On non-Hopfian groups of fractions

Olga Macedońska 1
  • 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100, Gliwice, Poland
Olga Macedońska
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  • Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100, Gliwice, Poland
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Abstract

The group of fractions of a semigroup S, if exists, can be written as G = SS−1. If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS−1. One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S, that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev.

A group G is called Hopfian if each epimorphizm GG is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [, Problem 15]. We prove here that if G = SS−1 is an n-generator group of fractions of a relatively free semigroup S, satisfying m-variable (m < n) non-transferable identity, then G is the non-Hopfian group.

1 Useful facts and definitions [2, 10]

  1. A semigroup S is called cancellative if for all a, b, cS either of equalities ca = cb, ac = bc implies a = b. All semigroups below are assumed to be cancellative.
  2. A semigroup S satisfies left (resp. right) Ore condition if for arbitrary a, bS there are a′, b′ ∊ S such that aa′ = bb′ (resp. aa = bb). The Ore conditions can be defined as aSbS≠ ∅, (resp. SaSb ≠∅).
  3. If a semigroup S satisfies both Ore conditions it embeds in the group G of its fractions G = SS−1 = S−1S.
  4. A subsemigroup S of a group G is called generating if elements of S generate G as a group. The group of fractions of S is unique in the sense that if a group G contains S as a generating semigroup then G is naturally isomorphic to S−1S = SS−1.
  5. A group G satisfies an identity u(x1,…,xm)≡ v(x1,…,xm) if for every elements g1,…,gm in G the equality u(g1,…, gm) = v(g1,…,gm) holds.
  6. An identity in a semigroup has a form u(x1,…,xn)≡ v(x1,…, xn) where the words u and v are written without inverses of variables, u, v ∊ ℱ. Such an identity in a group is called a semigroup identity or a positive identity. By the word “identity” we mean a non-trivial identity.

2 GB-Problem

Proposition 2.1

If a generating semigroup S of a group G = F/N satisfies an identity, then for every s, tS there exist s′, t′ ∊ S such that ss′ = tt′ (left Ore condition), G = SS−1 and F = ℱℱ−1N.

Proof

An n-variable identity a(x1,…,xn)≡ b(x1,…,xn) in S implies a 2-variable identity if we replace the i-th variable by xyi. In view of the cancellation property, it can be written as xu(x, y) = yv(x, y). For x = s, y = t, s′ := u(s, t), t′ := v(s, t), we have ss′ = tt′. Since each gG is a product of elements in SS−1, and for every s, tS, s−1t = st−1, we obtain G = SS−1. Since by assumption G = F/N, we have F = ℱℱ−1 N.  □

Remark

Note, that if we use the last letters on both sides of the identity, we get right Ore condition and the equality G = S−1S.

In 1981 G. Bergman [3], [4] posed the following natural question mentioned later in a different form in [5, Question 11.1]. It asks whether the group of fractions G = SS−1 of a semigroup S must satisfy every semigroup identity satisfied by S.

Note that the behavior of the identities may depend on additional properties of the group. A group G is called semigroup respecting (S-R group) if all of the identities holding in any generating semigroup S of G, hold in G. It is shown that extensions of soluble groups by locally finite groups of finite exponent are S-R groups [6, Theorem C]. For more results see [2].

Definition 2.2

We say that a semigroup identity ab is transferable if being satisfied in S, it is necessary satisfied in G = SS−1.

It is clear that the abelian identity is transferable. The nilpotent semigroup identities in S found by A. I. Mal’tsev [7] are transferable. The identities xn yn = yn xn are transferable [8].

So the Problem is whether every semigroup identity is transferable. This question we address as the GB- Problem. A group G = SS−1 where S satisfies a non-transferable identity we call GB-counterexample.

The first non-transferable identity and hence GB-counterexample (actually a family of them) was found in 2005 by S. Ivanov and A. Storozhev [9]. To speak of GB-counterexamples we suggest a new approach to the GB-Problem.

3 Another approach to the GB-Problem

Let F : = Fn be a free group and ℱ be a free semigroup with the unity, both freely generated by the set X = {x1, x2, x3,…,xn}.

By [10, Construction 12.3], if S = ℱ/ρ is a semigroup and G  = F/N is its group of fractions then there is a natural homomorphism φ   : F   →   F/N such that ℱφ   =   ℱ/ρ, that is

FF/N=:GFF/ρ=:S.
Each relation a = b in S is defined by the pair (a, b)∊ρ. We consider the set
A:=NFF1={ab1,(a,b)ρ},
which consists of words corresponding to relations a = b in S.

Proposition 3.1

Let S be a generating semigroup in a group G = F/N, A = N ∩ ℱℱ−1 and AF be the normal closure of the set A in F. Then if S satisfies an identity, the following equality holds

F=FF1AF.

