Generation of diagonal acts of some semigroups of transformations

Generation of diagonal acts of some semigroups of transformations and relationsPrinted in Australia by Pirion Printing Print Post approved - PP229219/00095
ISSN 0004-9727
142
pp. 139–146 : Peter Gallagher and Nik Ruskuc Generation of diagonal acts of some semigroups of transformations and relations.
(MathReviews) (Zentralblatt)
@article {GaRu2005, author="Peter Gallagher and Nik Ru\v{s}kuc", title="{Generation of diagonal acts of some semigroups of transformations and relations}", journal="Bull. Austral. Math. Soc.", fjournal={Bulletin of the Australian Mathematical Society}, volume="72", year="2005", number="1", pages="139--146", issn="0004-9727", coden="ALNBAB", language="English", date="5th April, 2005", classmath="20M20, 20M30", publisher={AMPAI, Australian Mathematical Society}, MRID="MR2162299", ZBLID="02212191", url="http://www.austms.org.au/Publ/Bulletin/V72P1/721-5106-GaRu/index.shtml", acknowledgement={The authors are grateful to an anonymous referee for his/her suggestions for streamlining the proof of Theorem \hyperref [ixbi]{4.3}.}, abstract={ The \emph {diagonal right} (respectively, \emph {left}) \emph {act} of a semigroup $S$ is the set $S \times S$ on which $S$ acts via $(x,y)s=(xs,ys)$ (respectively, $s(x,y)=(sx,sy)$); the same set with both actions is the \emph {diagonal bi-act}. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set $A \subseteq S \times S$ such that $S \times S=AS^1$ (respectively, $S \times S=S^1A$, $S \times S=S^1AS^1$). \par \par In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semigroups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts. } }
Bull. Austral. Math. Soc., Vol 72, No 1
Metadata in BibTeX format
Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ Generation of diagonal acts of some semigroups of transformations and relations Peter Gallagher Nik Ruškuc 7 June 2008 2008 6 7 139 146 721-5106-GaRu-2005 10.wxyz/C2005V72P1p139 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5106-GaRu/ The \emph {diagonal right} (respectively, \emph {left}) \emph {act} of a semigroup $S$ is the set $S \times S$ on which $S$ acts via $(x,y)s=(xs,ys)$ (respectively, $s(x,y)=(sx,sy)$); the same set with both actions is the \emph {diagonal bi-act}. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set $A \subseteq S \times S$ such that $S \times S=AS^1$ (respectively, $S \times S=S^1A$, $S \times S=S^1AS^1$). \par \par In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semigroups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts. 20M20, 20M30 MR2162299 02212191 The authors are grateful to an anonymous referee for his/her suggestions for streamlining the proof of Theorem 4.3. S. Bulman-Fleming and K. McDowell Problem e3311 Amer. Math. Monthly MR1541456 S. Bulman-Fleming and K. McDowell; Problem e3311, \textit{Amer. Math. Monthly} \textbf{96} (1989). P. Gallagher On the finite and non-finite generation of diagonal acts Comm. Algebra MR2252661 P. Gallagher; On the finite and non-finite generation of diagonal acts, \textit{Comm. Algebra} \textbf{34\year 2006}, pp.~3123--3137. S. Lipscomb Symmetric inverse semigroups American Mathematical Society MR1413301 S. Lipscomb; \textit{Symmetric inverse semigroups} (American Mathematical Society, Providence R.I., 1996). E.F. Robertson, N. Ru{š}kuc and M.R. Thomson On diagonal acts of monoids Bull. Austral. Math. Soc. MR1812320 E.F. Robertson, N. Ru{\v{s}}kuc and M.R. Thomson; On diagonal acts of monoids, \textit{Bull. Austral. Math. Soc.} \textbf{63} (2001), pp.~167--175. E.F. Robertson, N. Ru{š}kuc and M.R. Thomson On finite generation and other finiteness conditions of wreath products of semigroups Comm. Algebra MR1922315 E.F. Robertson, N. Ru{\v{s}}kuc and M.R. Thomson; On finite generation and other finiteness conditions of wreath products of semigroups, \textit{Comm. Algebra} \textbf{30} (2002), pp.~3851--3873. M.R. Thomson Finiteness conditions of wreath products of semigroups and related properties of diagonal acts (Ph.D. Thesis) University of St Andrews M.R. Thomson; \textit{Finiteness conditions of wreath products of semigroups and related properties of diagonal acts}, (Ph.D. Thesis) (University of St Andrews, St. Andrews, Scotland, 2001).
