Sierpi´nski proved that every countable set of mappings on an infinite set X is
contained in a 2-generated subsemigroup of the semigroup of all mappings on X. In this
paper we prove that every countable set of endomorphisms of an algebra A which has an
infinite basis (independent generating set) is contained in a 2-generated subsemigroup of
the semigroup of all endomorphisms of A

Araújo, João

Mitchell, James D.

Silva, Nuno

Algebra Universalis

Repositório Aberto da Universidade Aberta

On embedding countable sets of endomorphismsJ. Arau´jo, J.D. Mitchell and N. SilvaAbstract. Sierpin´ski proved that every countable set of mappings on an infinite set X iscontained in a 2-generated subsemigroup of the semigroup of all mappings on X. In thispaper we prove that every countable set of endomorphisms of an algebra A which has aninfinite basis (independent generating set) is contained in a 2-generated subsemigroup ofthe semigroup of all endomorphisms of A.1. IntroductionLet X be an infinite set. In [20] Sierpin´ski proved the following result:Proposition 1.1. Any countable subset of the semigroup TX of all transformationson X is contained in a 2-generated subsemigroup of TX .A short proof of this result was immediately given by Banach in [3] (see also [13]).In [13] it was shown that the method of Banach can be modified slightly to proveanalogous results for countable subsets of the semigroup of all partial bijectionsand the semigroup of partial mappings on an infinite set X. As may be expectedthe situation is more complicated for bijections of an infinite set X. However, in[9, Theorem 3.3] it was proven that every countable set of permutations of X iscontained in a two-generated subgroup of the symmetric group SX .There are several important corollaries of these results, including the result thatevery countable semigroup can be embedded in a two-generated semigroup. Thisresult was proven, independently of Sierpin´ski’s result, by Evans in [6]. Severalother proofs of this result appeared, including [19] and [22].The full transformation semigroup can be considered as the semigroup of endo-morphisms of the (unstructured) set X. It is natural to ask if a result, analogousto Proposition 1.1, holds when X is endowed with some structure. It was shownin [15] that every countable set of endomorphisms of an infinite dimensional vectorspace V, over an arbitrary field, is contained in a two-generated subsemigroup ofthe semigroup of all endomorphisms of V. This result was motivated by resultsin [21] relating to the semigroup of continuous self-maps S(X) of a topologicalspace X. In [21] it was shown that any countable subset of S(X) is contained in atwo-generated subsemigroup of S(X) when X is the rationals, the irrationals, thecountable discrete space, an m-dimensional closed unit cube, or the Cantor set.1991 Mathematics Subject Classification: 20M20, 20M10, 08A35.Key words and phrases: Endomorphisms, universal algebras, independence, semigroups.12 J. ARAU´JO, J.D. MITCHELL AND N. SILVAThe main aim of this paper is to prove that a result analogous to Proposition 1.1holds for the endomorphism semigroups of a large class of algebras.For the remainder of this section we introduce the main notions and terminologywe shall use in this paper. Let A be a non-empty set. For n 2 N, an n-aryoperation on A is a mapping ↵ : An  ! A, where An denotes the Cartesianproduct A ⇥ A ⇥ · · · ⇥ A with n terms. As usual a nullary operation on A is amapping a : ;  ! A. These operations may be referred to as finitary operations. Amapping from AI to A, where I is an infinite set, is called an infinitary operation.Throughout this paper we shall follow the convention of writing operations on theleft and mappings on the right. A (universal) algebra with universe A is a pairA = (A,⌦) where ⌦ is a set of (finitary or infinitary) operations on A. For a subsetB ✓ A we shall denote by hB i the subalgebra generated by B.Let A and B be algebras. Then a morphism from A to B is any mapping ↵ fromA to B that preserves the operations of A. Note that such a mapping can onlyreasonably exist if A and B have the ‘same’ operations, in some sense. However,in this paper we shall only consider