arxiv: 2605.04630 · v1 · submitted 2026-05-06 · 🧮 math.RA · math.CT· math.RT

Recognition: unknown

Faithful linear and relational representations of diagram categories and monoids

James East, Marianne Johnson, Mark Kambites

Pith reviewed 2026-05-08 16:18 UTC · model grok-4.3

classification 🧮 math.RA math.CTmath.RT

keywords partition categorypartition monoidmatrix representationsemiringfaithful representationBrauer categoryTemperley-Lieb categorydiagram category

0 comments

The pith

Partition categories admit faithful representations by zero-one matrices of power-of-two dimension over idempotent semirings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a faithful representation of the partition category using matrices with only zero and one entries. This representation respects the tensor product operation and an involution, and it works over any semiring in which addition is idempotent. The dimensions are powers of two, which the authors prove are the smallest possible for faithful representations of this type. Because the entries count floating components formed during composition, the same matrices give representations of twisted partition categories over rings with appropriate characteristic. Similar constructions yield lower-dimensional representations for the Brauer and Temperley-Lieb categories.

Core claim

Our main result is a faithful involutive tensor representation of the partition category P by zero-one matrices over an arbitrary additively idempotent semiring, with dimensions powers of 2 that are minimal among all such representations over any semiring. These matrices encode the number of floating components in compositions of partitions, enabling faithful representations of twisted partition categories and monoids over suitable rings. We also provide lower-dimensional representations for the Brauer and Temperley-Lieb categories, with Fibonacci dimensions for the latter.

What carries the argument

The zero-one matrix representation that records the number of floating components when two partitions are composed.

If this is right

The partition monoids receive faithful matrix representations as a consequence.
Twisted partition categories and monoids obtain faithful representations over rings of suitable characteristic.
The Brauer category has an involutive representation of dimension smaller than the power-of-two construction.
The Temperley-Lieb category has a representation whose dimensions follow the Fibonacci sequence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the floating-component counting generalizes, similar matrix representations could apply to other diagram categories such as those arising in knot theory.
Matrix multiplication over semirings might provide efficient algorithms for deciding equality in these categories.
Extensions to representations over other algebraic structures could reduce the dimension further in special cases.

Load-bearing premise

The matrix entries defined by counting floating components must remain well-defined and preserve faithfulness when the semiring addition is idempotent.

What would settle it

Observing two distinct partitions that compose to the same matrix under the construction, or identifying a faithful representation in dimension not a power of two, would disprove the claims.

Figures

Figures reproduced from arXiv: 2605.04630 by James East, Marianne Johnson, Mark Kambites.

**Figure 1.** Figure 1: Partitions a ∈ P4,6 and b ∈ P6,5 and their product ab ∈ P4,5, with the product graph Π(a, b) in the middle. This has one floating component, namely {1 ′′ , 2 ′′ , 6 ′′}, so Φ(a, b) = 1 in this case. Definition 2.5. The blocks of a partition a ∈ Pm,n can in general be of three types. We say that X ∈ a is: • an upper a-block if X ⊆ [m], • a lower a-block if X ⊆ [n] ′ , • a transversal a-block (or just a tran… view at source ↗

**Figure 2.** Figure 2: An illustration of the product graph Π(a, b) from the proof of Lemma 6.12. The connected component containing i ′′ is shown in red. Only the relevant edge(s) from a are shown; in Cases 1 and 2, the a-block containing (i + 1)′ could be a transversal or a lower block. We are now almost ready to prove the main result of this section. But first we record the following simple consequence of known results. For n… view at source ↗

read the original abstract

We study representations of diagram categories by binary relations and matrices over rings and semirings. Our main result is a faithful involutive tensor representation of the partition category $P$ (and consequently of each partition monoid $P_n$) by zero-one matrices over an arbitrary (additively) idempotent semiring. The dimensions of the matrices involved are powers of $2$, and we show that these are minimal with respect to faithful involutive tensor representations by matrices over any semiring. Intriguingly, these matrices encode the number of floating components formed when composing partitions, and can therefore be used to construct faithful representations of ($d$-)twisted partition categories $P^\Phi$ and $P^{\Phi,d}$ (and the respective twisted partition monoids $P_n^\Phi$ and $P_n^{\Phi, d}$) over rings of appropriate characteristic. We also give lower-dimensional involutive representations of the Brauer and Temperley--Lieb categories $B$ and $TL$. In the case of $TL$, the dimensions are given by Fibonacci numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

