Monoids, Categories & Monotone Partial...

Monoids,Categories &Monotone Partial Injections

Why Computer Scientists should knowand love Inverse Semigroup Theory

peter.hines@york.ac.uk Peter Hines — University of York

Some motivation ...

This talk is about some inverse semigroup theory

closely associated with theoretical computer science.

It is not a C.S. talk as such – hopefully the structures used are

of interest for their own sake!

We will take a bottom-up approach.

A relevant quotation

“Many reversible functional programminglanguages, as well as categorical models thereof,come equipped with a tacit assumption of totality, aproperty that is neither required nor necessarilydesirable”.

Axelsen & Kaarsgaard (2016 ... to appear)

A sample question

Fixed point operators are important in many fields of computerscience, from lambda calculus to functional programming.

Can we introduce fixed points into models ofreversible functional programming?

A disclaimer

We may or may not reach the final destination

but there is a lot of fun stuff on the way

However, the answer is YES

Our categorical setting

We work with the prototypical inverse category pInjof partial injections on sets.

Wagner-Preston in the the multi-object case

(R. Cockett 2002)Every small inverse category has a representation withinthe category pInj.

(C. Heunen 2013)The same holds for locally small inverse categories.

The basics ...

We will be mixing order theory & partial injections.

By analogy with Kleene equality

In a poset pP,ďPq, we write a ÀP b for

“a ďP b provided both a and b are defined”.

A partial injection f : pP,ďq Ñ pQ,ďq is monotone (mono.)when

a ď b ñ f paq À f pbq

Denote the mono.s from P to Q by monopP,Qq

A word of warning!

Monotone partial injections form categories / monoids

— these are not generally inverse categories / monoids.

The generalised inverse of a monotone partial injectionis not, in general, monotone.

Obvious example:

The natural numbers with distinct partial orderings.

Consider some non-zero

f P monoppN,“q, pN,ďqq

Then f´1 is monotone precisely when f “ 1txu, for some x P N.

From the inverse semigroup viewpoint:

We may consider several interesting subsets of monotonepartial injections.

tf : A Ñ B : f´1 P monopB,Aqu Ď monopA,Bq

tf : A Ñ B : dompf q is a chain.u Ď monopA,Bq

tf : A Ñ B : @ a K a1 P dompf q, f paq K f pa1qu Ď monopA,Bq

These all coincide with monopA,Bq itself when A,B arecountable, totally ordered sets.

Neither too simple, nor too complex

As the obvious example, consider N, with the usual ordering.

Denote the set of monotone partial injections on N by

monopN,Nq ď pInjpN,Nq

This set is closed under composition and generalised inverse

and contains all partial identities, so forms an inverse monoid.

A boolean inverse monoid?

monopN,Nq is not a boolean inverse monoid:

apnq “

n ` 1 n even

K n odd, bpnq “

K n even

n ´ 1 n odd

a K b P monopN,Nq, but a_ b is definitely not monotone.

Closure under joins of orthogonal elements fails!

A Σ-monoid structure nevertheless

An indexed family tajujPJ Ď monopN,Nq is summable iff

@i ‰ j P J,ai K aj

aj P pInjpN,Nq is monotone,

in which case, their sum is:ÿ

ajdef .“

aj P monopN,Nq.

We have a Σ-monoid structure, together with (infinite)

distributivity — as used by E. Haghverdi (2000)

Some intuition ...

Think of monotone partial injections on N as ‘planar diagrams’.

Monotonicity as planarity for partial injections on N

non-planar planar...

......

6 6qq5 5 5 5

3 32 2qq

1 1qq0 0oo 0 0

non-monotone monotone

Why planarity?

1 The quantum Jones polynomial algorithm(Aharanov, Jones, Landau)

A QM algorithm for computing Jones polynomials at e2kπi

Classically, a (presumably) P# problem.Based on the Temperley-Lieb algebra“Knot theory without crossings” – L. Kaufmann.