morphisms from an algebra to itself and sothis concern does not a↵ect us. A morphism which is injective and surjective iscalled an isomorphism. An endomorphism of A is a morphism from A to A. Sincecomposition of endomorphisms is associative the set of all endomorphisms of A isa semigroup, which we denote by End(A).We now introduce the notion that is central to the results contained in this paper.Let A be an algebra with universe A and let X ✓ A.Definition 1.2. We say that X is independent if every mapping ↵ : X  ! A canbe extended to a morphism ✏↵ : h X i  ! A such that the restriction of ✏↵ to Xequals ↵✏↵|X = ↵.For an arbitrary mapping ↵ from an independent set I to A we shall denote by✏↵ the extension of ↵ to an endomorphism from h I i to A.There are many definitions of independence in di↵erent mathematical structures.Here we follow the definition of Marczewski in [16] and [17]. We say that B is abasis for A if B is an independent generating set for A. For ↵,  2 End(A) it isclear that↵ =   if and only if ↵|B =  |B .There are many well-known algebraic structures, such as vector spaces, whichhave bases. There are also many algebraic structures without bases. We givean example of such a structure. Let p be a prime and let Zp denote the cyclicgroup of order p. Inductively, let Zpi be the cyclic group of order pi, where Zpi 1is a subgroup of Zpi . We denote the countable union of this ascending chain ofsubgroupsS1i=1 Zpi by Zp1 . This group can also be described as the set of all thepnth roots of unity with the usual complex multiplication. It is easy to verify thatthis group is not finitely generated and that the maximum size of an independentset is one.For more information about semigroups or algebras see [4], [5], [12] or [14].ON EMBEDDING COUNTABLE SETS OF ENDOMORPHISMS 32. The main resultThroughout this section let A denote an algebra with infinite basis B. In or-der to prove our main result we partition B into countably many disjoint setsB0, B1, B2, . . . so that each Bi has cardinality equal to that of B. Similarly, wepartition B0 into countably many disjoint sets B0,1, B0,2, B0,3, . . . so that each B0,ihas cardinality equal to that of B.Remark 2.1. Let ↵ 2 End(A) be an arbitrary endomorphism, let i 2 N and let  be any bijection from B into B0,i. Then the mapping  i from B0,i to A defined byb i = b  1↵|B (b 2 B0,i)satisfies   i = ↵|B , and hence ✏ ✏ i = ↵.We now prove our main result.Theorem 2.2. Let A be an algebra which has an infinite basis. Then any countablesubset of End(A) is contained in a 2-generated subsemigroup of End(A).Proof. Let {↵1,↵2,↵3, . . . } be our countable subset of elements of End(A). Wefind two elements of End(A) which generate a semigroup that contains this set.Let   be any mapping from B to B\B0 which maps Bi to Bi+1 bijectively for eachi 2 {0, 1, 2, . . . }. Since B is independent   can be extended to an endomorphism✏  of A.We define the second of our mappings   from B to A in two stages. First, wedefine   on B \ B0 to be any mapping which takes Bi to B0,i bijectively, for eachi 2 {1, 2, 3, . . . }. We observe that  i =    i  is a (well-defined) bijection from Bto B0,i.By Remark 2.1 it follows that for each i 2 N there exists a mapping  i such that i i = ↵i|B . We now complete the definition of   so that for b 2 B0,i we haveb  = b i. Hence for each i 2 N we have↵i|B =  i i =  i  =    i 2.It follows that ↵i = ✏ ✏ ✏i ✏2  , and so {↵1,↵2,↵3, . . . } ✓ h ✏  , ✏  i. ⇤For the remainder of this section we shall give examples of algebras which satisfythe condition of our theorem, i.e. algebras which have infinite bases.Let A be a class of algebras. Then every free A-algebra A on an infinite setX has an infinite basis, namely X. That X generates A is evident and that X isindependent follows immediately from the definition of a free algebra.Corollary 2.3. Let A be a non-finitely generated free A-algebra. Then every count-able subset of End(A) is contained in a 2-generated subsemigroup of End(A).Examples of such algebras are the non-finitely generated free algebras of any variety.Another example of a type of algebra which satisfies our condition is providedby a class of algebras which includes finite and infinite dimensional