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Jorge Almeiday Abstract

This paper determines all pseudovarieties of nite semigroups whose powers are contained in DA. As an application, it is shown that it is computable, for a decidable pseudovariety V of aperiodic semigroups, the least exponent n such that the iterated power P

n

V is the pseudovariety of all nite semigroups.

1. Introduction

Given an algebraic structure A, its power setP(A) carries a natural structure of the same type.

For instance, in group theory, when talking about subgroup cosets, one is really dealing with some structural aspects of the power set of a group.

In the realm of Universal Algebra, a basic question about power algebras which is raised concerns the equational properties which are preserved under the power set construction. It is then natural to deal with equationally de ned classes (i.e., Birkho 's varieties) instead of individual algebraic structures and to de ne the power operator on such a class Cby taking the variety PC generated

by allP(A) withA2C. The varieties stable under the power operator have been characterized in

[9, 19]. In [12], the semigroup case is considered and the identities satis ed by power semigroups are determined, and this led to the complete description of the power operator on varieties of semigroups given in [2].

In nite semigroup theory, the study of the power operator has been further developed in view of the motivation coming from language theory. In fact, certain natural operators on Eilen-berg's varieties of languages correspond to the power operator on pseudovarieties of semigroups (or monoids). This and further connections of speci c values of the power operator with problems such asdot-depth two [17] and thetype II conjecture [11, 16], prompted a systematic study of the power operator. The most comprehensive review of this area to this date can be found in [6]. There are some similarities and common features between the monoid and semigroup cases but it turns out that the latter is more complicated.

In particular, it was shown in [1] that the iteration of the power operator on pseudovarieties of semigroups stabilizes at the end of three steps. Moreover, for non-permutative pseudovarieties, the stabilization occurs at the pseudovariety

S

of all nite semigroups. The exponent of a non-permutative pseudovariety

V

is then the numbernof times the power operator needs to be applied

to obtain P

V

=

S

. A characterization of exponent one pseudovarieties was given in [13].

The present paper shows how to compute the exponent of a non-permutative pseudovariety of aperiodic semigroups. The monoid case was previously settled in [5] where all pseudovarieties of the form P

V

with

V

a pseudovariety of aperiodic monoids were identi ed (without actually

This work was supported, in part, by the project Praxis/2/2.1/MAT/63/94.

yAuthor's address: Centro de Matematica, Faculdade de Ci^encias da Universidade do Porto, P. Gomes Teixeira, 4050 Porto, Portugal.

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\computing" them). In view of previous work, the major step to achieve such a result in the semigroup case consists in determining all pseudovarieties whose powers are contained in

DA

, the pseudovariety of all nite semigroups whose regular elements are idempotent. It turns out that, just as in the monoid case, there are three maximal such pseudovarieties.

2. Preliminaries

In general, for background and unde ned notation see [6, 15]. Only some essential preliminaries are introduced here.

Multiplicative notation will be used for all semigroups. For a semigroup S, denote by P(S)

the semigroup of subsets of S under the multiplication given by AB = fab : a 2 A; b 2 Bg for A;B 2P(S).

Apseudovariety is a class of nite semigroups which is closed under taking homomorphic images, subsemigroups and nitary direct products. The pseudovariety generated by a class C of nite

semigroups is the smallest pseudovariety containing C. For a pseudovariety

V

, P

V

denotes the

pseudovariety generated by the class consisting of all power semigroupsP(S) withS2

V

.

For a pseudovariety

V

and a nonempty setA, anA-ary implicit operation on

V

is a

V

-indexed

family ( S)S2V of A-ary operations  S : S A

! S such that, for every homomorphism ':S ! T

with S;T 2

V

,  T

' n =

'

S. The set A

V

of all

A-ary implicit operations on

V

is denoted

by A

V

and constitutes a semigroup under pointwise multiplication. For each

a2A, there is an

associated projection (orvariable) ade ned bya S(

f) =f(a) forf 2S

A, where the same notation

is adopted to represent the projection and the element ofAwhich gives rise to it. The subsemigroup

of A

V

generated by all such projections is denoted by A

V

and is the

V

-free semigroup on A, its

elements being called A-ary explicit operations on

V

. Since

A

V

is completely determined by

V

and jAj, we write

n

V

for A

V

with

jAj=n.

