On the efficiency of semi-direct products of finite cyclic monoids by one-relator monoids

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On the Efﬁciency of Semi-Direct Products of Finite Cyclic Monoids by

One-Relator Monoids

Article · September 2010 DOI: 10.1063/1.3497846 CITATIONS 0 READS 23 5 authors, including:

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On the Efﬁciency of Semi-Direct Products of Finite Cyclic

Monoids by One-Relator Monoids

1

_{Fırat Ate¸s}

∗

_{, Eylem G. Karpuz}

2∗

_{, A. Dilek Güngör}

3†

_{, A. Sinan Çevik}

4†

_and

5

_{I. Naci}

Cangül

∗∗

∗_{Balikesir University, Department of Mathematics, Faculty of Science and Art, Cagis Campus, 10145, Balikesir,} Turkey

†_{Selçuk University, Department of Mathematics, Faculty of Science, Alaaddin Keykubat Campus, 42075, Konya,} Turkey

∗∗_{Uludag University, Department of Mathematics, Faculty of Science and Art, Görükle Campus, 16059, Bursa,} Turkey

Abstract. In this paper we give necessary and sufﬁcient conditions for the efﬁciency of a standard presentation for the

semi-direct product of ﬁnite cyclic monoids by one-relator monoids.

Keywords: Efﬁciency, ﬁnite cyclic monoids, one-relator monoids. PACS: 2010 MSC: 20L05, 20M05, 20M15, 20M50.

INTRODUCTION

LetP = [x;r] be a ﬁnite presentation for a monoid M. Then the Euler characteristic of P is deﬁned byχ(P) =

1−|x|+|r| and an upper bound of M is deﬁned byδ(M) = 1−rkZ(H1(M))+d(H2(M)). In an unpublished work, S.J.

Pride has shown thatχ(P) ≥δ(M). With this background, one can deﬁne a monoid presentation P to be efﬁcient if

χ(P) =δ(M), and then M is called efﬁcient if it has an efﬁcient presentation.

It is well known that one of the effective way to show efﬁciency for the monoid M is to use spherical monoid

pictures overP. These geometric conﬁgurations are the representative elements of the Squier complex denoted by

D(P) (see, for example [4], [5], [7]). Suppose Y is a collection of spherical monoid pictures over P. Two monoid

pictures P and Pare equivalent relative to Y if there is a ﬁnite sequence of monoid pictures P= P0,P1,···,Pm= P

where, for 1≤ i ≤ m, the monoid picture Pi is obtained from the picture Pi−1 either by the insertion, deletion and

replacement operations. By deﬁnition, a set Y of spherical monoid pictures overP is a trivializer of D(P) if every

spherical monoid picture is equivalent to an empty picture relative to Y. The trivializer is also called a set of generating pictures.

For any monoid picture P overP and for any R ∈ r, expR(P) denotes the exponent sum of R in P which is the

number of positive discs labelled by R₊, minus the number of negative discs labelled by R₋. For a non-negative

integer n,P is said to be n-Cockcroft if expR(P) ≡ 0 (mod n), (where congruence (mod 0) is taken to be equality) for

all R∈ r and for all spherical pictures P over P. Then a monoid M is said to be n-Cockcroft if it admits an n-Cockcroft

presentation. In fact to verify the n-Cockcroft property, it is enough to check for pictures P∈ Y, where Y is a trivializer

(see [4], [5]). The 0-Cockcroft property is usually just called Cockcroft. The following result is also an unpublished result by S.J. Pride.

Theorem 1 LetP be a monoid presentation. Then P is efﬁcient if and only if it is p-Cockcroft for some prime p.

1 _{Corresponding author, ﬁ[email protected]} 2 _{[email protected]}

3 _{[email protected]} 4 _{[email protected]}

5 _{The ﬁfth author is supported by Uludag University Research Fund, Project Nos: 2006-40, 2008-31 and 2008-54. [email protected]}

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CP1281, ICNAAM, Numerical Analysis and Applied Mathematics, International Conference 2010 edited by T. E. Simos, G. Psihoyios, and Ch. Tsitouras

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Let K be a ﬁnite cyclic monoid with the presentationPK = [y ; yk= yl(l < k)], and let A be one-relator monoid

with presentationPA= [x ; R+= R−]. Also letψbe an endomorphism of K. Then we have a mapping

x−→ End(K), x −→ψx,

where x∈ x.