Proof

There is the natural homomorphism FF/AF, such that ℱ→ ℱ/μ, where the congruence μ consists of pairs (a, b) for ab−1AF ∩ ℱℱ−1

AFFF1={ab1,(a,b)μ}.
Since by [10, Corollary 12.8] AF ∩ ℱℱ−1 = A, we have μ = ρ. So S is the generating semigroup in the group F/AF. Then, if S satisfies an identity, we have by Proposition 2.1 F/AF = SS−1, which implies F = ℱℱ−1 AF.  □

Lemma 3.2

Let S be a generating semigroup in a group G = F/N, and A = N ∩ ℱℱ−1. If S satisfies an identity, then N = AF.

Proof

Since F = ℱℱ−1 AF. and AFN, we have by Dedekind’s law

N=NF=NFF1AF=(NFF1)AF=AAF=AF.
 □

4 Positive endomorphisms in F

The words in ℱ are called positive words. We say that an endomorphism in F is a positive endomorphism if it maps generators to positive words X → ℱ The set of positive endomorphisms in F we denote by End+, the set of all endomorphisms - by End. The positive endomorphisms act as endomorphisms of ℱ.

Lemma 4.1

Let G = SS−1 = F/N. The normal subgroup N is positively invariant if and only if S is a relatively free semigroup.

Proof

If N is positively invariant subgroup in F then the set A = N ∩ ℱℱ−1 is also positively invariant. If a = b is a relation in S, then the word ab−1 is in A. Since A is positively invariant, the relation is the identity in S. So if N is positively invariant, then S is the relatively free semigroup.

Conversely, if S is a relatively free semigroup then each relation a = b in S is the identity, hence the set A is positively invariant and by Lemma 3.2, the normal subgroup N = AF is positively invariant.  □

We recall now the following facts similar to those in ([11] 12.21, 12.22).

Notation. Let w := ab−1∊ ℱℱ−1. By wEnd we denote the End-invariant subgroup (fully invariant subgroup) corresponding to the word w in F which is generated by all images of w under endomorphisms in F.

By wEnd+ we denote the End+-invariant subgroup (positively invariant subgroup) corresponding to the word w in F which is generated by all images of w under positive endomorphisms in F. This subgroup need not be normal.

Corollary 4.2

If G = SS−1 = F/N, then:

  1. S satisfies an identity ab if (ab−1)End+N.
  2. G satisfies an identity ab if (ab−1)EndN.
  3. G is an S-R group if for each identity ab−1:
    theinclusion(ab1)End+Nimplies(ab1)EndN.
  4. A semigroup identity ab is transferable if for every NF:
    theinclusion(ab1)End+Nimplies(ab1)EndN.
In particular we obtain

Corollary 4.3

A semigroup identity ab is transferable if

((ab1)End+)F=(ab1)End.

If S is a relatively free semigroup satisfying only transferable identities, then its group of fractions is the relatively free group.

If S is a relatively free semigroup defined by one identity ab, then A = (ab−1)End+ and by Lemma 3.2 the group of fractions of S is G0 := F/N0, where

N0=AF=((ab1)End+)F.

Corollary 4.4

For every semigroup S satisfying an identity ab its group of fractions G = F/N is a quotient group of G0 = F/N0.

5 GB-counterexamples and non-Hopfian groups

Definition 5.1

A group G = SS−1 = F/N is a GB-counterexample if S satisfies a non-transferable identity ab. By another words, if N contains (ab−1)End+, but does not contain (ab−1)End.

The first and the only known family of the GB-counterexamples was found by S.V.Ivanov and A.M.Storozhev [9]. Their groups G = SS−1 do not satisfy any identity, while S satisfies a 2-variable non-transferable identity with sufficiently large parameters, similar to that introduced by A. Yu. Ol’shanskii in [12].

Our aim is to show which GB-counterexamples are non-Hopfian groups.

Definition 5.2

A group G is called Hopfian if each epimorphizm GG is the automorphism, or in other words, if G is not isomorphic to its proper quotient.

Remark

If G is a non-Hopfian group, there is KG such that GG/K. Then for every quotient G/M we have

G/MG/KM(G/M)/(KM/M),
which implies that G/M is the non-Hopfian group unless KM.

It may help to see that the group of fractions G = F/N of a semigroup S is non-Hopfian since by Corollary 4.4, it is the quotient of G0 = F/N0, the group of fractions of a relatively free semigroup.

So we are interested in the question: when a relatively free semigroup satisfying at least one non-transferable identity, must have a non-Hopfian group of fractions?