The most common example of implicit operation which is (usually) not explicit is the unary operation x ! de ned by letting s ! = ( x !) S(

s) denote the only idempotent which is a power of s

for each elementsof a nite semigroup S. The implicit operation x !

x is also denoted byx !+1.

The semigroup A

V

is endowed with the initial topology for the homomorphisms A

V

! S

withS2

V

, where nite semigroups are taken with the discrete topology. The subsemigroup A

V

is dense in A

V

(cf. [18, 6]).

A pseudoidentity is a formal equality  =  where ; 2

n

S

for some

n  1,

S

denoting

the pseudovariety of all nite semigroups. A nite semigroup is said to satisfy a pseudoidentity

 =  and we write S j=  =  if  S =



S. For a set  of pseudoidentities, [[]] denotes the

pseudovariety consisting of all nite semigroups which satisfy all pseudoidentities from . By Reiterman's Theorem [18], all pseudovarieties are of this form. To avoid an excessively heavy notation, we adopt the convention that, in a pseudoidentity, the letterse;f;g denote the!-powers

of variables which do not appear elsewhere. For instance, [[ef=fe]] represents the class of all nite

semigroups in which idempotents commute. A nite semigroup S is permutative if S j= exyf = eyxf (cf. [6]).

The (left-right)dual of a semigroupSis the semigroupS

whose multiplication table is obtained

by transposition of the table of S. The dual of a pseudovariety

V

is the pseudovariety consisting

of the duals of the semigroups in

V

. Thedual of a word wis the word w

 obtained by reading w

backwards. Thedual of an implicit operation2

A

S

is the limit of any sequence ( w



n)nof words

such that (w

n)n converges to

. Often, the dual of an implicit operation is obtained by reading

it backwards, just as for words. The dual of a pseudoidentity is obtained by taking the formal equality of the duals of its sides.

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We next recall some results concerning the operator P. To state them, the following notation

is convenient.

Consider the semigroups presented by

B 2 =

ha;b; aba=a; bab=b;a 2 = b 2 = 0 i Y = he;s;f; e 2 = e; f 2= f;esf =s; ef =fe= 0i Q = he;s;t; e 2 = e;es=s; te=t;se=et= 0i

and the pseudovarieties

DA

= [[(xy) !( yx) !( xy) ! = ( xy) ! ; x !+1 = x !]]

K

= [[ex=e]]

D

= [[xe=e]]

M

V

= pseudovariety generated by all S 1 with S2

V

V

(S) = pseudovariety generated byS

B

= [[x 2 = x !]] = f nite bandsg

A

= [[x !+1 = x !]] =

f nite aperiodic semigroupsg

Perm

= [[exyf =eyxf]]:

We say that a pseudovariety has a certain property if all semigroups in it enjoy that property. In particular, a pseudovariety

V

is aperiodic if

V



A

and

V

is non-permutative if it contains some

non-permutative semigroup.

Lemma 2.1

[13]. The following are equivalent for an aperiodic pseudovariety

V

of semigroups: i)

V



DA

; ii) P

V



A

; iii) B 2 = 2

V

.

Lemma 2.2

[6, Lemma 11.6.4]. B 2 2P

V

(Q).

Lemma 2.3

[6, Prop. 11.6.7]. B 2 2P

V

(Y).

Lemma 2.4

[6, Lemma 6.5.14]. A pseudovariety

V

does not contain Q if and only if it satis es

at least one of the following pseudoidentities:

exeye=exye or (exe) !+1 =

exe:

Lemma 2.5

[6, Prop. 11.8.1]. A pseudovariety

V

does not contain Y if and only if it satis es at

least one of the following pseudoidentities: (exf) !+1 = exf; exf(ef) ! = exf or (ef) ! exf =exf:

Using Lemmas 2.4 and 2.5, it is now easy to establish the following result.

Proposition 2.6.