2 Lambek pregroups (From categorical linguistics)

Becoming used in Natural Language ProcessingDiagrams determined by planarity & acyclicity.

3 Complexity theory (Planarity provides bounds to complexity).

Matchgates and classical simulation of quantum circuits– R. Jozsa, A. MiyakeRestricting swap gates allows for efficient classicalsimulation of QM circuits.

4 . . .

First ... some simple theory!

Initial & final idempotents

Viewed graphically, the following is straightforward:

Every f P monopN,Nq is uniquely determined by its initial & finalidempotents.

planar...

6 65 54 43 32 21 10 0

monotone

dompf q, impf q Ď N are increasing sequences with bottom elements.

Initial & final idempotents

Viewed graphically, the following is straightforward:

Every f P monopN,Nq is uniquely determined by its initial & finalidempotents.

planar...

6 6qq5 5

3 32 2mm

1 1qq0 0

monotone

dompf q, impf q Ď N are increasing sequences with bottom elements.

An equivalent viewpoint

Every monotone partial bijection is uniquely determined by apair of points of the Cantor set C.

Formally, one-sided countably infinite strings over t0,1u,or equivalently, C “ FunpN, t0,1uq.

What we will not do!

We all know: the Cantor set is isomorphicto two copies of the Cantor set.

Why pairs of points?

Working with single points, rather than pairs of points, wouldobscure the intuition.

Formalising a little more ...

Given E Ď N, denote the indicator function for membership by

indEpnq “"

1 n P E ,0 otherwise.

Abusing notation slightly: given e2 “ e “ 1E P monopN,Nq,

we write Inde : N Ñ t0,1u.

A trivial observation:

For arbitrary partial injections f P pInjpN,Nq,

indff ´1pnq “8ÿ

indf ´1f pnq P NY t8u

A few simple definitions

A pair of Cantor points pd , cq P Cˆ C is balanced when

dpjq “8ÿ

cpjq P NY t8u

We denote the set of balanced Cantor points by B Ď Cˆ C.

There is a 1 : 1 correspondence B ” monopN,Nq.

Giving this explicitly:

Balanced Cantor pairs ” monotone partial injections

ð Given f P monopN,Nq, the balanced pair is:

pIndff ´1 , Indf ´1f q P B

ñ Given pt , sq P B, define mpt ,sq P monopN,Nq by

mpt ,sqpnq “

K spnq “ 0

minxPN

řxj“0 tpjq “

řnj“0 spjq

spnq “ 1

An illustration:

A balanced pair of Cantor points:

t “ 1001110 , s “ 0110101 . . .

n “ 0 1 2 3 4 5 6 . . .

spnq “ 0 1 1 0 1 0 1 . . .

tpnq “ 1 0 0 1 1 1 0 . . .

An illustration:

A balanced pair of Cantor points:

t “ 1001110 , s “ 0110101 . . .

n “ 0 1 2 3 4 5 6 . . .

spnq “ 0 1 1 0 1 0 1 . . .ř

jďn spjq “ 0 1 2 2 3 3 4 . . .

jďn tpjq “ 1 1 1 2 3 4 4 . . .

tpnq “ 1 0 0 1 1 1 0 . . .

A monoid operation on balanced pairs?

There exists some operation

¨ : BˆBÑ B

such that pB, ¨q – monopN,Nq.

What does this look like?

Normal forms (I)

A monotone partial injection is reducing when @x P N,

Indff ´1pxq “ 1 ñ Indff ´1pkq “ 1 @k ď x

Dually, it is expanding when @x P N,

Indf ´1f pxq “ 1 ñ Indf ´1f pkq “ 1 @k ď x

An illustrative example:

Expanding Reducing...

......

6 6 6 6

5 5 5 54 4 4 4

3 32 2

2 2qq1 1

1 1qq0 0oo 0 0

Cantor points as reducing / expanding arrows

Reducing (resp. expanding) arrows are uniquely determined by

their initial (resp. final) idempotents.