Metadata in XML formal
Peter Gallagher and Nik Ruškuc, 7 June 2008, Bull. Austral. Math. Soc., vol. 72, pp.139--146 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5106-GaRu/
HTML for RSS feed.
President: M.G. Cowling Department of Pure Mathematics, The University of New South Wales, Sydney NSW 2052, Australia.
Secretary: E.J. Billington Department of Mathematics, The University of Queensland, Queensland 4072, Australia.
Treasurer: A. Howe Department of Mathematics, The Australian National University, Canberra ACT 0200, Australia
Membership and correspondence. Applications for membership, notices of change of address or title or position, members’ subscriptions and correspondence related to accounts should be sent to the Treasurer. Correspondence about the distribution of the Society’s BULLETIN, GAZETTE, and JOURNALS, and orders for back numbers should be sent to the Treasurer. All other correspondence should be sent to the Secretary.
The Bulletin. The Bulletin of the Australian Mathematical Society began publication in 1969. Normally two volumes of three numbers are published annually. The BULLETIN is published for the Australian Mathematical Society by the Australian Mathematical Publishing Association Inc.
Editor: Alan S. Jones Department of Mathematics, Deputy Editor: Graeme A. Chandler The University of Queensland,
Queensland 4072, Australia.
ASSOCIATE EDITORS
Robert S. Anderssen B.D. Craven B.D. Jones M. Murray R. Bartnik Brian A. Davey Owen D. Jones J.H. Rubinstein Elizabeth J. Billington J.R. Giles G.I. Lehrer Jamie Simpson G. Cairns J.A. Hempel K.L. McAvaney Brailey Sims J. Clark B.D. Hughes A.G.R. McIntosh Ross Street G.L. Cohen G. Ivanov Terry Mills R.P. Sullivan John Cossey B.R.F. Jefferies Sidney A. Morris H.B. Thompson
N.S. Trudinger A.J. van der Poorten
c©Copyright Statement Where necessary, permission to photocopy for internal or personal use or the internal or personal use of specific clients is granted by the Treasurer, Australian Mathematical Publishing Association, Inc., for libraries and other users regis- tered with the Copyright Clearance Center (CCC), provided that the base fee of $A2.00 per copy of article is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed to the Treasurer, Australian Mathematical Publishing Association, Inc., School of Mathematical Sciences, ANU, Canberra ACT 0200 Australia. Serial–fee code: 0004-9727/05 $A2.00 + 0.00.
INFORMATION FOR AUTHORS
The Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. The Editors receive more than three times as much material as can be published in the BULLETIN; many meritorious papers can, therefore, not be accepted. Authors are asked to avoid, as far as possible the use of mathematical symbols in the title. Manuscripts are accepted for review with the understanding that the same work is not concurrently submitted elsewhere.
To ensure speedy publication, editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. Papers are accepted only after a careful evaluation by the Editor and an Associate Editor or other expert in the field. As even minor revisions are generally not permitted, authors should read carefully all the details listed below. For a paper to be acceptable for publication, not only should it contain new and interesting results but also
(i) the exposition should be clear and attractive; (ii) the manuscript should be in publishable form, without revision. Authors should submit three clean, high quality copies to
The Editorial Office, Bulletin of the Australian Mathematical Society, Department of Mathematics, The University of Queensland, Queensland 4072, Australia.
Unless requested at the time, material submitted to the BULLETIN will usually not be returned.