Let

V

be a pseudovariety contained in

A

. Then neither Y nor Q lie in

V

if

and only if

V

satis es at least one of the following pseudoidentities: (exf)

2 =

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Proof.

Taking f =einexeyf =exyf, we obtain exeye=exye, while taking x =xf and y =f,

yields exfef =exf. Conversely, assuming the pseudoidentities exeye= exye and exfef = exf,

we deduce

exeyf =e(xey)f =exeyfef =exyfef =exyf:

Hence the pseudoidentity exeyf = exyf is equivalent to the conjunction of the pseudoidentities exeye = exye and exfef = exf. Since (exf)

2 =

exf implies exexe = exe, taking into account

the six possibilities resulting from chosing a pseudoidentity from each of Lemmas 2.4 and 2.5, the aperiodicity of

V

assumed in the hypothesis, and left-right duality, it remains to consider the case of the conjunction

exexe=exe and exfef =exf:

Now, these two pseudoidentities imply

exfexf =e(xf)e(xf)ef =exfef =exf:

The last two cases in Proposition 2.6 have been identi ed recently by Azevedo and Zeitoun [8], thereby con rming the guess in [6, Problem 24(b)].

Proposition 2.7

[Azevedo and Zeitoun]. a) M

K

_

D

= [[exeyf =exyf;x !+1= x !]] = [[ x ! yxzf =x ! yzf]]; a)

K

_M

D

= [[exfyf =exyf;x !+1= x !]] = [[ eyxzx ! = eyzx !]].

In these two dual cases, we establish in the following result that the powers remain within

DA

.

Proposition 2.8.

The pseudovariety P[[x !

yxzf = x !

yzf]] is contained in

A

and satis es all

pseudoidentities of the form

u ! ye=v ! ye with c(u) =c(v) and jj x= 1 for allx2c(u). (1) In particular, it is contained in

DA

.

Proof.

Let

V

= [[x ! yxzf =x !

yzf]]. To show thatP

V



A

, it suces, by Lemma 2.1, to observe

that

V



DA

. Indeed, by Proposition 2.7(a), since

K

_

D



DA

, we haveM

K

_

D

M

DA

=

DA

.

We proceed to show that, if S 2

V

, then P(S) satis es every pseudoidentity of the form (1).

Given an implicit operation  such that jj x =

1, evaluating the variables from c() in the

semigroupP(S), for every chosen elementsof the resulting set for, we can factorize it repeating n = jSj times the same element t of x. In view of [6, Lemma 7.2.4] applied to the monoid S

1,

which satis es the pseudoidentityexeye=exye,scan be factorized witht

n as a factor. Using the

pseudoidentity de ning

V

, we deduce that

P(S)j=yexye whenever jj x=

1. (2)

Hence, ifjj x =

1 for allx2c(u) =c(v), thenP(S) satis es u ! yev ! u ! yev ! u ! v ! ye:

To be able to add the equalityv ! u ! v ! ye=v !

yeto the above chain of inclusions, it remains to

establish that P(S) 2

DA

, which is a particular case of P(S) j= u !

ye= v !

ye as in (1), but

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Remark 2.9.

For every T 2

DA

, P(T) satis es the inclusion (xy) ! (xy) !( yx) !( xy) !.

Proof.

Let x;y 2 P(T) and suppose that n is a positive integer such that P(T) j= x ! = x n. Take s 1 ;:::;s n 2 x and t 1 ;:::;t n

2 y. By [6, Proposition 5.4.1(a)], there exist i and j such that

1<ij <nand s 1 t 1 s n t n= s 1 t 1 s i?1 t i?1( s i t i s j t j) ! s j+1 t j+1 s n t n : Hence, sinceT 2

DA

, s 1 t 1 s n t n= s 1 t 1 s i?1 t i?1( s i t i s j t j) !( t j s i t i s j) !( s i t i s j t j) ! s j+1 t j+1 s n t n showing that s 1 t 1 s n t n 2(xy) !( yx) !( xy) ! in view of Lemma 2.1. 