Given c P C, define Redc P monopN,Nq by

Redcpnq “

K n “ 0

řnj“0 cpnq ´ 1 n “ 1

Dually, define Expc P monopN,Nq by

Expc “ Red´1c

Normal forms (II)

Given an arbitrary monotone partial injection f P monopN,Nq,

then the balanced pair pt , sq “ pIndff ´1 , Indf ´1f q P B

is the unique balanced pair satisfying

f “ ExptReds

(The only non-trivial point is uniqueness, which follows

since pt , sq is required to be balanced).

By considering normal forms (or directly)

Given pv ,uq, pt , sq P B, define a composition by:

px ,wq “ pv ,uq ¨ pt , sq

where wpnq “ spnq.upjq.tpjq P t0,1u,

j “ minjPN

α“0

tpαq “nÿ

α“0

and similarly, xpnq “ vpnq.upkq.tpkq P t0,1u,

k “ minkPN

α“0

upαq “nÿ

α“0

The generalised inverse is immediate: pt , sq´1 “ ps.tq.

This givespB, ¨q – monopN,Nq as required.

Duals and self-encodings

Recall the complement / dual operation on the Cantor set:

cKpnq “ cpnq ` 1 pmod 2q @ c P C

E.g. c “ 0100101 . . . has complement cK “ 1011010 . . ..

A key definition

A Cantor point c P C is dual-balanced when pc, cKq is abalanced pair.

α“0

cpαq “ 8 “

α“0

cKpαq

Question: Topologically / graphically, what does the set of suchpoints look like?

Composing / inverting ??

Basic properties:

The set of balanced pairs of the form pc, cKq is:

closed under generalised inverse: pc, cKq´1 “ pcK, cq

decidedly not closed under composition!

pc, cKqpc, cKq “ 0 P monopN,Nq

The intuition / motivation

Dual-balanced Cantor points correspond to

Riffle Shuffles of two countably infinite decks of cards.

Credit: Johnny Blood Photography

How to shuffle (infinite) decks of cards

Cards from Deck A and Deck B are placed one-by-one onto asingle stack.

At each step, a single card is taken from the bottom1 of either Aor B, and placed on top of the stack.

D.-B. Cantor points provide instructions

At step j , take the next card from Deck

A cpjq “ 0

B cpjq “ 1

1Important: deals as opposed to shuffles are antimonotone!

Shuffles, threads & races

In theoretical / practical C.S. ‘shuffles’ are studied in parallelprocessing.

Two or more threads require processor time to perform aseries of tasks.

The ‘shuffle’ describes the actual order in which thesetasks are executed.

This is important whenever ‘race conditions’ occur.

Studying infinite shuffles is needed when we considernon-terminating processes.

pc, cKq P B ensures fair play!

The Dual-Balanced condition on the Cantor point

cpjq “ 8 “

ensures that no cards are ‘withheld’ in the process.

The alternating Cantor point

apnq “ n pmod 2q

gives the ‘perfect riffle shuffle’.

From D.-B. Cantor points to Young Tableaux

There is an obvious correspondence between1 D.-B. Cantor points,2 p8,8q Young tableaux.

c “ 1 0 1 0 1 0 1 1 0 0 . . . P C

cpnq “ 0 1 3 5 8 9 . . .

cpnq “ 1 0 2 4 6 7 . . .

The obvious question:

What about standard Young tableaux?

We first need some standard inverse semigroup theory.

D.-B. Cantor points as inverse monoids

Proposition: D.-B. Cantor points uniquely determine

effective representations of Nivat & Perrot’s polycyclic

monoid P2 as inverse submonoids of monopN,Nq.

Recall – the polycyclic monoid P2

Two generators, tp,quRelations:

pq´1 “ 0 “ q´1p and pp´1 “ 1 “ qq´1

Useful fact: polycyclic monoids are congruence-free.

Monotone representations of P2

Let c P C be dual-balanced. Looking at normal forms,

ExpcRedcK “ pc, cKq and pcK, cq “ ExpcKRedc

By construction, RedcExpc “ 1N “ RedcKExpcK .