vector spaces asspecial cases. In order to define this class we require a slightly di↵erent definitionof independence. Let A be an algebra with universe A and let X ✓ A. Then we4 J. ARAU´JO, J.D. MITCHELL AND N. SILVAsay X is T -independent if x 62 hX \ {x} i for each x 2 X. An algebra A is calledan independence algebra if it satisfies the following properties:(i) for every subset X of A and every element u of A, if X is T -independentand u 62 hX i, then X [ {u} is T -independent;(ii) for any basis X of A and any function ↵ from X to A, there is an endo-morphism ✏↵ of A such that ✏↵|X = ↵.The notions of T -independence and independence are not equivalent in general, butin the case of independence algebras they coincide. Independence algebras were firstintroduced in [18] and the basic structure of their endomorphism semigroups wasdescribed in [11] . This study, for both finite and infinite independence algebras,was continued in [1], [2], [7], [8] and [10]. Examples of independence algebras are(unstructured) sets, vector spaces and for any group G, free G-sets. It is clearthat any independence algebra A has a basis, the cardinality of which is called thedimension of A.Corollary 2.4. Let A be an infinite dimensional independence algebra. Then anycountable subset of End(A) is contained in a 2-generated subsemigroup of End(A).Since every vector space is an independence algebra this corollary provides ashorter proof of Theorem 3.1 in [15].3. Concluding remarksWe conclude the paper by giving an example of a finitely generated free algebraA for which it is not true that every countable subset of End(A) is contained ina two-generated subsemigroup of End(A), thereby demonstrating that we cannotomit the condition that A is non-finitely generated from Corollary 2.3.Let X be a non-empty set. Recall that an element ↵ of the semigroup TX , of allmappings from X to X, is called a proper idempotent if ↵2 = ↵ and ↵ 6= 1X , theidentity map on X. For a subsemigroup T of TX and a set Y ✓ X we say that Xis T -isomorphic to Y if there exist mappings  ,   2 T such that X  ✓ Y , Y   ✓ X,   = 1X and   |Y = 1Y , where 1X and 1Y denote the identity maps on X and Yrespectively.We require the following result:Lemma 3.1. Let X be an infinite set and let T be a subsemigroup of TX whichsatisfies the following:(i) the identity 1X is in T ;(ii) there exists ↵ 2 T such that ↵ is not injective;(iii) there exists   2 T such that   is not surjective but is injective;If every countable subset of T can be embedded in a 2-generated subsemigroup of Tthen X is T -isomorphic to the image of a proper idempotent in T .For a proof see [15, Theorem 2.4].Let A = {a, b} be an arbitrary two element alphabet. Then we denote by A+the free semigroup on A. That is all finite, non-empty words on A with the productof two words u and v being the juxtaposition uv of u and v.ON EMBEDDING COUNTABLE SETS OF ENDOMORPHISMS 5Proposition 3.2. It is not possible to embed every countable set of endomorphismsof A+ in a 2-generated subsemigroup of End(A+).Proof. It is obvious that End(A+)   TA+ . We verify that End(A+) satisfies theproperties (i) to (iii) of Lemma 3.1. Clearly 1A+ 2 End(A+). Let ↵ be anyendomorphism satisfying a↵ = b and b↵ = b; it is evident that ↵ is not injective.Let   be any endomorphism satisfying a  = ab and b  = ab2; it is clear that   isnot surjective but is injective.It is easy to verify that every proper idempotent endomorphism of A+ has image{a}+ or {b}+. Suppose that every countable subset of End(A+) can be embeddedin a 2-generated subsemigroup of End(A+). Then, by Lemma 3.1, there exist  ,   2End(A+) such that A+  ✓ {a}+, {a}+  ✓ A+,    = 1A+ and   |{a}+ = 1{a}+ .But it is clear that no subset of {a}+ has A+ as a homomorphic image, and hencewe have a contradiction. ⇤The analogue of Sierpin´ski’s result for subsets of the symmetric group was shownto be true in [9]. As we went from mappings of unstructured sets to endomorphismsof algebras, we ask if we can move from permutations of sets to automorphisms ofalgebras.Open Problem 3.3. Find algebras A for which every countable subset of the groupof automorphisms Aut(A) of A is contained in a 2-generated subgroup of