Back to the proof of Proposition 2.8, it remains to verify that

P(S)j= (xy) !( yx) !( xy) ! (xy) ! :

By (2), P(S) satis es the inclusions

(xy) !( yx) !( xy) ! (xy) !( yx) ! y(xy) ! (xy) ! x(yx) ! y(xy) ! = ( xy) !

by aperiodicity. This shows that P(S)2

DA

.

For the purposes of the main result of this paper, we might prefer to show only thatP[[x !

yxzf = x

!

yzf]] 

DA

. We chose to give a better upper bound since it will also be used below in a side

remark and since it is in fact sharp. For a proof, it suces to adapt the arguments in [6, section 11.7] which will, more generally, provide the values of P

V

for all

V

[[x

!

yxzf =x !

yzf]].

3. The semigroup

I

and its identities

To proceed with the characterization of all pseudovarieties

V

such that P

V



DA

, we must still

determine necessary and sucient conditions for

V

 [[(exf) 2 =

exf]] to satisfy P

V



DA

. We

rst note that clearly [[(exf) 2=

exf]]

A

. Then, say by [6, Exercise 6.5.13] together with Lemma

2.5, [[(exf) 2 =

exf]]

DA

.

Consider next the semigroup

I =he;s;t; e 2 = e; es=s; s 2 = se;et=st;te=ts=t 2 = ti:

This is a six-element semigroup whose structure, in terms of Green's relations, is described by the following diagram, where a marks a group H-class:

set  st  t  se  s e 

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The semigroup I is also obtained as the syntactic semigroup of the language a  bfa;bg + cA  over

the alphabetA=fa;b;cg whose minimal (trim) automaton is the following:

a,b,c a,b c a b a,b

Lemma 3.1.

P

V

(I)*

DA

.

Proof.

We must show that some generator P(II) of the pseudovariety P

V

(I) fails the

inclusion (xy) !( yx) !( xy) !  (xy)

! (cf. Lemma 2.1 and Remark 2.9). In fact, this inclusion fails

forP(I I) under the evaluation

x=f(e;e)g and y=f(e;e);(s;t);(t;s)g:

Indeed, a little calculation yields (xy)

! =

f(e;e);(s;st);(st;s);(se;st);(set;st);(st;se);(st;set)g

and so (st;st)2= (xy) !, while (st;st) = (e;e)(e;e)(e;e)(s;t)(t;s)(e;e)(e;e)(e;e)(e;e)(e;e) 2 (xy) !( yx) !( xy) ! :

By Lemma 3.1, I is excluded from every pseudovariety

V

such that P

V



DA

. Note that I satis es the pseudoidentity xexe = xe and, therefore, also (exf)

2 =

exf. We proceed by

es-tablishing for the semigroup I a result similar to Lemmas 2.4 and 2.5, which deal with Q and Y,

respectively.

Lemma 3.2.

The semigroup I satis es the identities x 3 = x 2 and xyx=xy 2 :

Proof.

This amounts to straightforward calculations.

We next view the identities x 3 =

x 2 and

xyx = xy

2 as word rewrite rules x

3 ! x

2 and xyx ! xy

2, where the second rule is only applied in case

x 6= y. It is then easy to show that

these rules constitute a Noetherian locally con uent system so that, by applying them successively in any order, every word is eventually rewritten in a unique canonical form (cf. [6, Prop. 4.2.6]). Moreover, the words in canonical form are precisely those which do not have a factor of the form

xuxwherexis a letter anduis a nonempty word. Put in another way, the words in canonical form

are those of the formx "1 1 :::x "r r where the x i (

i= 1;:::;r) are distinct letters and each " i

2f1;2g.

Denote the canonical form of a word wby r(w).

The subsemigroup fe;st;setg of I is a three-element band monoid whose minimal ideal is a

two-element left-zero semigroup. It is well known that such a band generates the pseudovariety

LeRegB

= [[xyx=xy;x 2=

x]];

of all nite left regular bands, and that an identity u = v holds in

LeRegB

if and only if the

leftmost occurrences of variables in u and v appear in the same order (see, e.g., [10]). Denoting

by F(w) the word obtained from a given wordw by retaining only the leftmost occurrence of each

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Lemma 3.3.