EDITORIAL POLICY
1. References. Arrange references alphabetically (by surname of the first author) and cite them numerically in the text. Ensure the accuracy of the references: authors’ names should appear as in the work quoted. Include in the list of references only those works cited, and avoid citing works which are “in preparation” or “submitted”. Where the work cited is not readily accessible (for example, a preprint) a photocopy of the title page and relevant sections of the copy that you have used should be included with your submission.
2. Abstracts.
1. Each paper must include an abstract of not more than 200 words, which should contain a brief but informative summary of the contents of the paper, but no inessential details.
2. The abstract should be self-contained, but may refer to the title. 3. Specific references (by number) to a section, proposition, equation or biblio-
graphical item should be avoided.
3. Subject Classification. Authors should include in their papers one or more classification numbers, following the 2000 Mathematics Subject Classification. De- tails of this scheme can be found in each Annual Index of Mathematical Reviews or on the web at http://www.ams.org/msc .
4. Abstracts of Ph.D. Theses. The Bulletin endeavours to publish abstracts of all accepted Australasian Ph.D. theses in mathematics. One restriction, however, is that the abstract must be received by the Editor within 6 months of the degree being approved.
5. Electronic Manuscripts. The Bulletin is produced using AMS-TEX. Authors who are able to do so are invited to prepare their manuscripts using TEX. (We accept Plain TEX, AMS-TEX or LaTEX.) Hard copy only should be submitted for assessment, but if the paper is accepted the author will be asked to send the text on an IBM PC compatible diskette or via e-mail to [email protected]. [Typed manuscripts are, of course, still acceptable.]
Bulletin of the Australian Mathematical Society
A note on the lattice of density preserving maps Sejal Shah and T.K. Das . . . . . . . . . . . . . . . . . . 1
A strong excision theorem for generalised Tate cohomology N. Mramor Kosta . . . . . . . . . . . . . . . . . . . . 7
Linear geometries on the Moebius strip: a theorem of Skornyakov type Rainer Lowen and Burkhard Polster . . . . . . . . . . . . . . 17
Div-curl type theorems on Lipschitz domains Zengjian Lou . . . . . . . . . . . . . . . . . . . . . . 31
A nonlinear map for midpoint locally uniformly rotund renorming S. Lajara and A.J. Pallares . . . . . . . . . . . . . . . . . . 39
A remarkable continued fraction David Angell and Michael D. Hirschhorn . . . . . . . . . . . . . . 45
A new variational method for the ()-Laplacian equation Marek Galewski . . . . . . . . . . . . . . . . . . . . 53
Boundary unique continuation theorems under zero Neumann boundary conditions Xiangxing Tao and Songyan Zhang . . . . . . . . . . . . . . 67
On the Ky Fan inequality and related inequalities II Edward Neuman and Josef Sandor . . . . . . . . . . . . . . 87
Finite presentability of some metabelian Hopf algebras Dessislava H. Kochloukova . . . . . . . . . . . . . . . . . . 109
On the monotonicity properties of additive representation functions Yong-Gao Chen, Andras Sarkozy, Vera T. Sos and MinTang . . . . . . . . 129
Generation of diagonal acts of some semigroups of transformations and relations Peter Gallagher and Nik Ruskuc . . . . . . . . . . . . . . . . 139
Subalgebras of free restricted Lie algebras R.M. Bryant, L.G. Kovacs and Ralph Stohr . . . . . . . . . . . . 147
A multiple character sum evaluation Dae San Kim . . . . . . . . . . . . . . . . . . . . . . 157
Implicit vector equilibrium problems via nonlinear scalarisation Jun Li and Nan-Jing Huang . . . . . . . . . . . . . . . . 161
ABSTRACTS OF AUSTRALASIAN Ph.D. THESES
Numerical methods for quantitative finance Jamie Alcock . . . . . . . . . . . . . . . . . . . . . . 173
Volume 72 Number 1 August 2005
Vol. 72 (2005) [139–146]
GENERATION OF DIAGONAL ACTS OF SOME SEMIGROUPS OF
TRANSFORMATIONS AND RELATIONS
Peter Gallagher and Nik Ruskuc
The diagonal right (respectively, left) act of a semigroup is the set × on which acts via (, ) = (, ) (respectively, (, ) = (, )); the same set with both actions is the diagonal bi-act. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set ⊆ × such that × = 1
(respectively, × = 1, × = 11). In this paper we consider the question of finite generation for diagonal acts of
certain infinite semigroups of transformations and relations. We show that the semigroups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite- to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts.