By definition, of p qK : C Ñ C,

cpnq “ 0 ðñ cKpnq “ 1

and soRedcExpcK “ 0N “ RedcKExpc

The assignment p ÞÑ Redc , q Ñ RedcK gives a monotonerepresentation of P2.

Such representations are effective, since domppq Y dompqq “ N.

As always ... an example

For the alternating Cantor point, or perfect riffle shuffle,

apnq “ n pmod 2q or a “ 010101010101 . . .

we derive the representation of P2 corresponding to the Cantorpairing:

p´1pxq “ 2x and q´1pxq “ 2x ` 1

On to Standard Young tableaux

In standard Young tableaux, the cells are well-ordered

both horizontally and vertically.

x ay b

b ď y

Which dual-balanced Cantor points / riffle shuffles /

representations of P2 satisfy such conditions?

Ballot sequences & monoids

A (binary) ballot sequence is an element w P t0,1u˚ where,for every prefix u of w ,

#1s in u ď #0s in u

Denote the set of all finite ballot sequences by Ballot— this forms a submonoid of t0,1u˚.

By contradiction: Consider v ,w P Ballot such that vw R Ballot .Then there exists some prefix u of vw satisfying #0s in u ă #1s in u.As v P Ballot , u is not a prefix of v , so u “ vl , for some prefix l of w .However, #0s in v ě #1s in v . Therefore, #1s in l ě #0s in l ,contradicting the assumption that w P Ballot .

Ballot sequences & monoids

A (binary) ballot sequence is an element w P t0,1u˚ where,for every prefix u of w ,

#1s in u ď #0s in u

Denote the set of all finite ballot sequences by Ballot— this forms a submonoid of t0,1u˚.

By contradiction: Consider v ,w P Ballot such that vw R Ballot .Then there exists some prefix u of vw satisfying #0s in u ă #1s in u.As v P Ballot , u is not a prefix of v , so u “ vl , for some prefix l of w .However, #0s in v ě #1s in v . Therefore, #1s in l ě #0s in l ,contradicting the assumption that w P Ballot .

A surprisingly simple monoid

Ballot sequences are very heavily studied

... but only in combinatorics!

Proposition The monoid of binary ballot sequences is notfinitely generated.

By contradiction: Assume a finite generating set G forBallot ď t0,1u˚. As G is finite, the longest contiguous string of 1s inany member of G is bounded by some finite K P N. No composite ofmembers of G can account for the ballot sequence 0K`11K`1.

A non-minimal generating set is given by t0n`k1nupn,kqPNˆN

Boring monoid – interesting embedding

To see that this set is non-minimal:

p0x10qp0y1zq “ 0x`y1z

Consider the submonoid where

“All non-empty sequences contain 1.”

This submonoid:

has minimal generating set t0n`k`11n`1upn,kqPNˆN

is the free monoid on its generating set.

We derive an embedding of free monoids:

ufe : pNˆ Nq˚ Ñ t0,1u˚

From the finite to the infinite:

A Cantor point c P C is ballot provided

cpjq ď

cKpjq @ N P N

Denote the ballot Cantor points by CB.

(Standard example: the alternating Cantor point).

“ Every finite prefix is a member of the Ballot monoid”

Question:

How do such Cantor points behave under the usual meet and join?

The ballot Scott domain

Key properties:

There is no top element & they are not closed under the join

pc _ dqpnq “ maxtcpnq,dpnqu.

They are closed under the meet, pc ^ dqpnq “ cpnqdpnq

There is a bottom element zpnq “ 0, for all n P N.

The supremum of every chain c0 ď c1 ď c2 ď . . . is also in CB

There is a notion of compact element: c P CB is compact iffř8

j“0 cpjq ă 8

Every element is the supremum of a chain of compact elements.

There is also a nice diagrammatic representation:

The ballot Cantor points

� � � � �

��

Combining two properties:

Let us remove the compact points!