If the semigroup I satis es an identity u=v, then F(u) =F(v).

Proposition 3.4.

The following conditions are equivalent for an identity u=v:

i) I j=u=v; ii) fx 3 = x 2 ;xyx=xy 2 g`u=v; iii) r(u) =r(v).

Proof.

Since, by de nition of the reduction rules, fx 3 = x 2 ;xyx = xy 2 g ` u = r(u), we have

(iii))(ii). On the other hand, (ii))(i) follows from Lemma 3.2. So, it remains to establish that

(i))(iii) for which we may, by the above, assume that u and v are words in canonical form such

thatI j=u =v. We have to show that u=v.

By Lemma 3.3, the rst occurrences of variables in u=r(u) andv =r(v) must appear in the

same order. Henceu=x " 1 1 :::x "r r and v =x  1 1 :::x r r with the x i (

i= 1;:::;r) distinct letters and " i ; i 2 f1;2g (i = 1;:::;r). If, say, " j <  j for a given j  r, then evaluate x j by s,x i by efor i<j, andx i by

tfor i>j. This yieldsst(respectively s) for u and set (respectivelyse) forv in

case j<r (respectively j=r). Hence " j =  j ( j= 1;:::;r) and sou=v.

Corollary 3.5. V

(I) = [[x 3= x 2 ;xyx=xy 2]]. 

Lemma 3.6.

Let w be a word and let x2 c(w). Then x appears in r(w) with exponent 2 if and

only if

i) the rst occurrence of x in w is not the last letter of w; and

ii) the letter which immediately follows the rst occurrence ofx does not occur there for the rst

time.

Proof.

Recall that I j=w=r(w).

()) Suppose that (i) fails. Then, evaluating inI the letterxby sand all other letters bye, we

obtain the valuesforwand se forr(w). Hence (i) holds. On the other hand, if (ii) fails, then the

letter y in question (which exists by (i)) must be di erent fromx. So evaluating inI the letter x

by s, y by t, and all other letters by e, we obtain the value stfor w and set for r(w). Hence (ii)

holds.

(() Supposexoccurs only once inr(w). Ifxis the last letter ofr(w), then, evaluating inI the

letter xby sand all other letters bye, we obtain the valuesforr(w) and, by (i), se forw. Hence xis not the last letter of r(w). Lety be the letter which follows xinr(w). Evaluate inI the letter x by s,y by tand all other letters by e to obtain the valuest forr(w). However, by Lemma 3.3,

the rst occurrences of letters must be found in the same order in w and r(w). Hence the value

forw isset. This shows that x must occur twice inr(w).

4. Some technical lemmas

For the sequel, we will need some technical lemmas.

Lemma 4.1.

The pseudoidentity exexf = exf implies (exf) 2 =

exf and it is equivalent to exexyf =exyf.

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Proof.

Taking x = xf in exexf = exf gives exfexf = exf. On the other hand, assuming the

pseudoidentity exexf =exf, we obtain exexyf = exe(xy)f

= exexyfexyf exyf::: sinceexyf is idempotent

= exex(yfex) ! yf in fact, (yfex) != ( yfex) 2 = ex(yfex) !

yf by the pseudoidentity exexf =exf

= exyfexyfexyf as above

= exyf:

The converse is obtained by taking y=f.

Lemma 4.2.

Let u = v be an identity with c(u) 6= c(v). Then f(exf) 2 =

exf;u = vg implies exexf =exf =exfxf.

Proof.

Suppose that the variable y occurs in u but not in v. Substitutef fory and efor every

other variable, multiply both sides on the left and on the right bye, and use the fact that ef and feare idempotents to obtain the pseudoidentity efe=e. Now (exf)

2=

exf and efe=e imply exexf =eexeexf =exf

sinceexe is an idempotent. Dually, exfxf =exf.

Lemma 4.3.

Letu=vbe an identity such thatF(u)6=F(v). Thenf(exf) 2 =

exf;u=vgimplies exexf =exf.

Proof.