1. Introduction
If is a semigroup and is a set then we say that acts on ( is a right -act)
via : × → ( : (, ) → ∈ ) if (1)2 = (12) for all ∈ , 1, 2 ∈ .
Left acts are defined analogously. We say that is a bi-act if acts on it both on the
left and on the right and () = () for all ∈ , 1, 2 ∈ .
If is a right (respectively, left, bi) -act then ⊆ is a generating set for this
act if 1 = (respectively, 1 = , 11 = ). If can be chosen to be finite
then is said to be finitely generated. If can be chosen to be a singleton then we say
that is cyclic.
The diagonal right act of is the set = × with acting via (, ) = (, ).
The diagonal left act is defined analogously, via (, ) = (, ). The diagonal bi-act
is the same set × on which acts from both left and right.
Received 5th April, 2005 The authors are grateful to an anonymous referee for his/her suggestions for streamlining the proof of Theorem 4.3.
Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/05 $A2.00+0.00.
139
Semigroup Right act Left act Bi-act
cyclic (Thm 2.1) cyclic (Thm 3.1) cyclic cyclic (Cor 2.2) cyclic (Cor 3.3) cyclic cyclic (Cor 2.3) cyclic (Thm 3.2) cyclic cyclic (Cor 2.4) infinite (Thm 3.4) cyclic infinite (Thm 2.5) infinite (Thm 3.5) cyclic (Thm 4.3) infinite infinite infinite (Thm 5.1)
Table 1: Summary of results.
Diagonal acts were first mentioned, implicitly, in [1]. It was then known that the
diagonal right act of , the full transformation semigroup on the natural numbers,
is cyclic. The notion of diagonal acts was then considered by Robertson, Ruskuc and
Thomson [5] in relation to wreath products. Necessary and sufficient conditions for the
finite generation and presentability of a restricted wreath product were proved, and these
referred to the finite generation of diagonal acts. In [4] the same authors considered
diagonal acts in their own right and made some interesting connections with finitary
power semigroups. It was shown that the diagonal left act of is cyclic, and that
both (the monoid of partially recursive functions of one variable) and a particular
homomorphic pre-image of have both cyclic diagonal left and right acts. We now
search for more semigroups with these properties.
In [2] several statements were proved regarding the classes of infinite semigroups
which admit finitely generated diagonal acts. The diagonal bi-act of a non-trivial semi-
group is not cyclic if the semigroup is either finite or cancellative. The diagonal bi-act
of an infinite semigroup is not finitely generated if it is in any of the following classes:
commutative; idempotent; or Bruck–Reilly extensions. Neither the diagonal right nor
left act of an infinite semigroup is finitely generated if it is in any of the following classes:
left cancellative; right cancellative; inverse; completely regular; completely zero-simple;
or locally finite. Clearly there are many families of transformation semigroups that are
not in any of these classes, so this is a good place to search for more examples of infinite
semigroups with finitely generated diagonal acts.
Throughout the paper we let be an infinite set. We consider the question of finite
generation of the diagonal right, left and bi-acts of the followings semigroups: of
binary relations; of partial transformations; of full transformations; of full
finite-to-one transformations (that is, no infinite subset is mapped to a single point);
of partial injective transformations (also called partial bijections); and of bijections
on . Our findings are summarised in Table 1.
[3] Diagonal acts of semigroups 141
2. Diagonal Right Acts
Theorem 2.1. The diagonal right act of , the semigroup of binary relations
on an infinite set , is cyclic.
