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Monoids,Categories &Monotone Partial Injections
Why Computer Scientists should knowand love Inverse Semigroup Theory
[email protected] 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.
[email protected] Peter Hines — University of York
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)
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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!
[email protected] Peter Hines — University of York
A Σ-monoid structure nevertheless
An indexed family tajujPJ Ď monopN,Nq is summable iff
@i ‰ j P J,ai K aj
aiŽ
aj P pInjpN,Nq is monotone,
in which case, their sum is:ÿ
jPJ
ajdef .“
ł
jPJ
aj P monopN,Nq.
We have a Σ-monoid structure, together with (infinite)
distributivity — as used by E. Haghverdi (2000)
[email protected] Peter Hines — University of York
Some intuition ...
Think of monotone partial injections on N as ‘planar diagrams’.
Monotonicity as planarity for partial injections on N
non-planar planar...
......
...
6 6
tt
6 6qq5 5 5 5
4 4
ll
4 4oo
3 3mm
3 32 2qq
2 2mm
1 1mm
1 1qq0 0oo 0 0
non-monotone monotone
[email protected] Peter Hines — University of York
Why planarity?
1 The quantum Jones polynomial algorithm(Aharanov, Jones, Landau)
A QM algorithm for computing Jones polynomials at e2kπi
5
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 . . .
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First ... some simple theory!
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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
4 4oo
3 32 2mm
1 1qq0 0
monotone
dompf q, impf q Ď N are increasing sequences with bottom elements.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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,
8ÿ
n“0
indff ´1pnq “8ÿ
n“0
indf ´1f pnq P NY t8u
[email protected] Peter Hines — University of York
A few simple definitions
A pair of Cantor points pd , cq P Cˆ C is balanced when
8ÿ
j“0
dpjq “8ÿ
j“0
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.
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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 . . .
[email protected] Peter Hines — University of York
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 . . .
ř
jďn tpjq “ 1 1 1 2 3 4 4 . . .
tpnq “ 1 0 0 1 1 1 0 . . .
[email protected] Peter Hines — University of York
A monoid operation on balanced pairs?
There exists some operation
¨ : BˆBÑ B
such that pB, ¨q – monopN,Nq.
What does this look like?
[email protected] Peter Hines — University of York
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
uu
5 5 5 54 4 4 4
ss3 3
kk
3 32 2
kk
2 2qq1 1
kk
1 1qq0 0oo 0 0
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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).
[email protected] Peter Hines — University of York
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
#
jÿ
α“0
tpαq “nÿ
α“0
spαq
+
and similarly, xpnq “ vpnq.upkq.tpkq P t0,1u,
k “ minkPN
#
kÿ
α“0
upαq “nÿ
α“0
vpαq
+
The generalised inverse is immediate: pt , sq´1 “ ps.tq.
This givespB, ¨q – monopN,Nq as required.
[email protected] Peter Hines — University of York
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.
8ÿ
α“0
cpαq “ 8 “
8ÿ
α“0
cKpαq
Question: Topologically / graphically, what does the set of suchpoints look like?
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
The intuition / motivation
Dual-balanced Cantor points correspond to
Riffle Shuffles of two countably infinite decks of cards.
Credit: Johnny Blood Photography
[email protected] Peter Hines — University of York
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!
[email protected] Peter Hines — University of York
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.
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pc, cKq P B ensures fair play!
The Dual-Balanced condition on the Cantor point
8ÿ
j“0
cpjq “ 8 “
8ÿ
j“0
cKpjq
ensures that no cards are ‘withheld’ in the process.
The alternating Cantor point
apnq “ n pmod 2q
gives the ‘perfect riffle shuffle’.
[email protected] Peter Hines — University of York
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.
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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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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
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On to Standard Young tableaux
In standard Young tableaux, the cells are well-ordered
both horizontally and vertically.
x ay b
x ď
ď
aď
b ď y
Which dual-balanced Cantor points / riffle shuffles /
representations of P2 satisfy such conditions?
[email protected] Peter Hines — University of York
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 .
[email protected] Peter Hines — University of York
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 .
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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
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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˚
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From the finite to the infinite:
A Cantor point c P C is ballot provided
Nÿ
j“0
cpjq ď
Nÿ
j“0
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?
[email protected] Peter Hines — University of York
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:
[email protected] Peter Hines — University of York
The ballot Cantor points
� � � � �
�
�
�
�
�
�
�
�
���
���
���
���
���
���
���
���
���
������
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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!
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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
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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.
[email protected] Peter Hines — University of York
The infinite perfect shuffle
y “ 0 1 2 3 4 5 . . .
x“
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 . . ....
......
......
......
. . .
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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?
[email protected] Peter Hines — University of York
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 ‚
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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
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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!
[email protected] Peter Hines — University of York
The standard Young property
It is straightforward that rows and columns are well-ordered:
k m
l
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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
8č
j“0
q´1pNq “ H
and so the embedding of P8 is effective.
[email protected] Peter Hines — University of York
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
0 0
1
0 2
1
0 2
1
3
0 2 4
1
3
0 2 4
1 5
3
Generators of irreducible representations of symmetric groups
. . . or just a complicated way of counting in binary?
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Thinking more categorically
From counting in binary to categorical models
of fixed points for reversible computing.
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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
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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
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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 //
α
��
¨
¨pfbgqbh
// ¨
α´1
OO
It is strict when f b pg b hq“pf b gq b h.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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 “
$
’
’
&
’
’
%
2f` n
2
˘
n pmod 2q “ 0
2g` n´1
2
˘
` 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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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. ?
[email protected] Peter Hines — University of York
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 ...)
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
The conclusion ...
We are stuck dealing with non-strict associativity
¨fbpgbhq //
α
��
¨
¨pfbgqbh
// ¨
α´1
OO
However, dealing with it is not complex, despite appearances.
[email protected] Peter Hines — University of York
Another example
Let’s look at another tensor on pInjpN,Nq
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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
[email protected] Peter Hines — University of York
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 “
8ł
j“0
p´1upjq v pj “
8ł
j“0
q´upjqp´1 v pqj
[email protected] Peter Hines — University of York
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 “
8ł
j“0
p´1upjq v pj “
8ł
j“0
q´upjqp´1 v pqj
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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
j“0
p´1j f pj
In terms of P2, rather than P8,
!pf q “8ł
j“0
q´jp´1 f pqj
[email protected] Peter Hines — University of York
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
j“0
p´1f pjq pj
In terms of P2, rather than P8,
?pf q “8ł
j“0
q´f pjqp´1 pqj
[email protected] Peter Hines — University of York
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
¯
q
“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.
[email protected] Peter Hines — University of York
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
¯
q
“ p´1fp _´
Ž8j“1 q´jp´1fpqj
¯
“ !pf q
The homomorphism 1b provides right fixed points for ‹ .
[email protected] Peter Hines — University of York
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 . . .
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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.
[email protected] Peter Hines — University of York
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 !
[email protected] Peter Hines — University of York