A dual-balanced ballot Cantor point c P C satisfies:ř8

j“0 cpjq “ř8

j“0 cKpjq

řNj“0 cpjq ď

řNj“0 cKpjq.

There is a 1:1 correspondence:

DBB Cantor points ” Standard p8,8q Young tableaux

The ballot property in C.S.

Recall the intuition of

Shuffles ” order of execution of threads

The ballot property rules out:

“We have executed more tasks from thread B than thread A”

Canonical example:

Thread A tasks push data onto a stack.

Thread B tasks remove data from a stack.

The ballot condition ensures that we are never trying toremove data from an empty stack.

The ballot property in C.S.

Recall the intuition of

Shuffles ” order of execution of threads

The ballot property rules out:

“We have executed more tasks from thread B than thread A”

Canonical example:

Thread A tasks push data onto a stack.

Thread B tasks remove data from a stack.

The ballot condition ensures that we are never trying toremove data from an empty stack.

Another interpretation

As card-shuffling:

The only way we can see: . . . x

. . . y zwith z ď x is when

“More cards have been laid from Deck B than from Deck A”

As DBB Cantor points are dual-balanced, they uniquely determine

representations of P2, as monotone partial injections on N.

Call these standard monotone representations.

Fun & games with polycyclic monoids

A very standard result N. & P. (1970)

There exists an embedding of P8 into P2.

Recall:

The infinite-generator polycyclic monoid P8 has generating

set tpjujPN, with relations pjp´1k “

1 j “ k

0 j ‰ k

The embedding is given by

pj ÞÑ pqj , p´1j ÞÑ q´jp´1

Straightforward to check that the required relations aresatisfied!

Polycyclic monoids as bijections

A slightly lesser-known result H.& L. ( ... a while back)

Representations of P8 within pInjpN,Nq correspond toinjections

Nˆ N ãÑ N

which are bijections when the representation is effective

A very simple construction

For a given representation, we define

Ψpx , yq “ p´1x pyq @px , yq P Nˆ N

A worked example:

Let’s do this for the standard monotone representation

determined by the alternating Cantor point a P C.

p´1pnq “ 2n and q´1pnq “ 2n ` 1

Expanding out, we get

Ψapx , yq “ q´xp´1pyq “ 2x`1y ` 2x ´ 1

A bijection from Nˆ N to N.

The infinite perfect shuffle

y “ 0 1 2 3 4 5 . . .

0 0 2 4 6 8 10 . . .1 1 5 9 13 17 21 . . .2 3 11 19 27 35 43 . . .3 7 23 39 55 71 87 . . .4 15 47 79 111 143 175 . . .5 31 94 159 223 287 351 . . ....

......

A few observations:

Simple observations

This table contains every natural number.

Both rows and columns appear to be well-ordered

— an p8,8,8, . . .q standard Young tableaux?

There seems to be some ‘underlying fractal structure’ ...

More practically — how easy is it to perform this shuffle?

Deep fractal structure ??On the nth step, we play from Deck x :

¨ ¨ ¨ Deck4 Deck3 Deck2 Deck1 Deck0

n “ 1 ‚

n “ 2 ‚

n “ 3 ‚

n “ 4 ‚

n “ 5 ‚

n “ 6 ‚

n “ 7 ‚

n “ 8 ‚

n “ 9 ‚

n “ 10 ‚

n “ 11 ‚

n “ 12 ‚

n “ 13 ‚

n “ 14 ‚

n “ 15 ‚

n “ 16 ‚

This looks kind of familiar!

¨ ¨ ¨ 24 23 22 21 20

n “ 1 1n “ 2 1 0n “ 3 1 1n “ 4 1 0 0n “ 5 1 0 1n “ 6 1 1 0n “ 7 1 1 1n “ 8 1 0 0 0n “ 9 1 0 0 1

n “ 10 1 0 1 0n “ 11 1 0 1 1n “ 12 1 1 0 0n “ 13 1 1 0 1n “ 14 1 1 1 0n “ 15 1 1 1 1n “ 16 1 0 0 0 0

Performing the perfect infinite riffle

A very simple rule

1 Count in binary ...

2 What is the most significant bit that has just changed?

3 Play a card from that deck!

The standard Young property

It is straightforward that rows and columns are well-ordered:

k “ Ψapx , yq for somepx , yq P Nˆ N.

l “ 2k ` 1 ą k

m “ k ` 2y`1 ą k .They also contain all natural numbers.