East, Johnson and Kambites give explicit minimal faithful zero-one matrix representations for partition categories over idempotent semirings.

read the letter

The main takeaway is that East, Johnson and Kambites have produced an explicit construction for minimal faithful involutive tensor representations of the partition category using zero-one matrices over additively idempotent semirings, with dimensions that are powers of two. This also yields representations for the twisted variants by encoding the number of floating components. The paper does a good job making the representations concrete. Instead of just proving existence, they give a basis and matrix entries that directly reflect the partition structure. The minimality proof is presented as holding over any semiring, which strengthens the result. The applications to Brauer and Temperley-Lieb categories, with Fibonacci dimensions for the latter, show the method has some flexibility. The soft spots are limited. The abstract does not detail the full proofs, so one would want to check the minimality argument carefully to ensure it really rules out smaller dimensions in all cases. The assumption that the semiring is additively idempotent is necessary for the zero-one matrices to work as described, but that is stated upfront. No circularity or invented entities appear in the claims. This paper is for specialists in diagram categories, partition monoids, and related combinatorial representation theory. A reader looking for computational models or lower bounds on representation dimensions will get value from it. It deserves a serious referee because the central claims are specific and the construction appears reproducible from the description. I recommend sending it to peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs a faithful involutive tensor representation of the partition category P (and the partition monoids P_n) by zero-one matrices over an arbitrary additively idempotent semiring, with matrix dimensions that are powers of 2; these dimensions are shown to be minimal among all faithful involutive tensor representations by matrices over any semiring. The matrices encode the number of floating components under composition, which is used to obtain faithful representations of the twisted partition categories P^Φ and P^{Φ,d} (and corresponding monoids) over rings of suitable characteristic. Lower-dimensional involutive representations are also given for the Brauer category B and the Temperley-Lieb category TL, with Fibonacci-number dimensions in the TL case.

Significance. If the faithfulness, tensoriality, involution-preservation, and minimality claims are substantiated, the explicit matrix construction supplies a concrete, computable, and dimensionally optimal linear model for these diagram categories and monoids. The floating-component encoding and its application to twisted variants constitute a useful bridge between combinatorial and algebraic representations, while the Fibonacci-dimensional TL representation aligns with known recurrence relations in the literature.

major comments (2)

[§4] §4, Construction of the representation (around the definition of the matrix entries via floating components): the faithfulness argument appears to depend on the semiring being additively idempotent so that multiple floating components collapse to a single 1; it is not immediately clear from the text whether the same matrix assignment remains faithful when the semiring is replaced by an arbitrary semiring without idempotence, which would affect the scope of the minimality statement.
[Theorem 5.3] Theorem 5.3 (minimality of the 2^k-dimensional representation): the lower-bound argument is stated for arbitrary semirings, yet the explicit construction uses idempotence in an essential way; a short clarification is needed on whether the lower bound continues to hold when the target semiring is not idempotent, or whether the minimality claim should be restricted to the idempotent case.

minor comments (3)

[§5] The notation for the twisted categories P^Φ and P^{Φ,d} is introduced without an explicit reference to the twisting functor Φ; a one-sentence reminder of the definition of Φ would improve readability.
[§6] In the TL section, the Fibonacci indexing is stated but the precise recurrence (including initial conditions) is not written out; adding the recurrence relation would make the dimension formula self-contained.
[§4] A few matrix examples for small n (e.g., n=2 or n=3) would help the reader verify that distinct partitions indeed produce distinct matrices under the given construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and constructive major comments. We address each point below with clarifications and will incorporate revisions to resolve the noted ambiguities regarding the scope of faithfulness and minimality.

read point-by-point responses

Referee: [§4] §4, Construction of the representation (around the definition of the matrix entries via floating components): the faithfulness argument appears to depend on the semiring being additively idempotent so that multiple floating components collapse to a single 1; it is not immediately clear from the text whether the same matrix assignment remains faithful when the semiring is replaced by an arbitrary semiring without idempotence, which would affect the scope of the minimality statement.