By Lemma 4.2, we may assume thatc(u) =c(v). Consider then two variablesxandysuch

that the rst occurrence of y comes before the rst occurrence of x in u but not inv. Substitute exf for y, ex for x, ande for every other variable in u = v, multiply both sides on the right by exf, and use the facts thatexe and exf are idempotents andexfwexf =exf for every wordwin e,x and f (since [[(exf)

2 =

exf]]

DA

), to obtain exf for the left side and exexf for the right

side, i.e., the pseudoidentityexexf =exf.

Proposition 4.4.

Let u = v be an identity such that r(u) 6= r(v). Then f(exf) 2 =

exf;u =vg

implies exexf =exf.

Proof.

By Lemma 4.3, we may assume that F(u) = F(v) which implies that r(u) = x " 1 1 :::x "r r and r(v) =x  1 1 :::x r r where the x i (

i= 1;:::;r) are distinct letters and each" i

; i

2f1;2g. Since r(u) 6= r(v), there is some j 2 f1;:::;rg such that "

j

6

= 

j. Without loss of generality, we may

assume that "

j = 1 and 

j = 2. Then by Lemmas 3.6 and 3.3, the rst occurrence of x

j in v is

not the last letter of v and the letter which immediately follows it in v is not x

j+1, whereas, for u

either

i) the rst occurrence ofx

j is the last letter of u, or

ii) the letter which immediately follows the rst occurrence of x j in

u isx j+1.

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In case (i), where j=n, substituteexfor x j and

efor every other letter in u=v to obtain ex

for u and either exe orexex for v (in both cases taking into account that exe is an idempotent).

Since exe is an idempotent, ex = exe implies ex = exex. Thus, in both cases, we deduce the

pseudoidentity exf =exexf.

So, we may assume that (i) fails and (ii) holds. Then, substituting ex forx j,

f for x

j+1 and e

for every other letter in u =v, and multiplying both sides on the right byexf, we obtain one of

the pseudoidentities exf =exexf orexf =exef. In the second case, we have exf =exef=exexef =e(xex)ef =exexf:

Lemma 4.5.

[[exeyxfyf =exyf]]

DA

.

Proof.

Taking in the pseudoidentity exeyxfyf =exyf,e=y=f =x

!, we obtain x !+2 = x !+1 and, thereforex !+1= x !. Taking in (3) e=f = (xy) ! and y=y(xy) ! gives (xy) ! = ( xy) ! xy(xy) ! (xy) ! = ( xy) ! x(xy) ! y(xy) ! x(xy) ! y(xy) ! = (xy) ! x(xy) !( yx) !( xy) ! y(xy) ! :

This shows that, in the compact semigroup A[[

exeyxfyf =exyf]], (xy) ! J(xy) !( yx) !( xy) ! and, therefore, (xy) ! H(xy) !( yx) !( xy)

! (cf. [6, Exercise 5.1.3]) which implies ( xy) != ( xy) !( yx) !( xy) !

by aperiodicity. This proves the lemma.

Lemma 4.6.

The set of pseudoidentities

fexexf =exf =exfxf;efege=efgeg

is equivalent to the single pseudoidentity exeyxfyf =exyf.

Proof.

Assume rst the pseudoidentities in the set. Then

exyf = exyfexyf by Lemma 4.1

= e(ex) !( yf) ! fe(ex) !( yf) !

f by Lemma 4.1 and its dual

= e(ex) ! e(yf) ! fe(ex) ! f(yf) ! f using efege=efge = exeyfexfyf = (ex) !( ey) ! f(ex) ! fyf = (ex) !( ey) !( ex) ! f(ex) ! fyf usingefege=efge

= exeyexfexfyf

= exeyexfyf:

Dually, exyf =exeyfxfyf and so exyf = exyf exyf

= exeyexfyf exeyfxfyf = e(exey) ! e(xfyf) ! fe(exey) ! f(xfyf) ! f = e(exey) !( xfyf) ! fe(exey) !( xfyf) !

f using efege=efge

= exeyxfyfexeyxfyf

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and this completes half of the proof.