It is not known whether each non-transferable identity implies a 2-variable non-transferable identity. So we assume that S satisfies a non-transferable identity of the form a(x1,…,xm)≡ b(x1,…,xm). We show that if S is the n-generator semigroup and n > m, then the group of fractions of S is the non-Hopfian group. For this purpose we start with the following “Common denominator Lemma”.

Lemma 5.3

Let a generating semigroup S of a group G satisfy an identity. Then for all g1, g2,…,gm in G there are s1, s2,…,sm, and r in S such that gi = sir−1, i = 1, 2,…,m.

Proof

In view of Proposition 2.1, G = SS−1. So for m = 1 the statement is clear. Let gi = tiq−1 for im−1 and gm = ab−1. By left Ore condition there exist q′, b′ ∊ S such that qq′ = bb′. We denote r : = qq′ = bb′, si : = tiq′ and sm := ab′. Then

gi=tiq1=ti(qq1)q1=(tiq)(q1q1)=sir1,gm=ab1=a(bb1)b1=(ab)(b1b1)=smr1,
as required.  □

Theorem 5.4

Let G = F/N = SS−1 be an n-generator group of fractions of a relatively free semigroup S, satisfying a non-transferable identity a(x1,…,xm)≡ b(x1,…,xm), n > m. Then G is the non-Hopfian group.

Proof

In view of Lemma 4.1 we can assume that N is the normal, positively invariant subgroup in F, such that by Corollary 4.2,

(ab1)End+N,(ab1)EndN.
where a : = a(x1,…,xm), and b:= b(x1,…,xm) are the words in ℱ.

Let α be an automorphism in the free group F, which maps x1x1 and xixix1−1 for i > 1. Then

F/NF/Nα.

Note that α−1 maps xixi x1, i > 1, hence α−1 is the positive endomorphism. Since N is the positively invariant subgroup in F, we have Nα−1N, and hence

NNα.
To show that the inclusion (2) is proper, assume the contrary, that N = Nα. By assumption the word w := ab−1, where w := w(x1, x2, …, xm)∊ N defines the identity ab in S but not in G, so the condition (1) holds. It means that for some words g2, g3,…,gm+1 in F, the value of w is not in N:
w(g2,g3,gm+1)N.
Together with the word w(x1, x2, …, xm), N must contain the word w(x2, x3,…,xm+1). Since we assumed that (2) is: Nα = N, we have
w(x2x11,x3x11,,xm+1x11)N.
By Lemma 5.3 for every g2, g3,…,gm+1 in G there are s2, s3,…,sm+1, and r in S such that gi = sir−1. The map x1r and xisi is positive. So since N is positively invariant, we get w(g2, g3,… gm+1)∊ N, which contradicts (3).

So the inclusion (2) is proper NNα. Then

(F/N)/(Nα/N)F/NαF/N,
that is F/N is isomorphic to its quotient, which proves that G = F/N is the non-Hopfian group.  □

Corollary 5.5

  1. The n-generator (n > 2)GB-counterexamples constructed in [9] by S. V.Ivanov and A.M. Storozhev are the non-Hopfian groups.
  2. An infinitely generated GB-counterexample must be the non-Hopfian group.

6 Remark: When AF = 〈A

In the book [10], just after Theorem 12.10 there is a wrong statement saying that it is shown in [13] that the right Ore condition implies thatA〉 = AF. Our next Theorem describes the conditition for this equality to hold.

Theorem 6.1

Let G = F/N where N is positively invariant, A = N ∩ ℱℱ−1. The equalityA〉 = AF holds if and only if G is a relatively free group of finite exponent.

Proof

If 〈A〉 = AF then by Lemma 3.2, N = 〈A〉 is generated as a subgroup by elements from the set A. Since N is normal, together with ab−1 where a, b ∊ ℱ, it must contain the word (ab−1)b = b−1 a as a product of words st−1A. That is b−1a = s1t11s2t21sntn1, s, t ∊ ℱ, and hence

a=bs1t11s2t21sntn1A.
We can write a=xi1k1xirkr,ki>0. Since by assumption N is positively invariant, N contains xn, n = Σi ki, xX. Then N contains xn. Now, since x−1 = xn−1 xn ∊ ℱ N, we have ℱ−1⊆ ℱN. Then by Proposition 3.1 and Lemma 3.2, F = ℱ N. Hence each endomorphism ϕEnd coincides modulo N with a positive endomorphism φ∊ End+. If wN, then wϕwφ N = N. So N is fully invariant and must contain Fn, which implies that G is a relatively free group of finite exponent.