Proof: Let 1 and 2 be disjoint subsets of with = 1 ∪ 2 and || = |1| = |2|. We fix bijections : → 1 and : → 2, consider them as
binary relations
} .
Select arbitrary , ∈ and construct ∈ as
= {(
} .
Then (, ) = (, ) and hence the diagonal right act of is cyclic.
We note that if and are both partial (respectively, full, full finite-to-one) trans-
formations then our construction yields which is also a partial (respectively, full, full
finite-to-one) transformation. Hence we have the following results.
Corollary 2.2. The diagonal right act of , the semigroup of partial trans-
formations on an infinite set , is cyclic.
Corollary 2.3. The diagonal right act of , the semigroup of full transfor-
mations on an infinite set , is cyclic.
(This is an extension of the known facts for .)
Corollary 2.4. The diagonal right act of , the semigroup of full finite-to-
one transformations on an infinite set , is cyclic.
However, it is not possible to apply this reasoning to extend this result to partial (or
full) injective mappings. [2, Theorem 7.4] states that no infinite inverse semigroup has a
finitely generated diagonal right act. Therefore we have the following result.
Theorem 2.5. The diagonal right act of , the semigroup of partial injective
transformations on an infinite set , is not finitely generated.
We also know, from Theorem 5.1 below, that the diagonal right act of , the
semigroup of bijections on an infinite set , is not finitely generated.
3. Diagonal Left Acts
In an analogous manner to the proof of Theorem 2.1 (reversing each pair in each
element of ) the next theorem follows.
142 P. Gallagher and N. Ruskuc [4]
Theorem 3.1. The diagonal left act of , the semigroup of binary relations
This time, however, we cannot use this proof to extend this result to , or ;
reversing a transformation does not yield a transformation. We turn to , and use a
different method of proof. We note that this is another extension of the known facts for
.
Theorem 3.2. The diagonal left act of , the semigroup of full transformations
Proof: We fix a bijection : → ×, let be its inverse and define 1, 2 :
× → as projections onto the first and second co-ordinates, respectively. We let
, ∈ be defined as = 1 and = 2, and claim that × =
{ (, )
} . For
, ∈ , we define ∈ as () = ( (), ()
) for all ∈ . Then (, ) = (, ),
so the diagonal left act of is cyclic.
We now turn to . As is customary we shall write () = − when is not defined
on ; that is, when /∈ Dom(). Thus we may consider as the subsemigroup of
∪{−} which consists of all the transformations with (−) = −. If we apply the proof
of Theorem 3.2 to ∪{−} ensuring that (−) = (−,−), then it is clear that , ∈ ,
and that if , ∈ then ∈ . Thus we have the following result.
Corollary 3.3. The diagonal left act of , the semigroup of partial trans-
formations on an infinite set , is cyclic.
By way of contrast, we have the following statement.
Theorem 3.4. The diagonal left act of , the semigroup of full finite-to-one
Proof: Assume that the diagonal left act of is finitely generated, so there
is a finite ⊆ × such that × = . For all , ∈ there are
∈ , (, ) ∈ such that (, ) = (, ). It is clear that{( (), ()
) : ∈ , , ∈
} = ×,
} = ×.
{} × ⊆ {(
(), () )
: ∈ , (, ) ∈ } .
As is finite, there is a particular (, ) ∈ and infinite ⊆ such that
{} × ⊆ {(
(), () )
: ∈ } ,
so ∈ has infinitely many pre-images under , and hence /∈ , a contradiction.
Thus is the only one of our semigroups for which the diagonal right and left
acts behave differently. Other known examples of semigroups with finitely generated
diagonal right but not left acts are defined by using certain semigroup constructions, in
particular, ‘constant extensions’ (discussed in [4]) and direct products with finite left zero
semigroups (discussed in [2]).
The dual of [2, Theorem 7.4] states that no infinite inverse semigroup has a finitely
generated diagonal left act. Therefore we have the following result.
Theorem 3.5. The diagonal left act of , the semigroup of partial injective
We also know, from Theorem 5.1 below, that the diagonal left act of , the semi-
group of bijections on an infinite set , is not finitely generated.