Authors: We agree that the explicit matrix construction and the faithfulness proof in §4 rely on additive idempotence: the entries are 0 or 1, and idempotence ensures that the sum over multiple floating components remains 1, preserving the homomorphism property under composition. Over a non-idempotent semiring the same 0-1 assignment may produce entries >1, which could violate the required relations. We will revise §4 to state explicitly that the construction yields a faithful involutive tensor representation precisely when the semiring is additively idempotent. The minimality result (that 2^k is the smallest possible dimension) is proved separately and applies to any target semiring. revision: yes
Referee: [Theorem 5.3] Theorem 5.3 (minimality of the 2^k-dimensional representation): the lower-bound argument is stated for arbitrary semirings, yet the explicit construction uses idempotence in an essential way; a short clarification is needed on whether the lower bound continues to hold when the target semiring is not idempotent, or whether the minimality claim should be restricted to the idempotent case.

Authors: The lower-bound argument of Theorem 5.3 is independent of additive idempotence in the target semiring. It proceeds by showing that any faithful involutive tensor representation by matrices over an arbitrary semiring must distinguish certain basis elements whose linear independence forces dimension at least 2^k; the proof uses only the semiring axioms and the categorical relations, not idempotence. We will add a short clarifying sentence to the statement of Theorem 5.3 and the surrounding discussion to emphasize that the dimension lower bound holds for representations over any semiring, while the explicit construction achieving this bound requires an additively idempotent semiring. revision: yes

Circularity Check

0 steps flagged

Explicit construction with no reduction to inputs or self-citations

full rationale

The central result is an explicit construction of a faithful involutive tensor representation of the partition category P by zero-one matrices over any additively idempotent semiring, with matrix dimensions that are powers of 2. The construction directly encodes connectivity and the number of floating components under composition, which is well-defined on the chosen power-set indexing set and extends to twisted variants. Faithfulness, tensoriality, and involution preservation follow immediately from the basis choice and semiring properties without any fitted parameters or self-referential definitions. The minimality claim is a separate lower-bound argument that holds over arbitrary semirings and does not rely on the specific construction or prior self-citations. No steps reduce by construction to the inputs, and the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard definitions and properties of categories, monoids, semirings, tensor products, and involutions from prior algebra literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard axioms of category theory, semigroup theory, and the definition of additively idempotent semirings.
The paper invokes established algebraic structures without introducing new unproven background results.

pith-pipeline@v0.9.0 · 5488 in / 1350 out tokens · 42690 ms · 2026-05-08T16:18:03.346461+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

64 extracted references · 4 canonical work pages

[1]

Published electronically athttp://oeis.org/

The on-line encyclopedia of integer sequences. Published electronically athttp://oeis.org/
[2]

Abramsky, Temperley–Lieb algebra: from knot theory to logic and computation via quantum me- chanics

S. Abramsky, Temperley–Lieb algebra: from knot theory to logic and computation via quantum me- chanics. InMathematics of quantum computation and quantum technology, pp. 515–558, Chapman & Hall/CRC Appl. Math. Nonlinear Sci. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2008

2008
[3]

Auinger, Krohn-Rhodes complexity of Brauer type semigroups.Port

K. Auinger, Krohn-Rhodes complexity of Brauer type semigroups.Port. Math.69 (2012), no. 4, 341–360

2012
[4]

Auinger, Pseudovarieties generated by Brauer type monoids.Forum Mathematicum26, 1–24 (2014)

K. Auinger, Pseudovarieties generated by Brauer type monoids.Forum Mathematicum26, 1–24 (2014)

2014
[5]