Claim These properties follow generally from:

1 The fact that representations of P2 are monotone(since they are derived from DB Cantor points).

2 The ballot property on these Cantor points.

A quick outline

Let c P C be a dual-balanced ballot Cantor point. This determines

an effective monotone representation P2c

ãÑ pInjpN,Nq which

corresponds to an p8,8q Young tableau:

p´1p0q p´1p1q p´1p2q p´1p3q p´1p4q . . .

q´1p0q q´1p1q q´1p2q q´1p3q q´1p4q . . .

By the ballot property, p´1pnq ď q´1pnq, so this is standard.

A quick outline (cont.)

By the same properties, q´k pnq ă q´pk`1qpnq, so the following table ishas well-ordered rows and columns:

p´1p0q p´1p1q p´1p2q p´1p3q p´1p4q . . .

q´1p´1p0q q´1p´1p1q q´1p´1p2q q´1p´1p3q q´1p´1p4q . . .

q´2p´1p0q q´2p´1p1q q´2p´1p2q q´2p´1p3q q´2p´1p4q . . .

q´3p´1p0q q´3p´1p1q q´3p´1p2q q´3p´1p3q q´3p´1p4q . . ....

......

Finally, as q´1 is monotone and q´1pxq ą p´1pxq, we deduce that

q´1pNq “ H

and so the embedding of P8 is effective.

From the infinite to the finite

Each p8,8,8, . . .q standard Young tableau can be written as aconsistent sequence of finite standard Young tableaux.

For the alternating Cantor point:

n=0 n=1 n=2 n=3 n=4 n=5

Generators of irreducible representations of symmetric groups

. . . or just a complicated way of counting in binary?

Thinking more categorically

From counting in binary to categorical models

of fixed points for reversible computing.

Thinking categorically ...

An old result (H. & L.):

Every effective representation of P2 within pInjpN,Nqdetermines a (semi-) monoidal tensor on this monoid.

The construction: For all partial bijections f ,g P pInjpN,Nq, wetake

f ‹ g “ p´1fp _ q´1fq

Explaining terms!

A monoidal tensor on a category C is a functor,

p b q : C ˆ C ÝÑ C

satisfying additional properties.

Functoriality implies:

1 Identities are preserved . . .

1X b 1Y “ 1XbY

2 . . . as is composition:

pf b gqph b kq “ pfh b gkq

About those additional properties ...

1 There exists a unit object I satisfying

Ab I – A – I b A

for all objects A.

2 The tensor b is associative up to a (well-behaved)family of isomorphisms:

¨fbpgbhq //

��

¨pfbgqbh

It is strict when f b pg b hq“pf b gq b h.

A semi- suffices

A semi-monoidal category satisfies the exioms for a monoidal

category, except perhaps for the existence of a unit object.

Some standard results

1 We can freely adjoin a unit object to any semi-monoidalcategory.

2 At the unit object (i.e. in the monoid CpI, Iq)

The tensor and composition coincide.

The monoid CpI, Iq is thus (E.-H. argument) abelian.

We are working with monoids (single-object categories)

— things become uninteresting if our unique object is the unit.

Semi-monoidal monoids:

A semi-monoidal monoid consists of:A single-object category (i.e. a monoid) M

A monoid homomorphism

‹ : MˆMÑM

An invertible associativity element α PM satisfying:

αpf b pg b hqq “ ppf b gq b hqα

MacLane’s coherence condition

α2 “ pα ‹ 1qαp1 ‹ αq

A concrete example (H. & L.)Based on the alternating Cantor point apnq “ n` 1 pmod 2q, we have:

The monoid: pInjpN,Nq

The tensor

pf ‹ gqpnq “

n pmod 2q “ 0

2g` n´1

` 1 n pmod 2q “ 1

The associativity element:

αpnq “

2n n pmod 2q “ 0,

n ` 1 n pmod 4q “ 1,

n´12 n pmod 4q “ 3.