4. Diagonal Bi-Acts
As the diagonal right acts of , , and ( infinite) are cyclic, it follows
that the diagonal bi-acts of these semigroups are also cyclic. We now consider , the
semigroup of partial injective transformations on an infinite set , and show that, in
contrast to its diagonal right and left acts, its diagonal bi-act is cyclic.
This semigroup is important in inverse semigroup theory; just as all groups are
subgroups of for some set , all inverse semigroups are subsemigroups of some .
In [3], the notion of cycle representations of permutations is extended to that of path
representations of partial injective transformations. We shall use this notion, in an infinite
context, in the proof of our result.
For ∈ we let Γ be a digraph with vertex set and edges specified by
→ ⇔ () = .
Let Ω be the set of all digraphs on in which every vertex has in-degree either 0 or 1
and out-degree either 0 or 1. Then there is a natural bijection : → Ω , defined as
() = Γ.
For graphs Λ1 and Λ2, a graph isomorphism : Λ1 → Λ2 is a bijection between the
vertex sets such that
→ in Λ1 ⇔ () → () in Λ2.
If there exists a graph isomorphism from Λ1 to Λ2 then they are isomorphic, which is
denoted Λ1 ∼= Λ2.
We say that , ∈ are conjugate if there exists ∈ (the group of bijections
on ) such that = −1. In [3] it is shown, for a finite set , that , ∈ are
conjugate if and only if they have the same path structure. We state and prove this in
terms of digraphs, and where may be infinite or finite.
Lemma 4.1. Elements , ∈ are conjugate if and only if Γ and Γ are
isomorphic.
Proof: (⇒) If there is ∈ such that = −1, then
→ in Γ ⇔ () = ⇔ [ ()−1
] −1 = ()−1
] = ()−1 ⇔ ()−1 → ()−1 in Γ,
so −1 : Γ → Γ is a graph isomorphism.
(⇐) A graph isomorphism : Γ → Γ is a bijection on , so may be considered as
some ∈ . Also,
() = ⇔ → in Γ ⇔ () → () in Γ
⇔ [ ()
so = −1 and they are conjugate.
The components of a digraph are the connected components of its underlying undi-
rected graph. Up to isomorphism, the only components which may appear in Λ ∈ Ω
are:
1. finite paths of size (for all ∈ ) 1 → 2 → · · · → ;
2. finite cycles of size (for all ∈ ) 1 → 2 → · · · → → 1;
3. left infinite paths · · · → −2 → −1 → 0;
4. right infinite paths 1 → 2 → 3 → · · · ; 5. bi-infinite paths · · · → −1 → 0 → 1 → · · · .
In particular, there are only countably many isomorphism classes for components,
so these can be indexed by . To be more precise, we fix a family { : ∈ } where
is a component in some digraph Λ ∈ Ω and such that for any component in any
Λ ∈ Ω there is a unique ∈ such that ∼= . In this case we refer to as the
isomorphism class of .
The restriction of ∈ to ⊆ is denoted and defined as () = () if
, () ∈ and () = − otherwise. It is clear that ∈ .
Lemma 4.2. There exists ∈ such that for all ∈ there exists ∈
such that = −1.
Proof: We let be an index set with || = || and partition as the disjoint
union
=
,
where each |,| = ||. We let Υ ∈ Ω be arbitrary such that each , is the vertex
set of a component of isomorphism class and let ∈ be such that Γ = Υ.
Let ∈ be arbitrary and let ( ∈ ) be the components of Γ. As every graph
in Ω has at most || components of each isomorphism class, we may assume without
loss of generality that ⊆ . For each we consider ,, where is the isomorphism
class of . Clearly ∼= ,, so the restrictions
and , are conjugate, by Lemma
4.1. That is, there exists ∈ such that = ,
−1 . We define : →
as () = () when ∈ . Then is a well-defined full injective (but not necessarily
surjective) transformation and = −1.
We are now ready to prove the main result.