Aut(A).Acknowledgements: The first author would like to acknowledge the support of‘Financiamento Plurianual do CAUL’ and Fundac¸a˜o para a Cieˆncia e a Tecnologia(FCT) and Fundo Europeu de Desenvolvimento Regional (FEDER) project num-ber POCTI/32440/MAT/2000. The second author would like to acknowledge thesupport of FCT and FEDER project ‘Financiamento Programa´tico de CAUL’. Thethird author would like to acknowledge the support of ‘Financiamento Plurianualdo CAUL’, FEDER and FCT grant number SFRH/BD/6081/2001.References[1] Arau´jo, J., Generators for the semigroup of endomorphisms of an independence algebra,Algebra Colloquium 9:4 (2002), 375-382.[2] Arau´jo, J., Normal semigroups of endomorphisms of proper independence algebras areidempotent generated, Proc. Edinburgh Math. Soc. 45 (2002), 205-217.[3] Banach, S., Sur un theore`me de M. Sierpin´ski, Fund. Math. 25 (1935), 5-6.[4] Burris, S. and Sankappanavar, S.P., A course in universal algebra, Graduate Texts inMath. No. 78, Springer-Verlag, 1981.[5] Cohn, P.M., Universal algebra, Mathematics and its applications, Vol. 6, D. ReidelPublishing Co., 1981.[6] Evans, T., Embedding theorems for multiplicative systems and projective geometries, Proc.Amer. Math. Soc. 3 (1952), 614-620.[7] Fountain, J. and Lewin, A., Products of idempotent endomorphisms of an independencealgebra of finite rank, Proc. Edinburgh Math. Soc. 35 (1992), 493-500.[8] Fountain, J. and Lewin, A., Products of idempotent endomorphisms of an independencealgebra of infinite rank, Math. Proc. Cambridge Phil. Soc. 114 (1993), 303-319.[9] Galvin, F., Generating countable sets of permutations, J. London Math. Soc. 51 (1995),230-242.6 J. ARAU´JO, J.D. MITCHELL AND N. SILVA[10] Gomes, G.M.S. and Howie, J.M., Idempotent endomorphisms of an independence algebraof finite rank, Proc. Edinburgh Math. Soc. 38 (1995), 107-116.[11] Gould, V., Independence algebras, Algebra Universalis 33 (1995), 294-318.[12] Higgins, P.M., Techniques of semigroup theory, Oxford Publications, The Clarendon Press,Oxford University Press, New York, 1992.[13] Higgins, P.M., Howie, J.M., Mitchell, J.D. and Rusˇkuc, N., Countable versusuncountable ranks in full transformation semigroup, submitted.[14] Howie, J.M., Fundamentals of semigroup theory, Oxford University Press, 1995.[15] Magill, K.D., The countability index of the endomorphism semigroup of a vector space,Linear and multilinear algebra 22 (1988), 349-360.[16] Marczewski, E., A general scheme of the notions of independence in mathematics,Bulletin de l’academie polonaise des sciences VI (1958), 731-736.[17] Marczewski, E., Independence in algebras of sets and boolean algebras, Fund. Math. 48(1960), 137-145.[18] Narkiewicz, W., Independence in a certain class of abstract algebras, Fund. Math. 50(1961/62), 333-340.[19] Neumann, B.H., Embedding theorems for semigroups, J. London Math. Soc. 35 (1960),184-192.[20] Sierpin´ski, W., Sur les suites infinies de fonctions de´finies dans les ensembles quelconques,Fund. Math. 24 (1935), 209-212.[21] Subbiah, S., Some finitely generated subsemigroups of S(X), Fund. Math. 86 (1975),221-231.[22] Subbiah, S., Another proof of a theorem of Evans, Semigroup Forum 6 (1973), 93-94.J. Arau´jo, J.D. Mitchell, N. Silva,Universidade Aberta, Centro de A´lgebra, Mathematics Institute,R. Escola Polite´cnica, 147 Universidade de Lisboa, University of St Andrews,1269-001 Lisboa, 1619-003 Lisboa, North Haugh,Portugal Portugal St Andrews,& mitchell@cii.fc.ul.pt Fife,Centro de A´lgebra, KY16 9SS,Universidade de Lisboa, UK1619-003 Lisboa, &Portugal Centro de A´lgebra,mjoao@lmc.fc.ul.pt Universidade de Lisboa,1619-003 Lisboa,Portugalnelson@mcs.st-and.ac.uk

On embedding countable sets of endomorphisms

The Repositório Aberto

On embedding countable sets of endomorphisms
J. Arau´jo, J.D. Mitchell and N. Silva
Abstract. Sierpin´ski proved that every countable set of mappings on an infinite set X is
contained in a 2-generated subsemigroup of the semigroup of all mappings on X. In this
paper we prove that every countable set of endomorphisms of an algebra A which has an
infinite basis (independent generating set) is contained in a 2-generated subsemigroup of
the semigroup of all endomorphisms of A.