Dealing with associativity:

A consequence of M.V.L. (2004)

The group of isomorphisms generated by:

αpnq “

2n n pmod 2q “ 0,n ` 1 n pmod 4q “ 1,pn ´ 1q{2 n pmod 4q “ 3.

p1 ‹ αqpnq “

n n pmod 2q “ 0,2n ´ 1 n pmod 4q “ 1,n ` 2 n pmod 8q “ 3,pn ´ 1q{2 n pmod 8q “ 7.

generate a copy of Thompson’s group F

... as used by V. Shpilrain, A. Ushakov intheir public-key cryptographic protocol.

Can we please have a simpler example?

Can we chose a different effective representation of P2so that associativity becomes strict?

f ‹ pg ‹ hq “ pf ‹ hq ‹ h

How about if we relax conditions on the ballot property /monotonicity / effectiveness / etc. ?

A no-go result

Proposition: In any semi-monoidal monoid pM, ‹q,

p ‹ q : MˆMÑM is strictly associativeðñ

The unique object of M is the unit object.

Proof (ð) (Standard Theory ...)

The complicated direction!

(Proof) pðq

Some useful theory ...An alternative characterization of unit objects given by:

A. Saavedra (1972), J. Kock (2008), A. Joyal, J. Kock (2011)

In the monoid case, these state:

The unique object of pM, ‹q is a unit object precisely when

p1 ‹ q, p ‹ 1q : MÑM

are isomorphisms.

Is it because I is strict?

Assume ‹ is strictly associative, and define the injectivemonoid homomorphism:

η “ p1 ‹ ‹ 1q : M ãÑM

Define a semi-monoidal tensor on its image, by, for allηprq, ηpsq P ηpMq

ηprq d ηpsq “ 1 ‹ pr ‹ sq ‹ 1

By construction, pM, ‹q – pηpMq,dq.

The final step

By definition, for all ηpf q P ηpMq,

1d ηpf q “ 1 ‹ p1 ‹ f q ‹ 1“ 1 ‹ 1 ‹ f ‹ 1“ 1 ‹ f ‹ 1“ ηpf q

Thus 1d “ IdηpMq “ d 1.

The unique object of pηpMq,dq is a unit object.

However, pηpMq,dq – pM, ‹q.

Thus the unique object of M is a unit object. M is an abelianmonoid, and ‹ is simply composition.

A categorical reason why P2 is congruence-free & F has no non-abelian quotients.

The conclusion ...

We are stuck dealing with non-strict associativity

¨fbpgbhq //

��

¨pfbgqbh

However, dealing with it is not complex, despite appearances.

Another example

Let’s look at another tensor on pInjpN,Nq

One tensor is never enough!

Start with the alternating Cantor point

a “ 010101010 . . .

or any DBB Cantor point.

This determines a standard monotone representationof P2 within pInjpN,Nq.

Via the N. & P embedding P8 ãÑ P2,

pj ÞÑ pq j @j P N

this in turn determines an effective embedding of P8.

This is equivalent to a bijection Ψa : Nˆ N Ñ N, given by

Ψapx , yq “ p´1x pyq @px , yq P Nˆ N

a “ 010101010 . . .

pj ÞÑ pq j @j P N

a “ 010101010 . . .

pj ÞÑ pq j @j P N

a “ 010101010 . . .

pj ÞÑ pq j @j P N

Another H.-L. result

We define a tensor using the bijection Ψc : Nˆ N Ñ N by:

u b v “ Ψcpu ˆ vqΨ´1c @u, v P pInjpN,Nq

We may prove this is a semi-monoidal tensor by:

Direct calculations (tedious)

Categorical methods (abstract nonsense)

Unwinding the definitions:

u b v “

p´1upjq v pj “

q´upjqp´1 v pqj

Another H.-L. result

We define a tensor using the bijection Ψc : Nˆ N Ñ N by:

u b v “ Ψcpu ˆ vqΨ´1c @u, v P pInjpN,Nq

We may prove this is a semi-monoidal tensor by:

Direct calculations (tedious)

Categorical methods (abstract nonsense)

Unwinding the definitions:

u b v “

p´1upjq v pj “

q´upjqp´1 v pqj

A quick check ...

