchange between the operations (black to white, or reverse).

• These rewritings convert one monomial in Free to another monomial

which is equivalent to the first modulo associativity and interchange.

• Given an element X of DIA represented by a monomial T in Free,

we convert T to another monomial T ′ in the same inverse image as T with respect to the natural surjection Free ։ DIA.

• We use undirected rewriting: to pass from T to T ′, we may need to

reassociate left to right, apply interchange, reassociate right to left.

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Commutative diagram of operads and morphisms

Free BP DBP Inter AssocB AssocNB DIA

−/A△ ,A α −/⊞ χ Γ

• γ

isomorphism

• −/A△ +I,A +I

α ρ isomorphism

• −/⊞+A

χ inclusion ι

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Further details (if I haven’t run out of time already)

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Cuts and slices

Definition

• Subrectangle: any union of empty blocks forming a rectangle.
• Let P be a block partition of I 2, and let R be a subrectangle of P.
• A main cut in R is a horizontal or vertical bisection of R.
• Every subrectangle has at most two main cuts (horizontal, vertical).
• Suppose that a main cut partitions R into subrectangles R1 and R2.
• If either R1 or R2 has a main cut parallel to the main cut of R, we call

this a primary cut in R; we also call the main cut of R a primary cut.

• In general, if a subrectangle S of R is obtained by a sequence of cuts

parallel to a main cut of R then a main cut of S is a primary cut of R.

• Let C1, . . . , Cℓ be the primary cuts of R parallel to a given main cut Ci

(1 ≤ i ≤ ℓ) in positive order (left to right, or bottom to top) so that there is no primary cut between Cj and Cj+1 for 1 ≤ j ≤ ℓ−1.

• Define “cuts” C0, Cℓ+1 to be left, right (bottom, top) sides of R.
• Write Sj for the j-th slice of R parallel to the given main cut.

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Commutativity relations

Definition Suppose that for some monomial m of arity n in the operad Free, and for some transposition (ij) ∈ Sn, the corresponding cosets in DIA satisfy: m(x1, . . . , xi, . . . , xj, . . . , xn) ≡ m(x1, . . . , xj, . . . , xi, . . . , xn). In this case we say that m admits a commutativity relation. Proposition (Fatemeh Bagherzadeh) Assume that m is a monomial in Free admitting a commutativity relation which is not a consequence of a commutativity relation holding in (i) a proper factor of m, or (ii) a proper quotient of m. (Quotient refers to substitution of a decomposable factor for the same indecomposable argument on both sides of a relation of lower arity). Then the dyadic block partition P = Γ(m) contains both main cuts. In other words, it must be possible to apply the interchange law as a rewrite rule at the root of the monomial m (regarded as a binary tree).

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Border blocks and interior blocks

Definition Let P be a block partition of I 2 consisting of empty blocks R1, . . . , Rk. If the closure of Ri has nonempty intersection with the four sides of the closure I 2 then Ri is a border block, otherwise Ri is an interior block. Lemma Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic block partitions of I 2 such that m1 ≡ m2 in every double interchange semigroup. Then any interior (border) block of P1 is an interior (border) block of P2. Lemma If m admits a commutativity relation then in the corresponding block partition P = Γ(m) the two commuting empty blocks are interior blocks. Basic idea of the proofs: neither associativity nor the interchange law can change an interior block to a border block or conversely.

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Lower bounds on the arity of a commutativity relation

Lemma If m admits a commutativity relation then P = Γ(m) has both main cuts; hence P is the union of 4 subsquares A1, . . . , A4 (NW, NE, SW, SE). If a subsquare has 1 (2) empty interior block(s) then that subsquare has at least 3 (4) empty blocks. Hence P contains at least 7 empty blocks. Proposition (Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Free admits a commutativity relation then P = Γ(m) has n ≥ 8 empty blocks. Reflecting P in the horizontal and/or vertical axes if necessary, we may assume that the NW subsquare A1 has two empty interior blocks and has

• nly the horizontal main cut (otherwise we reflect in the NW-SE diagonal).

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We display the 3 partitions with 7 empty blocks satisfying these conditions: a b e c d f g a b c d e f g a b c d e f g None of these configurations admits a commutativity relation. The method used for the proof of the last proposition can be extended to show that a monomial of arity 8 cannot admit a commutativity relation, although the proof is rather long owing to the large number of cases: (a) 1 square Ai has 5 empty blocks, and the other 3 squares are empty; (b) 1 square Ai has 4 empty blocks, another square Aj has 2, and the

• ther 2 squares are empty (2 subcases: Ai, Aj share edge or only corner);

(c) 2 squares Ai, Aj each have 3 empty blocks, other 2 empty (2 subcases). This provides a different proof, independent of machine computation, of the minimality result of Bremner and Madariaga.

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Commutative block partitions in arity 10

Lemma Let m admit a commutativity relation in arity 10. Then P = Γ(m) has at least two and at most four parallel slices in either direction. Proof. By the lemmas, P contains both main cuts. Since P contains 10 empty blocks, it has at most 5 parallel slices (4 primary cuts) in either direction. If there are 4 primary cuts in one direction and the main cut in the other direction, then there are 10 empty blocks, and all are border blocks. In what follows, m has arity 10 and admits a commutativity relation. Hence P = Γ(m) is a dyadic block partition with 10 empty blocks. Commuting blocks are interior; P has either 2, 3, or 4 parallel slices. If P has 3 (resp. 4) parallel slices, then commuting blocks are in middle slice (resp. middle 2 slices). Interchanging H and V if necessary, we may assume parallel slices are vertical.

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Four parallel vertical slices

We have H and V main cuts, and 2 more vertical primary cuts. Applying horizontal associativity gives 2 rows of 4 equal empty blocks. This configuration has 8 empty blocks, all of which are border blocks. We need 2 more cuts to create 2 interior blocks. Applying vertical associativity in the second slice from the left, and applying a dihedral symmetry of the square (if necessary), reduces the number of configurations to the following A, B, C: A: B : C :

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Theorem Configuration A: In every double interchange semigroup, the following commutativity relation holds for all values of the arguments a, . . . , j: ((a △ b) (c △ (d e))) △ (((f g) △ h) (i △ j)) ≡ ((a △ b) (c △ (g e))) △ (((f d) △ h) (i △ j)) For configuration B we label only the two blocks which transpose. Applications of associativity and interchange can easily be recovered: c g c g c g c g c g g c

g c g c

g c

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Theorem Configuration B: In every double interchange semigroup, the following commutativity relation holds for all values of the arguments a, . . . , j: ((a △ (b c)) (f △ (g h))) △ ((d △ e) (i △ j)) ≡ ((a △ (b g)) (f △ (c h))) △ ((d △ e) (i △ j)) For configuration C we obtain no new commutativity relations. Concluding remarks: higher dimensions We have studied structures with two operations, representing orthogonal (horizontal and vertical) compositions in two dimensions. Most of our constructions make sense for any number of dimensions d ≥ 2. Major obstacle for d ≥ 3: monomial basis for AssocNB consisting of nonbinary trees with alternating white and black internal nodes does not generalize in a straightforward way.

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Thanks to all of you for your attention. Merci ´ a vous tous pour votre attention. Gracias a todos por su atenci´

• n.

Gr` acies a tots per la vostra atenci´

• .

Eskerrik asko guztioi zuen arretarengatik. Obrigado a todos por sua aten¸ c˜ ao. Ngibonga nonke ngokunaka kwakho.

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