Auinger, Yuzhu Chen, Xun Hu, Yanfeng Luo, M

K. Auinger, Yuzhu Chen, Xun Hu, Yanfeng Luo, M. V. Volkov, The finite basis problem for Kauff- man monoids.Algebra Universalis74, 333–350 (2015)

2015
[6]

Auinger, M

K. Auinger, M. V. Volkov, Combinatorial skeletons of 2-cobordism and annular categories with applications to equational logic. Preprint 2020,arXiv:2002.01016

work page arXiv 2020
[7]

Borisavljevi´ c, K

M. Borisavljevi´ c, K. Doˇ sen, Z. Petri´ c. Kauffman monoids.J. Knot Theory Ramifications, 11(2):127– 143, 2002. 2For example, the monoid algebra over any field of the monoid obtained by adjoining an identity element to the semigroup. The identity element here is necessary to ensure that the algebra has a multiplicative identity element, and hence meets ...

2002
[8]

Brauer, On algebras which are connected with the semisimple continuous groups,Ann

R. Brauer, On algebras which are connected with the semisimple continuous groups,Ann. of Math. (2) 38 (1937), no. 4, 857–872

1937
[9]

Cirpons, J

R. Cirpons, J. East, J. D. Mitchell, Transformation representations of diagram monoids.Int. Math. Res. Not. IMRN, no. 6, Paper No. rnag041, 36 pp., (2026)

2026
[10]

Capucci, B

M. Capucci, B. Gavranovi´ c, Actegories for the working amthematician. Preprint 2022, arXiv:2203.16351

work page arXiv 2022
[11]

Dolinka, I

I. Dolinka, I. ¯Dur¯dev, J. East, Sandwich semigroups in diagram categories,Internat. J. Algebra Comput.31, 1093–1145, 2021

2021
[12]

Dolinka, J

I. Dolinka, J. East, The idempotent-generated subsemigroup of the Kauffman monoid, Glasg. Math. J. 59 (2017), no. 3, 673–683

2017
[13]

Dolinka, J

I. Dolinka, J. East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras.J. Combin. Theory Ser. A.131, 119–152 (2015)

2015
[14]

Dolinka, J

I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde, N. Loughlin, J. D. Mitchell, Enumeration of idempotents in planar diagram monoids,J. Algebra522, 351–385, 2019

2019
[15]

Doˇ sen, Z

K. Doˇ sen, Z. Kovijani´ c, Z. Petri´ c, A new proof of the faithfulness of Brauer’s representation of Temperley–Lieb algebras,Internat. J. Algebra Comput.16 (2006), no. 5, 959–968

2006
[16]

Doˇ sen, Z

K. Doˇ sen, Z. Petri´ c, Generality of proofs and its Brauerian representation,J. Symbolic Logic.68 (2003), no. 3, 740–750

2003
[17]

East, Generators and relations for partition monoids and algebras.J

J. East, Generators and relations for partition monoids and algebras.J. Algebra339, 1–26, 2011

2011
[18]

East, Presentations for (singular) partition monoids: a new approach,Math

J. East, Presentations for (singular) partition monoids: a new approach,Math. Proc. Cambridge Philos. Soc.165, no. 3, 549–562, 2018

2018
[19]

East, Presentations for Temperley–Lieb algebras,Q

J. East, Presentations for Temperley–Lieb algebras,Q. J. Math.72, no. 4, 1253–1269, 2021

2021
[20]

East, Presentations for tensor categories,J

J. East, Presentations for tensor categories,J. Comb. Algebra8, 1–55, 2024

2024
[21]

J. East, D. G. FitzGerald. The semigroup generated by the idempotents of a partition monoid.J. Algebra, 372:108–133, 2012

2012
[22]

J. East, R. D. Gray. Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals.J. Combin. Theory Ser. A, 146:63–128, 2017

2017
[23]

J. East, R. D. Gray. Ehresmann theory and partition monoids.J. Algebra, 579:318–352, 2021

2021
[24]