1. Introduction
Let X be an infinite set. In [20] Sierpin´ski proved the following result:
Proposition 1.1. Any countable subset of the semigroup TX of all transformations
on X is contained in a 2-generated subsemigroup of TX .
A short proof of this result was immediately given by Banach in [3] (see also [13]).
In [13] it was shown that the method of Banach can be modified slightly to prove
analogous results for countable subsets of the semigroup of all partial bijections
and the semigroup of partial mappings on an infinite set X. As may be expected
the situation is more complicated for bijections of an infinite set X. However, in
[9, Theorem 3.3] it was proven that every countable set of permutations of X is
contained in a two-generated subgroup of the symmetric group SX .
There are several important corollaries of these results, including the result that
every countable semigroup can be embedded in a two-generated semigroup. This
result was proven, independently of Sierpin´ski’s result, by Evans in [6]. Several
other proofs of this result appeared, including [19] and [22].
The full transformation semigroup can be considered as the semigroup of endo-
morphisms of the (unstructured) set X. It is natural to ask if a result, analogous
to Proposition 1.1, holds when X is endowed with some structure. It was shown
in [15] that every countable set of endomorphisms of an infinite dimensional vector
space V, over an arbitrary field, is contained in a two-generated subsemigroup of
the semigroup of all endomorphisms of V. This result was motivated by results
in [21] relating to the semigroup of continuous self-maps S(X) of a topological
space X. In [21] it was shown that any countable subset of S(X) is contained in a
two-generated subsemigroup of S(X) when X is the rationals, the irrationals, the
countable discrete space, an m-dimensional closed unit cube, or the Cantor set.
1991 Mathematics Subject Classification: 20M20, 20M10, 08A35.
Key words and phrases: Endomorphisms, universal algebras, independence, semigroups.
1
2 J. ARAU´JO, J.D. MITCHELL AND N. SILVA
The main aim of this paper is to prove that a result analogous to Proposition 1.1
holds for the endomorphism semigroups of a large class of algebras.
For the remainder of this section we introduce the main notions and terminology
we shall use in this paper. Let A be a non-empty set. For n 2 N, an n-ary
operation on A is a mapping ↵ : An  ! A, where An denotes the Cartesian
product A ⇥ A ⇥ · · · ⇥ A with n terms. As usual a nullary operation on A is a
mapping a : ;  ! A. These operations may be referred to as finitary operations. A
mapping from AI to A, where I is an infinite set, is called an infinitary operation.
Throughout this paper we shall follow the convention of writing operations on the
left and mappings on the right. A (universal) algebra with universe A is a pair
A = (A,⌦) where ⌦ is a set of (finitary or infinitary) operations on A. For a subset
B ✓ A we shall denote by hB i the subalgebra generated by B.
Let A and B be algebras. Then a morphism from A to B is any mapping ↵ from
A to B that preserves the operations of A. Note that such a mapping can only
reasonably exist if A and B have the ‘same’ operations, in some sense. However,
in this paper we shall only consider morphisms from an algebra to itself and so
this concern does not a↵ect us. A morphism which is injective and surjective is
called an isomorphism. An endomorphism of A is a morphism from A to A. Since
composition of endomorphisms is associative the set of all endomorphisms of A is
a semigroup, which we denote by End(A).
We now introduce the notion that is central to the results contained in this paper.
Let A be an algebra with universe A and let X ✓ A.
Definition 1.2. We say that X is independent if every mapping ↵ : X  ! A can
be extended to a morphism ✏↵ : h X i  ! A such that the restriction of ✏↵ to X
equals ↵
✏↵|X = ↵.
For an arbitrary mapping ↵ from an independent set I to A we shall denote by
✏↵ the extension of ↵ to an endomorphism from h I i to A.
There are many definitions of independence in di↵erent mathematical structures.
Here we follow the definition of Marczewski in [16] and [17]. We say that B is a
basis for A if B is an independent generating set for A. For ↵,  2 End(A) it is
clear that
↵ =   if and only if ↵|B =  |B .
There are many well-known algebraic structures, such as vector spaces, which
have bases. There are also many algebraic structures without bases. We give
an example of such a structure. Let p be a prime and let Zp denote the cyclic
group of order p. Inductively, let Zpi be the cyclic group of order pi, where Zpi 1
is a subgroup of Zpi . We denote the countable union of this ascending chain of
subgroups
S1
i=1 Zpi by Zp1 . This group can also be described as the set of all the
pnth roots of unity with the usual complex multiplication. It is easy to verify that
this group is not finitely generated and that the maximum size of an independent
set is one.