For the converse, assume the pseudoidentity

exeyxfyf =exyf: (3)

Taking y=f and x=ex, or y=eyields, respectively,

exefexf = exf (4)

exexfef = exef: (5)

Next, taking in (4) x = xe gives that exef is an idempotent. Hence, by (4), (5) and Lemma

4.5, exf, exef and exexf are all R-equivalent idempotents (cf. [6, Exercise 5.1.3]). Since exf is

also L-equivalent to exexf, we obtain the pseudoidentity exexf =exf. Dually, exfxf =exf.

Finally, taking in (3) x=f,y=gand f =egives

efegfege=efge (6) and so efge = efegfege = efegefgefege by (6) again = efegeefgeefege = efege

sinceefege efgeare J-equivalent idempotents by Lemma 4.5.

Corollary 4.7.

The pseudoidentity exeyxfyf =exyf implies the pseudoidentity (exf) 2=

exf.

Lemma 4.8.

The power pseudovariety P[[exeyxfyf =exyf]]satis es the pseudoidentity

(exey) ! exyf(xfyf) != ( exey) !( xfyf) ! : (7)

Proof.

LetS be a nite semigroup satisfyingexeyxfyf =exyf. By Lemmas 4.6, 4.1 and its dual, S also satis es

exexyf = exyf (8)

exyfyf = exyf: (9)

We rst show that P(S) satis es the inclusion

exyf exeyxfyf: (10)

Indeed, givenA;X;Y;B 2P(S) andnjSjsuch thatP(S)j=x ! = x n, if a 1 ;:::;a n 2A,x2X, y 2Y, and b 1 ;:::;b n

2B, then (cf. [6, Prop. 5.4.1(a)]) there arei;j;k;l such that 1<ij<n,

1<k l<n, and a 1 a n = a 1 a i?1( a i a j) ! a j+1 a n b 1 b n = b 1 b k?1( b k b l) ! b l +1 b n :

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Hence a 1 a n = 1 e 2 and b 1 b n = 1 f 2 with 1 ;e; 2 2 A !, 1 ;f; 2 2 B !, and e and f

idempotents. We can now write

a 1 a n xyb 1 b n = 1 e 2 xy 1 f 2 = 1( e 2) ! xy( 1 f) ! 2 by (8) and (9) = 1( e 2) ! x(e 2) ! yx( 1 f) ! y( 1 f) ! 2 by (3)

and this shows that A ! XYB ! A ! XA ! YXB ! YB

!, where we are using freely that

P(S)2

A

.

Hence P(S) satis es the inclusion (10).

Using (10), we nd that P(S) satis es

(exey) !( xfyf) ! = ( exey) ! exeyxf yf(xfyf) ! by aperiodicity  (exey) ! exeyexyfxfyf(xfyf) ! = (exey) ! exyf(xfyf) ! again by aperiodicity  (exey) ! exeyxf yf(xfyf) ! = (exey) !( xfyf) ! ;

and so P(S) also satis es (7).

Lemma 4.9.

The power pseudovariety P[[exeyxfyf = exyf]] satis es all pseudoidentities of the

form (u ! vw !)! = ( u ! w !)! with c(v)c(u)\c(w): (11)

Proof.

Let S be a nite semigroup satisfying exeyxfyf = exyf. Evaluate the letters of c(uw)

to elements ofP(S) and denote by U,V and W the resulting values respectively for u,v and w.

Take n jSj such that P(S) j=x ! = x n and let a 1 ;:::;a n 2 U and c 1 ;:::;c n 2 W. As usual, we may write a 1 a n = 1 e 2, c 1 c n= 1 f 2 with 1 ;e; 2 2U !, 1 ;f; 2 2W !, and eand f

idempotent. This gives

a 1 a n c 1 c n = 1 e 2 1 f 2 = 1 e 2 1 fe 2 1 f 2 by Corollary 4.7 = 1( e 2) !( 1 f) !( e 2) !( 1 f) ! 2 by Lemma 4.6 = 1( e 2) ! e( 1 f) !( e 2) ! f( 1 f) ! 2 also by Lemma 4.6 :

Now, by Lemma 4.5 and since c(v)c(u)\c(w), we claim that there are b 0 ;b 00 2V such that e=eb 0 e and f =fb 00

f. Indeed, the condition c(v)c(u) guarantees that there is a productb 0 in V made up of factors which lie J-above e. By [6, Lemma 8.1.4], eb

0

eJeand soeb 0

e=esince the J-class ofe is a rectangular band. This establishes the claim concerning the existence of b

0; the

existence ofb

00 is proved similarly.