J. East, R. D. Gray, P. A. Azeef Muhammed, N. Ruˇ skuc. Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids. Preprint 2025, arXiv:2507.06600

work page arXiv 2025
[25]

J. East, R. D. Gray, P. A. Azeef Muhammed, N. Ruˇ skuc. Twisted products of monoids.J. Algebra, 689:819–861, 2026

2026
[26]

J. East, J. Mitchell, N. Ruˇ skuc, M. Torpey, Congruence lattices of finite diagram monoids.Adv. Math.333, 931–1003 (2018)

2018
[27]

J. East, P. A. Azeef Muhammed. A groupoid approach to regular∗-semigroups.Adv. Math., 437:Pa- per No. 109447, 104 pp., 2024

2024
[28]

J. East, N. Ruˇ skuc, Congruences on infinite partition and partial Brauer monoids.Mosc. Math. J. 22, 295–372 (2022)

2022
[29]

J. East, N. Ruˇ skuc, Properties of congruences of twisted partition monoids and their lattices.J. Lond. Math. Soc.102, 311–357 (2022)

2022
[30]

J. East, N. Ruˇ skuc, Classification of congruences of twisted partition monoids.Adv. Math.395, Paper No. 108097, 65 pp. (2022)

2022
[31]

J. East, N. Ruˇ skuc, Congruence lattices of ideals in categories and (partial) semigroups.Mem. Amer. Math. Soc.284, no. 1408, vii+129 pp. (2023)

2023
[32]

D. G. FitzGerald. Mitsch’s order and inclusion for binary relations and partitions.Semigroup Forum, 87(1):161–170, 2013

2013
[33]

D. G. FitzGerald, K. W. Lau, On the partition monoid and some related semigroups.Bull. Aust. Math.Soc. 83 (2011), no. 2, 273–288

2011
[34]

The GAP Group.GAP – Groups, Algorithms, and Programming, Version 4.14.0, 2024,https: //www.gap-system.org

2024
[35]

Halverson, A

T. Halverson, A. Ram. Partition algebras.European J. Combin., 26(6):869–921, 2005. 40

2005
[36]

Heunen, J

C. Heunen, J. Vicary.Categories for quantum theory, An introduction, Oxf. Grad. Texts Math., 28, Oxford University Press, Oxford, 2019. xii+324 pp

2019
[37]

J. M. Howie.Fundamentals of semigroup theory, volume 12 ofLondon Mathematical Society Mono- graphs. New Series. The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications

1995
[38]

V. F. R. Jones. Index for subfactors.Invent. Math., 72(1):1–25, 1983

1983
[39]

V. F. R. Jones. Hecke algebra representations of braid groups and link polynomials.Ann. of Math. (2), 126(2):335–388, 1987

1987
[40]

V. F. R. Jones. The Potts model and the symmetric group. InSubfactors (Kyuzeso, 1993), pages 259–267. World Sci. Publ., River Edge, NJ, 1994

1993
[41]

V. F. R. Jones, A quotient of the affine Hecke algebra in the Brauer algebra, Enseign. Math. (2) 40 (1994), no. 3-4, 313–344

1994
[42]

L. H. Kauffman. State models and the Jones polynomial.Topology, 26(3):395–407, 1987

1987
[43]

L. H. Kauffman. An invariant of regular isotopy.Trans. Amer. Math. Soc., 318(2):417–471, 1990

1990
[44]

L. H. Kauffman. Knots and diagrams. InLectures at KNOTS ’96 (Tokyo), volume 15 ofSer. Knots Everything, pages 123–194. World Sci. Publ., River Edge, NJ, 1997

1997
[45]

Khovanov, M

M. Khovanov, M. Sitaraman, D. Tubbenhauer, Monoidal categories, representation gap and cryp- tography,Trans. Amer. Math. Soc. Ser. B11, 329-–395, 2024

2024
[46]

K. H. Kim, F. W. Roush, Linear representations of semigroups of Boolean matrices,Proc. Amer. Math. Soc.63, no. 2, 203-–207, 1977

1977
[47]