In fact this induces a homomorphismθ: A−→ End(K), x −→ψx if and only ifψR₊=ψR₋ and(yk)ψx= (yl)ψx

where x∈ x. For R+= a1a2···arand R−= z1z2···zs, let us deﬁne(y)ψai by y

mi _and_(y)_ψ

zj by y

nj_{, where a}

i,zj∈ x

and mi,nj∈ Z+. Therefore in order to deﬁne semi-direct product, we must have

[ym1m2···mr] = [yn1n2···ns], ₍₁₎

and then we get the corresponding semi-direct product M= K ×_θA with the presentation

PM= [y,x ; yk= yl, R+= R−, Tyxi], (2)

where Tyxi : yxi= xiyti (xi∈ x,ti∈ Z+) such that we have the relation (1) for yti.

If U= x1x2···xnis a positive word on x, then we denote the word(y)ψUby(···(((y)ψx1)ψx2)ψx3···)ψxn) and this

can be represented by a monoid picture, say AU,y. For the relation R+= R−, we have two important special cases AR₊,y

and AR₋,yof this consideration. We should note that these non-spherical pictures consist of only Tyxi-discs (xi∈ x).

Also, since[(yk₎_ψ

x]PK= [(y l₎_ψ

x]PK, there exists a non-spherical picture, say BS,x, overPKwhere S : y

k_{= y}l_{. On the}

other hand, since we have R+= R−, we get[(y)ψR+]PK= [(y)ψR−]PK. Hence there is a non-spherical picture over

PK which is denoted by Cy,ψR withι(Cy,ψR) = (y)ψR+andτ(Cy,ψR) = (y)ψR−, where R : R+= R−.

Now, by considering the presentationPM in (2), we prove the following theorem as a main result in the present

paper.

Theorem 2 Let p be a prime or 0. Then the presentationPM is p-Cockcroft if and only if the following conditions

hold.

(i) k − l ≡ 0 (mod p),

(ii) For each i, ti≡ 1 (mod p),

(iii) m1m2···mr−n1n2···ns

k−l ≡ 0 (mod p),

(iv) expT_yxi(AR_+,y) ≡ expT_yxi(AR_−,y) (mod p).

We may refer [1, 2, 3, 4, 5, 6, 7] to the reader for most of the fundamental material (for instance, semidirect products of monoids, Squier complex, a trivializer set of the Squier complex, spherical and non-spherical monoid pictures) which will be needed here.

TRIVIALIZER SET

D(P

M

)

In [8], Wang constructed a ﬁnite trivializer set for the standard presentation for the semi-direct product. By using this paper, we can construct spherical monoid pictures, say PS,xand PR,y, by the non-spherical pictures BS,x, AR+,y, AR−,y

and Cy,ψR.

Let XAand XKbe trivializer sets ofD(PA) and D(PK), respectively. Also, let C1consists of the pictures PS,xand

C2consists of the pictures PR,ywhere S : yk= yl and R : R+= R−.

Let us consider the presentationPM, as in(2). Then, by [8], a trivializer set of D(PM) is

XA∪ XK∪ C1∪ C2.

In here, the subpictures Cy,ψRand B −1

S,xi contain just S : y

k_{= y}l_{(l < k) discs. Moreover, each of the subpictures A}

R_+,y

and A−1_R_−,y consists of just the Tyxi discs. The reason for us keeping work on the above monoid pictures is their usage

in the important connection between efﬁciency and p-Cockcroft property. Therefore, in the present paper, we will use this connection to get the efﬁciency. To do that we will count the exponent sums of the discs in these above pictures to

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PROOF OF THE MAIN RESULT AND ITS APPLICATION

Let us consider the pictures given in above. To prove Theorem 2, we will count the exponent sums of the discs in these pictures.

At ﬁrst, let us think the pictures PS,xi. It is easy to see that

expS(PS,xi) = 1 − expS(BS,xi), and to p-Cockcroft property be hold, we need to have

expS(PS,xi) ≡ 0 (mod p) ⇔ expS(BS,xi) ≡ 1 (mod p)

⇔ ti≡ 1 (mod p).

