Cancellative semigroups admitting conjugates

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J. Semigroup Theory Appl. 2015, 2015:9 ISSN: 2051-2937

CANCELLATIVE SEMIGROUPS ADMITTING CONJUGATES

G.I. MOGHADDAM∗, R. PADMANABHAN

Department of Mathematics, University of Manitoba, Winnipeg, Canada

Copyright c2015 Moghaddam and Padmanabhan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. We prove conjugate analogs of Mal’cev-Neumann-Taylor-Levi’s theorems. In other words, we char-acterize semigroups embeddable in nilpotent groups of classnby means of conjugacy laws involving semigroup laws described by Mal’cev, Neumann and Taylor. Moreover we prove that a cancellative semigroup admitting conjugates is embeddable in a nilpotent group of class 2 if and only if it satisfies the conjugacy law xyz ₌_xy_.

Also adapting Ore’s techniques we describe an exact procedure for embedding a cancellative semigroup admitting conjugates into a group.

Keywords:Cancellative semigroup; Conjugate; Embedding; Nilpotent. 2010 AMS Subject Classification:20M07, 20E10.

1. Introduction

While every subsemigroup of a group is cancellative, a famous theorem of A.I. Mal’cev [4] shows that not every cancellative semigroup is embeddable in a group. In fact, Mal’cev gave

∗_{Corresponding author}

E-mail addresses: [email protected] (G.I. Moghaddam), [email protected] (R. Padmanab-han)

Received July 14, 2015

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an infinite set of necessary and sufficient conditions for the possibility of imbedding a cancel-lation semigroup into a group and proved that no finite set of such conditions would suffice. This calls for finite sets of sufficient conditions that would guarantee the group embeddabilty. Patterned after the classical quotient construction, Oystein Ore [2] discovered the ”principle of common left multiple” to imbed a non-commutative domain into a division ring. Using this as a backdrop, Mal’cev, B.H. Neumann and Taylor developed semigroup equivalents of nilpotent groups of class n and proved that cancellative semigroups of nilpotent class n are embeddable in groups of the same nilpotency class. In this paper, we study some equational classes of enriched semigroups - i.e. semigroups admitting conjugates - and prove that all the valid group theory implications do carry over to the equational theory of semigroups admitting conjugates.

2. Preliminaries

Definition 2.1. LetSbe a semigroup. If for allx,yandzinS,xy=xzimpliesy=z, and also

yx=zximpliesy=z, then we saySis a cancellative semigroup.

Definition 2.2. Let S be a semigroup, and x,y∈S. If there exist an element z∈S such that

xy=yz, thenzis called conjugate ofxbyyand it is denoted byxy. If for all elementsxandyin

S,xyexists we saySadmits conjugates.

Therefore for anyxandyin a cancellative semigroupSthat admits conjugates,xy is unique and

xy=y xy. (_~)

Lemma 2.1.Let S be a cancellative semigroup which admits conjugates and let x, y and z be in S, then:

(1) xx=x ,

(2) (xy)z=xyz,

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Proof.(1) : Sincexx=xxxby(_~), so by left cancellation we getxx=x. (2) : Using(_~):

(yz)xyz=x(yz) = (xy)z= (yxy)z=y(xyz) =y z(xy)z= (yz) (xy)z,

and by left cancellation we get (xy)z=xyz. (3) : Using(_~):

z(xy)z= (xy)z=x(yz) =x(zyz) = (xz)yz= (zxz)yz=z(xzyz),

and by left cancellation we get (xy)z=xzyz.

Definition 2.3. Let Sbe a cancellative semigroup which admits conjugates. For any elements

a,b,canddinSwe define

a_rb={(x,y)|ay=xby, x,y∈S}.

The set of alla_rbis denoted byS, i.e.

S={a_rb| a,b∈S},

and inSwe define binary operation ∗as

(a_rb)∗(c_rd) =ac_rdbc.

3. Embedding into groups

First we prove an analog of the classical Ore’s condition [2].

Theorem 3.1. Let S be a cancellative semigroup which admits conjugates. For any a, b, x,y,u and v in S, if

ay=xby, cy=xdy, av=ubv,

then cv=udv.

Proof.First multiplying ay=xby by v we get (xby)v= (ay)v now using Lemma 2.1 and (_~) we get

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= (ubv)yv by hypothesis of the Theorem =u(bvyv)

=u(by)v by Lemma 2.1 =u(yby)v by(_~)

=u yv(by)v by Lemma 2.1 = (uyv)(by)v.

Also

(xby)v=x(byv) =x v(by)v= (xv)(by)v.

Therefore we have (xv)(by)v= (uyv)(by)vand by right cancellation we get

(1) xv=uyv.

On the other hand multiplying cy=xdy by v we get (xdy)v= (cy)v now using Lemma and (_~) we get

(xdy)v=x(dyv) =x v(dy)v= (xv)(dy)v

= (uyv)(dy)v by(1) =u yv(dy)v

=u yv(dyv)

by Lemma 2.1 =u yv(dvyv)

by(_~)

=u yv(dv)yv by Lemma 2.1 =u(dvyv) by(_~)

= (udv)yv.