For more information about semigroups or algebras see [4], [5], [12] or [14].
ON EMBEDDING COUNTABLE SETS OF ENDOMORPHISMS 3
2. The main result
Throughout this section let A denote an algebra with infinite basis B. In or-
der to prove our main result we partition B into countably many disjoint sets
B0, B1, B2, . . . so that each Bi has cardinality equal to that of B. Similarly, we
partition B0 into countably many disjoint sets B0,1, B0,2, B0,3, . . . so that each B0,i
has cardinality equal to that of B.
Remark 2.1. Let ↵ 2 End(A) be an arbitrary endomorphism, let i 2 N and let  
be any bijection from B into B0,i. Then the mapping  i from B0,i to A defined by
b i = b 
 1↵|B (b 2 B0,i)
satisfies   i = ↵|B , and hence ✏ ✏ i = ↵.
We now prove our main result.
Theorem 2.2. Let A be an algebra which has an infinite basis. Then any countable
subset of End(A) is contained in a 2-generated subsemigroup of End(A).
Proof. Let {↵1,↵2,↵3, . . . } be our countable subset of elements of End(A). We
find two elements of End(A) which generate a semigroup that contains this set.
Let   be any mapping from B to B\B0 which maps Bi to Bi+1 bijectively for each
i 2 {0, 1, 2, . . . }. Since B is independent   can be extended to an endomorphism
✏  of A.
We define the second of our mappings   from B to A in two stages. First, we
define   on B \ B0 to be any mapping which takes Bi to B0,i bijectively, for each
i 2 {1, 2, 3, . . . }. We observe that  i =    i  is a (well-defined) bijection from B
to B0,i.
By Remark 2.1 it follows that for each i 2 N there exists a mapping  i such that
 i i = ↵i|B . We now complete the definition of   so that for b 2 B0,i we have
b  = b i. Hence for each i 2 N we have
↵i|B =  i i =  i  =    i 2.
It follows that ↵i = ✏ ✏ ✏i ✏
2
  , and so {↵1,↵2,↵3, . . . } ✓ h ✏  , ✏  i. ⇤
For the remainder of this section we shall give examples of algebras which satisfy
the condition of our theorem, i.e. algebras which have infinite bases.
Let A be a class of algebras. Then every free A-algebra A on an infinite set
X has an infinite basis, namely X. That X generates A is evident and that X is
independent follows immediately from the definition of a free algebra.
Corollary 2.3. Let A be a non-finitely generated free A-algebra. Then every count-
able subset of End(A) is contained in a 2-generated subsemigroup of End(A).
Examples of such algebras are the non-finitely generated free algebras of any variety.
Another example of a type of algebra which satisfies our condition is provided
by a class of algebras which includes finite and infinite dimensional vector spaces as
special cases. In order to define this class we require a slightly di↵erent definition
of independence. Let A be an algebra with universe A and let X ✓ A. Then we
4 J. ARAU´JO, J.D. MITCHELL AND N. SILVA
say X is T -independent if x 62 hX \ {x} i for each x 2 X. An algebra A is called
an independence algebra if it satisfies the following properties:
(i) for every subset X of A and every element u of A, if X is T -independent
and u 62 hX i, then X [ {u} is T -independent;
(ii) for any basis X of A and any function ↵ from X to A, there is an endo-
morphism ✏↵ of A such that ✏↵|X = ↵.
The notions of T -independence and independence are not equivalent in general, but
in the case of independence algebras they coincide. Independence algebras were first
introduced in [18] and the basic structure of their endomorphism semigroups was
described in [11] . This study, for both finite and infinite independence algebras,
was continued in [1], [2], [7], [8] and [10]. Examples of independence algebras are
(unstructured) sets, vector spaces and for any group G, free G-sets. It is clear
that any independence algebra A has a basis, the cardinality of which is called the
dimension of A.
Corollary 2.4. Let A be an infinite dimensional independence algebra. Then any
countable subset of End(A) is contained in a 2-generated subsemigroup of End(A).
Since every vector space is an independence algebra this corollary provides a
shorter proof of Theorem 3.1 in [15].