Theorem 4.3. The diagonal bi-act of , the semigroup of partial injective
transformations on an infinite set , is cyclic.
Proof: We claim that × =
{ (, 1)
} where is as in Lemma 4.2. Let
, ∈ be arbitrary. By Lemma 4.2 there exists ∈ such that −1 = −1, and
we note that
() = () ⇔ ()−1 = ⇔ ()−1 = ⇔ () = ().(1)
Let : → be defined on Im()∪ Im() such that [ ()
] = () and
] = ().
Suppose that a contradiction is implicit here, in that is required to map one point
to two different image points. This could only be the case if () = () but () = ()
for some , ∈ , which by (1), cannot happen.
Suppose that is not injective. This is only the case if () = () but () = ()
for some , ∈ . Again, by (1), this does not happen.
So ∈ and (, ) = (, ) = (, 1), and the theorem is proved.
5. Further Remarks and Conclusion
In the course of this paper we considered many families of transformation semigroups.
For , the symmetric group on an infinite set , we use the following result, which
appears as [6, Proposition 6.7] and which we reprove here for completeness.
Theorem 5.1. If is a group then the diagonal bi-act of is finitely generated
if and only if has only finitely many conjugacy classes.
Proof: (⇒) If the diagonal bi-act of is finitely generated then there is a finite
⊆ such that × = ( × ). For all ∈ there are 1, 2 ∈ , 1, 2 ∈
such that (, 1) = 1(1, 2)2. Then = 1−1 = −1
2 −1 2 −1
1 112 = −1 2 (−1
2 1)2 is
conjugate to −1 2 1, so has only finitely many conjugacy classes.
(⇐) If has only finitely many conjugacy classes, then let be a set of conjugacy
class representatives in . For arbitrary , ∈ there are ∈ , ∈ such that
−1 = −1, so
(, ) = −1(, 1) ∈ ( × ),
and we see that × finitely generates the diagonal bi-act of .
It is now clear that the diagonal bi-act of is not finitely generated. Without
proof, we also remark that none of the following semigroups: injections; non-injections;
surjections; non-surjections; or partial surjections on an infinite set have finitely generated
diagonal bi-acts.
There are many open problems concerning which infinite semigroups have finitely
generated diagonal acts. We know that the semigroup of all × matrices over an
infinite field has no finitely generated diagonal acts, as (under multiplication) is
a homomorphic image of and would inherit these properties; it is clear from [2] that
this is not the case. We ask if the semigroup of all × matrices over an infinite
ring can have any finitely generated diagonal acts? Again, (under multiplication)
is a homomorphic image of so we ask: does there exist an infinite ring whose
multiplicative semigroup * has a finitely generated diagonal right, left or bi-act?
References
M [1] S. Bulman-Fleming and K. McDowell, ‘Problem e3311’, Amer. Math. Monthly 96 (1989).
M [2] P. Gallagher, ‘On the finite and non-finite generation of diagonal acts’, Comm. Algebra
34, 3123–3137.
M [3] S. Lipscomb, Symmetric inverse semigroups (American Mathematical Society, Providence
R.I., 1996).
M [4] E.F. Robertson, N. Ruskuc and M.R. Thomson, ‘On diagonal acts of monoids’, Bull.
Austral. Math. Soc. 63 (2001), 167–175.
M [5] E.F. Robertson, N. Ruskuc and M.R. Thomson, ‘On finite generation and other finiteness
conditions of wreath products of semigroups’, Comm. Algebra 30 (2002), 3851–3873.
[6] M.R. Thomson, Finiteness conditions of wreath products of semigroups and related properties of diagonal acts, (Ph.D. Thesis) (University of St Andrews, St. Andrews, Scotland, 2001).
School of Mathematics and Statistics University of St. Andrews North Haugh St. Andrews Scotland KY16 9SS
Index of Volume 72 No. 1
Generation of diagonal acts of some semigroups of transformations and relations
1. Introduction
2. Diagonal Right Acts
Theorem 5.1.

Documents

Generation of diagonal acts of some semigroups of transformations