Also

(cy)v=c(yv) =c(vyv) = (cv)yv.

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Lemma 3.1. Let S be a cancellative semigroup which admits conjugates and let a, b, c and d be in S then

(1) If(a_rb)∩(c_rd)6= _∅, then a_rb=c_rd,

(2) a_ra=b_rb .

Proof. To prove part 1 : First we note that sinceab=abb so(a,b)∈a_rbthat is a_rb6=_∅. Now if (a_rb)∩(c_rd)6=_∅, let assume (x1,y1) ∈(arb)∩(crd) then, ay1 =x1by1 and

cy₁=x₁dy1. Now for any(u,v)ina_rbsinceav=ubv by Theorem 3.1 mustcv=udv, which means(u,v)∈c_rdsoa_rb⊆c_rd. Similarly for any(u,v)inc_rdsincecv=udvby Theorem 3.1 mustav=ubv, soc_rd⊆a_rband thereforea_rb=c_rd.

To prove part 2 : Sinceax=xaxand alsobx=xbxso(x,x)∈a_ra∩b_rband therefore by part 1 musta_ra=b_rb.

Lemma 3.2. Let S be a cancellative semigroup which admits conjugates and let a, b, u and v be in S then

(1) au_rbu=a_rb ,

(2) ua_rbua=a_rb ,

(3) au_ru=av_rv .

Proof.(1) : For any(x,y)ina_rbwe have ay=xby now since

(au)y=a(uy) =a(yuy) = (ay)uy= (xby)uy=x(byuy) =x(bu)y,

so(x,y)∈au_rbuthat isa_rb⊆au_rbuand by Lemma 3.1 au_rbu=a_rb. (2) : For any(x,y)ina_rbwe have ay=xby now since

(ua)y=u(ay) = (ay)uay= (xby)uay=x(byuay) =x(bua)y,

so(x,y)∈ua_rbuathat isa_rb⊆ua_rbuaand by Lemma 3.1 ua_rbua=a_rb.

(3) : First we note that (au,u)∈(au_ru) because (au)u= (au)uu; now since (av)u=

a(vu) =a(uvu) = (au)vu so (au,u)∈(av_rv) that is au_ru∩av_rv6= _∅and by Lemma 3.1

au_ru=av_rv.

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Proof. (i) Closure : For all a_rb,c_rd ∈S, (a_rb)∗(c_rd) =ac_rdbc is a well defined element ofS.

(ii) Associativity: For alla_rb,c_rd,f_rg∈S,

(a_rb)∗(c_rd)∗(f_rg) = (ac_rdbc)∗(f _rg)

= (ac)f_rg(dbc)f

=a(c f)_r(gdf)bc f by Lemma 2.1 = (a_rb)∗(c f_rgdf)

= (a_rb)∗ (c_rd)∗(f_rg).

(iii) Identity: Since by of Lemma 3.1, for any u and v in S, u_ru=v_rv, let E=u_ru=v_rv

then E is the identity element because by Lemma 3.2

(a_rb)∗E= (a_rb)∗(u_ru) =au_rubu=au_rbu=a_rb;

also

E∗(a_rb) = (u_ru)∗(a_rb) =ua_rbua=a_rb.

(iv) Inverses: For eacha_rb∈Sthere existb_rain S such that

(a_rb)∗(b_ra) =ab_rabb=ab_rab=E;

also

(b_ra)∗(a_rb) =ba_rbaa=ba_rba=E,

so(a_rb)−1=b_ra. Therefore(S,∗)is a group.

Theorem 3.3. If S is a cancellative semigroup which admits conjugates, then it can be embed-ded into a group.

Proof. Define φ :S→S by φ(a) =au_ru where u∈S; then φ is a well defined function

because by by Lemma 3.2, au_ru=av_rv. φ is a one to one function because ifφ(a) =φ(b)

thenau_ru=bu_ruwhich means for any(x,y)∈au_ru=bu_ru,(au)y=xuyand(bu)y=xuy

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In order to show that φ is a homomorphism that is fora,b∈S, φ(ab) =φ(a)∗φ(b)since φ(ab) = (ab)u_ru=abu_ru and

φ(a)∗φ(b) = (au)_ru)∗(bu)_ru) = ((au)(bu)_ruubu) =aubu_rubu,

we need to show that abu_ru=aubu_rubu.But for any (x,y) in abu_ru we have abuy=

xuynow since

(aubu)y=a(u(buy))

=a((buy)ubuy) by(_~) = (abuy)ubuy

= (xuy)ubuy since(x,y)∈(abu_ru) =x(uyu(bu)y)

=x(uubu)y by Lemma 2.1

=x(ubu)y by(_~),

so(x,y)∈(aubu_rubu)that is(abu_ru)⊆(aubu_rubu)and by Lemma 3.1 abu_ru=aubu_r ubu.