Moreover, in PS,xi, we also have l times positive and k times negative Tyxi: yxi= xiy

ti _{discs. That means}

expT_yxi(PS,xi) = l − k,

and so we must have k− l ≡ 0 (mod p) to p-Cockcroft property be hold.

Secondly let us consider the pictures PR,y. In these pictures, we have only one positive and one negative S disc.

Therefore expS(PR,y) ≡ 0 (mod p). Now let us check the exponent sums in the subpictures AR_+,y and A−1_R_−,y. (We recall

that these two subpictures are dual of each other). Then, the exponent sum of Tyxidiscs in pictures PR,yare

expT_yxi(PR,y) = expT_yxi(AR_+,y) − expT_yxi(AR_−,y), (3)

So we have expT_yxi(AR_+,y) ≡ expT_yxi(AR_−,y) (mod p) to get efﬁciency. Thus it remains to check the subpictures Cy,ψR. By a simple calculations, one can show that the number of discs is m1m2···mr−n1n2···ns

k−l = c in this picture. Thus we have

c≡ 0 (mod p). Hence the result.

We note that, by considering the trivializer sets XA and XK of the Squier complexes D(PA) and D(PK),

respectively, it can be easily deduced thatPAandPKare p-Cockcroft, in fact Cockcroft, presentations.

These all above procedure give us sufﬁcient conditions to be the presentationPM in (2) is p-Cockcroft for any

prime p. In fact the converse part (necessary conditions) of the theorem is quite clear.

As an example, let us suppose that the monoid A is presented byPA= [x1,x2; xm1x2x1= xm2x1x2]. Then in order to

deﬁne semi-direct product with K, we must have[yt1m+1t2] = [yt2m+1t1] by replacing (y)ψ

x1 with y

t1 _and(y)ψ

x2 with y

t2_.

Hence we get the corresponding semi-direct product M with the presentation PM= [y,x1,x2; yk= yl, xm1x2x1= xm2x1x2,

yx1= x1yt1, yx2= x2yt2]. (4)

Let us consider presentation given in(4) .Then we can give the following corollary as a consequence of the main

result.

Corollary 3 Let p be a prime or 0. Then the presentationPMis p-Cockcroft if and only if the following conditions

hold.

(i) k − l ≡ 0 (mod p),

(ii) For each i ∈ {1,2}, ti≡ 1 (mod p),

(iii) t1m+1t2−t2m+1t1 k−l ≡ 0 (mod p), (iv) 1−tm1 1−t1 +t m 1t2−t2m≡ 0 (mod p) and 1−tm 2 1−t2 +t1t m 2 −t1m≡ 0 (mod p).

REFERENCES

1. F. Ate¸s and A. S. Çevik, “Minimal but inefﬁcient presentations for semidirect products of ﬁnite cyclic monoids", in Groups St.

Andrews 2005, Volume 1, edited by C. M. Campbell, M. R. Quick, E. F. Robertson and G. C. Smith, LMS Lecture Note Series

339, (2006), 170-185.

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2. A. S. Çevik, The p-Cockcroft property of the semidirect products of monoids, Int. J. Algebra and Comput. 13(1) (2003), 1-16. 3. A. S. Çevik, Minimal but inefﬁcient presentations of the semidirect products of some monoids, Semigroup Forum 66 (2003),

1-17.

4. S. J. Pride, “Geometric methods in combinatorial semigroup theory", in Semigroups, Formal Languages and Groups, edited by J. Fountain, Kluwer Academic Publishers, (1995), 215-232.

5. S. J. Pride, Low-dimensional homotopy theory for monoids, Int. J. Algebra and Comput. 5(6) (1995), 631-649. 6. S. J. Pride, Low-dimensional homotopy theory for monoids II, Glasgow Math. J. 41 (1999), 1-11.

7. C. C. Squier, Word problems and a homological ﬁniteness condition for monoids, Journal of Pure and Appl. Algebra 49 (1987), 201-216.

8. J. Wang, Finite derivation type for semi-direct products of monoids, Theoretical Computer Science 191(1-2) (1998), 219-228.