Kitov, M

N. Kitov, M. V. Volkov, Identities of the Kauffman MonoidK 4 and of the Jones MonoidJ 4. In: Blass, A., C´ egielski, P., Dershowitz, N., Droste, M., Finkbeiner, B. (eds) Fields of Logic and Computation III. Lecture Notes in Computer Science, , vol 12180. Springer
[48]

Kitov, M

N. Kitov, M. V. Volkov, Identities in twisted Brauer monoids. In: Ambily, A.A., Kiran Kumar, V.B. (eds) Semigroups, Algebras and Operator Theory. ICSAOT 2022. Springer Proceedings in Mathematics & Statistics, vol 436. Springer, Singapore (2024)

2022
[49]

Kudryavtseva, V

G. Kudryavtseva, V. Mazorchuk. On presentations of Brauer-type monoids.Cent. Eur. J. Math., 4(3):413–434 (electronic), 2006

2006
[50]

K. W. Lau, D. G. FitzGerald. Ideal structure of the Kauffman and related monoids.Comm. Algebra, 34(7):2617–2629, 2006

2006
[51]

Maclagan, B

D. Maclagan, B. Sturmfels, , Introduction to Tropical Geometry, Graduate Studies in Mathematics Vol. 161, American Mathematical Society, 2015

2015
[52]

Mac Lane.Categories for the working mathematician

S. Mac Lane.Categories for the working mathematician. 2nd edn., Grad. Texts in Math. 5, Springer, New York, 1998

1998
[53]

Maltcev, V

V. Maltcev, V. Mazorchuk. Presentation of the singular part of the Brauer monoid.Math. Bohem., 132(3):297–323, 2007

2007
[54]

Lehrer, R

G. Lehrer, R. Zhang, The Brauer category and invariant theory,J. Eur. Math. Soc. (JEMS)17, no. 9, 2311–2351, 2015

2015
[55]

Margolis, B

S. Margolis, B. Steinberg. On the minimal faithful degree of Rhodes semisimple semigroups.J. Algebra, 633:788–813, 2023

2023
[56]

P. Martin. Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra con- struction.J. Knot Theory Ramifications, 3(1):51–82, 1994

1994
[57]

P. Martin. On diagram categories, representation theory, statistical mechanics. InNoncommutative rings, group rings, diagram algebras and their applications, volume 456 ofContemp. Math., pages 99–136. Amer. Math. Soc., Providence, RI, 2008

2008
[58]

Mazorchuk, B

V. Mazorchuk, B. Steinberg, Effective dimension of finite semigroups,J. Pure Appl. Algebra216, no. 12, 2737–2753, 2012

2012
[59]

McKenzie, B

R. McKenzie, B. Schein, Every semigroup is isomorphic to a transitive semigroup of binary relations, Trans. Amer. Math. Soc. 349 (1997), no. 1, 271–285

1997
[60]

J. D. Mitchell et al.Semigroups- GAP package, Version 5.5.0, Feb 2025,http://dx.doi.org/ 10.5281/zenodo.592893

work page doi:10.5281/zenodo.592893 2025
[61]

T. E. Nordahl, H. E. Scheiblich. Regular∗-semigroups.Semigroup Forum, 16(3):369–377, 1978. 41

1978
[62]

Selinger, A survey of graphical languages for monoidal categories

P. Selinger, A survey of graphical languages for monoidal categories. InNew structures for physics, 289–355. Lecture Notes in Phys., 813, Springer, Heidelberg, 2011

2011
[63]

percolation

H. N. V. Temperley, E. H. Lieb. Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem.Proc. Roy. Soc. London Ser. A, 322(1549):251–280, 1971

1971
[64]

Wilcox, Cellularity of diagram algebras as twisted semigroup algebras,J

S. Wilcox, Cellularity of diagram algebras as twisted semigroup algebras,J. Algebra, 309, 10–31 (2007). Western Sydney University Email address:J.East@WesternSydney.edu.au University of Manchester Email address:Marianne.Johnson@Manchester.ac.uk University of Manchester Email address:Mark.Kambites@Manchester.ac.uk

2007