Conversely, let xn belong to N. To show that AF = 〈A〉 it suffices to check that for every ab−1A and each generator x, the word x−1(ab−1)x is a product of elements in A. So let ab−1A, then for each c ∊ ℱ, (ca)(cb)−1A, hence (xn−1a)(xn−1b)−1A. Now, since x± nN ∩ ℱℱ−1 =: A, we get x−1(ab−1)x = xn(xn−1a)(xn−1b)−1xn∊〈A〉. This implies that the subgroup 〈A〉 is normal, which finishes the proof.  □

Let G = F/N be a group constructed by S.V.Ivanov and A.M.Storozhev in [9]. Since G = SS−1 where S satisfies a positive identity, N is positively invariant and S satisfies the left Ore condition. However the equality 〈A〉 = AF does not hold because otherwise G would have a finite exponent, which is not so.

Acknowledgement

The author is grateful to Referee for suggestions on the text improvement.

References

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    Clifford A. H., Preston G. B., The Algebraic Theory of Semigroups, 1964, (Vol. I), Math. Surveys, Amer. Math. Soc. Providence, R.I.,

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    Macedońska O., Słanina P., GB-problem in the class of locally graded groups, Comm. Algebra, 2008, 36(3), 842–850.

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    Bergman G., Hyperidentities of groups and semigroups, 1981, Aequat. Math., 23, 55–65.

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    Bergman G., Questions in algebra, Preprint, Berkeley, U.D. 1986.

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    Shevrin L. N., Sukhanov E. V., Structural aspects of the theory of varieties of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat., 1989, 6, 3–39 (Russian). English translation: Soviet Math., 1989, Iz. VUZ 6., 33, 1–34.

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    • Crossref
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    Maľtsev A. I., Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uc. Zap., 1953, 4, 107–111. (Russian).

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    Krempa J., Macedońska O., On identities of cancellative semigroups, Contemporary Mathematics, 1992, 131, 125–133.

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    Ivanov S. V., Storozhev A. M., On Identities in groups of fractions of cancellative semigroups, Proc. Amer. Math. Soc., 2005, 133, 1873–1879.

    • Crossref
    • Export Citation
  • [10]

    Clifford A. H., Preston G. B., The Algebraic Theory of Semigroups, 1967, (Vol. II), Math. Surveys, Amer. Math. Soc. Providence, R.I.

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    Neumann H., Varieties of groups, 1967, Springer-Verlag, Berlin, Heidelberg, New York.

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    Oľshanskii A. Yu., Storozhev A., A group variety defined by a semigroup law, 1996, J. Austral. Math. Soc. Series A, 60, 255–259.

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    V. Ptak., Immersibility of Semigroups, 1949, Acta Fac. Nat. Univ. Carol. Prague, 192.

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  • [1]

    Clifford A. H., Preston G. B., The Algebraic Theory of Semigroups, 1964, (Vol. I), Math. Surveys, Amer. Math. Soc. Providence, R.I.,

  • [2]

    Macedońska O., Słanina P., GB-problem in the class of locally graded groups, Comm. Algebra, 2008, 36(3), 842–850.

  • [3]

    Bergman G., Hyperidentities of groups and semigroups, 1981, Aequat. Math., 23, 55–65.

  • [4]

    Bergman G., Questions in algebra, Preprint, Berkeley, U.D. 1986.

  • [5]

    Shevrin L. N., Sukhanov E. V., Structural aspects of the theory of varieties of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat., 1989, 6, 3–39 (Russian). English translation: Soviet Math., 1989, Iz. VUZ 6., 33, 1–34.

  • [6]

    Burns R. G., Macedońska O., Medvedev Y., Groups Satisfying Semigroup Laws, and Nilpotent-by-Burnside Varieties, 1997, J.of Algebra, 195, 510–525.

    • Crossref
    • Export Citation
  • [7]

    Maľtsev A. I., Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uc. Zap., 1953, 4, 107–111. (Russian).

  • [8]

    Krempa J., Macedońska O., On identities of cancellative semigroups, Contemporary Mathematics, 1992, 131, 125–133.

  • [9]

    Ivanov S. V., Storozhev A. M., On Identities in groups of fractions of cancellative semigroups, Proc. Amer. Math. Soc., 2005, 133, 1873–1879.

    • Crossref
    • Export Citation
  • [10]

    Clifford A. H., Preston G. B., The Algebraic Theory of Semigroups, 1967, (Vol. II), Math. Surveys, Amer. Math. Soc. Providence, R.I.

  • [11]

    Neumann H., Varieties of groups, 1967, Springer-Verlag, Berlin, Heidelberg, New York.

  • [12]

    Oľshanskii A. Yu., Storozhev A., A group variety defined by a semigroup law, 1996, J. Austral. Math. Soc. Series A, 60, 255–259.

    • Crossref
    • Export Citation
  • [13]

    V. Ptak., Immersibility of Semigroups, 1949, Acta Fac. Nat. Univ. Carol. Prague, 192.

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