3. Conjugacy laws

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Theorem 4.1. Let S be a cancellative semigroup which admits conjugates, then the following laws are equivalent in S :

(1) xy=yx (commutativity ) ,

(2) xy=x (conjugacy ),

(3) x(yz)= (xy)z (associativity of conjugates).

Proof.It is clear that 1⇒2 and 2⇒3 . Now 3⇒1 because if x(yz)= (xy)z is a law then for

y=x we get xxz = (xx)z=xz. Multiplying both sides by xz gives xzxxz =xzxz. Using(_~)for the left side we get xxz=xzxz; right cancellation gives x=xz therefore xz=zxz=zx.

Theorem 4.2. Let S be a cancellative semigroup which admits conjugates, then S is nilpotent of class2if and only if it satisfies the conjugacy law xyz=xy.

Proof. By Mal’cev [4] and or by Neumann and Taylor [5] it is enough to prove that xyzyx=

yxzxy⇔xyz =xy. If xyz =xy is a law in S, then first we note that using (_~) twice, gives

yx=xyx=yxxyx =yxxy. Now

yxzxy= (xyx)z(yxy) by(_~) = (xyx)(zy)xy

= (xyx)xy(zy)xy by(_~)

=x(yxxy)(zy)x by assumption =x(yx)(zy)x by above note =xy x(zy)x

=xy (zy)x by(_~). =xyzyx.

Conversely ifxyzyx=yxzxy is a law in S, then first we note that using(_~)and cancelingywe get

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In(2)replacingybyyzwe getxyzzyzx=xzxyzwhich means

xyzyzx=xzxyz.

Now multiplying both sides byuzywe get

(3) xyzyzxuzy = xzxyzuzy.

But

xyzyzxuzy=xyzyz(xu)zy

=xyzzy(xu)yz by assumption =xyz(zyxuyz).

Also

xzxyzuzy=xzx(yzuzy)

=xzx(yuyz) by(2) = (xzxy)uyz

= (xyzyx)uyz by(2) =xy(zyxuyz).

Therefore by substitution in (3) we get xyz(zyxuyz) =xy(zyxuyz) and right cancellation gives

xyz =xy.

Corollary 4.1. Let S be a cancellative semigroup which admits conjugates, then S is nilpotent of class2if and only if it satisfies the conjugacy law xyz=xzy.

Proof. Using the above theorem it is enough to prove that xyz = xzy is a law if and only if xyz =xy is a law in S. If S satisfies the law xyz =xzy then since xyz =xzyz = (xz)yz so (xz)yz = (xz)y. Let t=xz then tyz=ty.

Conversely if S satisfies the law xyz =xy then

xyz= (xz)yz

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=xzy.

Following [4] and [5], we define a semigroup S to be nilpotent of class 3 if it satisfies the law (xyzyx)u(yxzxy) = (yxzxy)u(xyzyx); inductively we say S is of nilpotent class n if it satisfies the law f ug=gu f where the law f =g defines semigroups of nilpotent class n−1 and u is a new variable not occurring in the terms f or g.

Theorem 4.3. Let S be a cancellative semigroup which admits conjugates, then S is nilpotent of class n if and only if it satisfies the (n+1)-variable conjugacy law xf =xgwhere x is a variable not occurring in the terms f or g .

Proof. Assume that S satisfies the law xf =xg. Let x= f t where t is a new variable, then since xf = (f t)f = f tf =t f so must t f = (f t)g. Therefore f ug= (f u)g=g(f u)g=g(u f) =

gu f which means S is nilpotent of classn.

Conversely assume that S is nilpotent of classn that is f ug=gu f is a law. Then by the very definition of conjugates, we have x f = f xf . Premultiplying both sides of this equation by

gy, where y is a new variable, we get gyx f =gy f xf . Using the nilpotent identity f ug=gu f

twice, we obtain f yxg= f ygxf . Left canceling the common term f y we get xg=gxf. But

xg= gxg, therefore gxg =gxf . Finally left canceling the common term g, we obtain the desired conjugacy law xf =xg.

Conflict of Interests

The authors declare that there is no conflict of interests. REFERENCES

[1] F.W. Levi, Groups in which the commutator operation satisfies certain algebraic conditions, J. Indian Math. Soc. (N.S.) 6 (1942), 87-97.

[2] Oystein Ore, Linear Equations in non-commutative fields, Ann. Math. 32 (1931), 463-477. [3] A.G. Kurosh, The Theory of Groups, Chelsea, New York, 1956.

[4] A.I. Mal’cev, Nilpotent semigroups, Ivanov. Gos. Ped. Inst. U ˇc, Zap. Fiz.-Mat. Nauki 4 (1953), 107-111. [5] B.H. Neumann, L. Taylor, L. Tekla, Subsemigroups of nilpotent groups, Proc. Roy. Soc. Ser. A 274 (1963),