3. Concluding remarks
We conclude the paper by giving an example of a finitely generated free algebra
A for which it is not true that every countable subset of End(A) is contained in
a two-generated subsemigroup of End(A), thereby demonstrating that we cannot
omit the condition that A is non-finitely generated from Corollary 2.3.
Let X be a non-empty set. Recall that an element ↵ of the semigroup TX , of all
mappings from X to X, is called a proper idempotent if ↵2 = ↵ and ↵ 6= 1X , the
identity map on X. For a subsemigroup T of TX and a set Y ✓ X we say that X
is T -isomorphic to Y if there exist mappings  ,   2 T such that X  ✓ Y , Y   ✓ X,
   = 1X and   |Y = 1Y , where 1X and 1Y denote the identity maps on X and Y
respectively.
We require the following result:
Lemma 3.1. Let X be an infinite set and let T be a subsemigroup of TX which
satisfies the following:
(i) the identity 1X is in T ;
(ii) there exists ↵ 2 T such that ↵ is not injective;
(iii) there exists   2 T such that   is not surjective but is injective;
If every countable subset of T can be embedded in a 2-generated subsemigroup of T
then X is T -isomorphic to the image of a proper idempotent in T .
For a proof see [15, Theorem 2.4].
Let A = {a, b} be an arbitrary two element alphabet. Then we denote by A+
the free semigroup on A. That is all finite, non-empty words on A with the product
of two words u and v being the juxtaposition uv of u and v.
ON EMBEDDING COUNTABLE SETS OF ENDOMORPHISMS 5
Proposition 3.2. It is not possible to embed every countable set of endomorphisms
of A+ in a 2-generated subsemigroup of End(A+).
Proof. It is obvious that End(A+)   TA+ . We verify that End(A+) satisfies the
properties (i) to (iii) of Lemma 3.1. Clearly 1A+ 2 End(A+). Let ↵ be any
endomorphism satisfying a↵ = b and b↵ = b; it is evident that ↵ is not injective.
Let   be any endomorphism satisfying a  = ab and b  = ab2; it is clear that   is
not surjective but is injective.
It is easy to verify that every proper idempotent endomorphism of A+ has image
{a}+ or {b}+. Suppose that every countable subset of End(A+) can be embedded
in a 2-generated subsemigroup of End(A+). Then, by Lemma 3.1, there exist  ,   2
End(A+) such that A+  ✓ {a}+, {a}+  ✓ A+,    = 1A+ and   |{a}+ = 1{a}+ .
But it is clear that no subset of {a}+ has A+ as a homomorphic image, and hence
we have a contradiction. ⇤
The analogue of Sierpin´ski’s result for subsets of the symmetric group was shown
to be true in [9]. As we went from mappings of unstructured sets to endomorphisms
of algebras, we ask if we can move from permutations of sets to automorphisms of
algebras.
Open Problem 3.3. Find algebras A for which every countable subset of the group
of automorphisms Aut(A) of A is contained in a 2-generated subgroup of Aut(A).
Acknowledgements: The first author would like to acknowledge the support of
‘Financiamento Plurianual do CAUL’ and Fundac¸a˜o para a Cieˆncia e a Tecnologia
(FCT) and Fundo Europeu de Desenvolvimento Regional (FEDER) project num-
ber POCTI/32440/MAT/2000. The second author would like to acknowledge the
support of FCT and FEDER project ‘Financiamento Programa´tico de CAUL’. The
third author would like to acknowledge the support of ‘Financiamento Plurianual
do CAUL’, FEDER and FCT grant number SFRH/BD/6081/2001.
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J. Arau´jo, J.D. Mitchell, N. Silva,
Universidade Aberta, Centro de A´lgebra, Mathematics Institute,
R. Escola Polite´cnica, 147 Universidade de Lisboa, University of St Andrews,
1269-001 Lisboa, 1619-003 Lisboa, North Haugh,
Portugal Portugal St Andrews,
& mitchell@cii.fc.ul.pt Fife,
Centro de A´lgebra, KY16 9SS,
Universidade de Lisboa, UK
1619-003 Lisboa, &
Portugal Centro de A´lgebra,
mjoao@lmc.fc.ul.pt Universidade de Lisboa,
1619-003 Lisboa,
Portugal
nelson@mcs.st-and.ac.uk


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On embedding countable sets of endomorphisms

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