Checking the Saavedra / Kock / Joyal conditions for units:

The endomorphisms p1b q, p b 1q : pInjpN,Nq Ñ pInjpN,Nq

are injective monoid homomorphisms.

are not isomorphisms!

Some previous work ...

These operations (in heavily disguised form!) were used byJ.-Y. Girard’s GoI series (1989 onwards).

Using Girard’s notation, define ?pf q “ pf b 1q and !pf q “ p1b f q.

A quick check ...

Checking the Saavedra / Kock / Joyal conditions for units:

The endomorphisms p1b q, p b 1q : pInjpN,Nq Ñ pInjpN,Nq

are injective monoid homomorphisms.

are not isomorphisms!

Some previous work ...

These operations (in heavily disguised form!) were used byJ.-Y. Girard’s GoI series (1989 onwards).

Using Girard’s notation, define ?pf q “ pf b 1q and !pf q “ p1b f q.

Writing things explicitly (I)

Let’s look at p1b q.

!pf q “ p1b f q “ Ψp1ˆ f qΨ´1

From the definition of Ψ,

!pf q “8ł

p´1j f pj

In terms of P2, rather than P8,

!pf q “8ł

q´jp´1 f pqj

Writing things explicitly (II)

Alternatively, consider p b 1q.

?pf q “ pf b 1q “ Ψpf ˆ 1qΨ´1

From the definition of Ψ,

?pf q “8ł

p´1f pjq pj

In terms of P2, rather than P8,

?pf q “8ł

q´f pjqp´1 pqj

Interacting tensors (I)

How do ‹ and ?p q interact??

By direct calculation:

0‹?pf q “ p´10p _ q´1?pf qq

“ q´1´

Ž8j“0 q´f pjqp´1pqj

“Ž8

j“0 q´pf pjq`1qp´1pqpj`1q

“ ?psucc.f .succ´1q

Where succpnq “ n ` 1 is the successor function.

A simple shift operation on the action of ?pf q.

Interacting tensors (II)

How do ‹ and !p q interact??

By direct calculation:

f‹!pf q “ p´1fp _ q´1!pf qq

“ p´1fp _ q´1´

Ž8j“0 q´jp´1fpqj

“ p´1fp _´

Ž8j“1 q´jp´1fpqj

“ !pf q

The homomorphism 1b provides right fixed points for ‹ .

More generally

In general, what is the interaction between ‹ and b ?

An extremely subtle interaction, determined by:

1 N. & P.’s embedding P8 ãÑ P2.

2 Countability of N.

3 The ballot property & monotonicity.

4 Categorical distributivity.

5 . . .

Are we nearly there yet ??

We still haven’t found what we’re looking for

We have exhibited: fixed points for a tensor.

What is needed: fixed points for functional application.

The two are related in a fundamental way,in monoidal closed categories.

Fortunately: There is a construction (JSV96,SA96) of (well-behaved)monoidal closed categories that preserves such fixed points.

This is of course(!) applicable to pInj and related categories.

A little bit of advertising ...

For this to work in general, we need:

(Semi-)monoidal equivalences between categories & monoids.

A related coherence theorem / strictification procedure.

(PMH 2016) Journal of Homotopy & Related Structures ... to appear.

Even more fortunately ...

That belongs in the realm of category theory

rather than (inverse) semigroup theory.

Although inverse categories

– particularly pInj –

provide particularly neat examples !

Monoids, Categories & Monotone Partial...

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