Coupling with the above calculations, we obtain

a 1 a n c 1 c n = 1( e 2) ! eb 0 e( 1 f) !( e 2) ! fb 00 f( 1 f) ! 2 = 1 ? (e 2) ! eb 0  ! e( 1 f) !( e 2) ! f ? b 00 f( 1 f) !  ! 2 = 1 ? (e 2) ! eb 0  ! ( 1 f) !( e 2) ! ? b 00 f( 1 f) !  ! 2 by Lemma 4.6 = ( 1 e 2 e)b 0( 1 f) !( e 2) ! b 00( f 1 f 2) 2 U ! VW ! U ! VW ! :

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Hence U ! W ! (U ! VW !)2 and so ( U ! W !)! (U ! VW !)!.

For the reverse inclusion, with the same notation as above and b 2 V, we rst claim that ebe=eae and fbf =fcf for some a2U

jvj and c2W

jvj. To prove this, write v=x 1 :::x n, with thex i 2c(v), and letb=b 1 b

nbe the corresponding factorization of

bwith each b

i belonging to

the subset of S to whichx

i evaluates. Since x

i

2c(u), there are products 1i and

2i of elements

which lie J-above e such that a i = 1i b i 2i

2 U. Again by Lemma 4.5 and [6, Lemma 8.1.4], e 1i e=e 2i e=eand so eb i e = e 1i eb i e 2i e = e 1i b i 2i

e using efege=efge

= ea i

e:

Hence, using the pseudoidentity efege=efge, ebe = eb 1 eeb n e = ea 1 eea n e = e(a 1 a n) e;

which establishes the claim concerning the existence of a; the claim for c is proved similarly. The

remainder of the proof is now obtained by an argument similar to the proof of the direct inclusion. The details are left to the reader.

Corollary 4.10.

P[[exeyxfyf =exyf]]

DA

.

Proof.

Takeu=w=xyandv= (yx)

nin (11) to obtain (( xy) !( yx) n( xy) !) ! = (xy) !for all n1. Hence ((xy) !( yx) !( xy) !) ! = (xy)

! which, together with aperiodicity (which follows from Lemmas

4.5 and 2.1), implies (xy) !( yx) !( xy) ! = ( xy) !. 

We recall in the following diagram part of the bottom of the lattice of pseudovarieties of bands (cf. [10, 6]).

I

Sl

=[[x 2 = x;xy=yx]]

K

1= [[ xy =x]]

D

1 = [[ xy=y]] [[x 2= x;xyz=yxz]]

NB

=

B

\

Perm

LeRegB

=M

K

1 M

D

1 =

RiRegB

[[x 2 = x;xyz=xyxz]]

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From the description of the lattice of pseudovarieties of bands, it follows that, if a pseudovariety

V

of bands is not contained in the pseudovariety

RegB

= [[x 2 =

x;efege=efge]]

of all niteregular bands, then

V

contains either the pseudovariety

LeSNB

= [[x 2 = x;x 3 x 1 x 2= x 3 x 1 x 2 x 3 x 2]] ;

of all nite left semi-normal bands, or its dual. For a word w, denote by s(w) the pre x of w up

to, but excluding, the last occurrence for the rst time of a letter.

The next lemma, as well as its proof, is a slight extension of [6, Lemma 11.10.9].

Lemma 4.11.

P

LeSNB

*

DA

.

Proof.

LetS be the

LeSNB

-free semigroup over the setfa;b;cg. Evaluatex=fag,y=fb;cgto

obtain w=abcab= (ab)(ca)(ab)2(xy) !( yx) !( xy) ! :

On the other hand, (xy)

! is the subsemigroup of

S generated byabandac, so that, foru2(xy) !, s(u)2fa;aba;acag while s(w) =ab. Hence w2= (xy)

! [10] and ( xy) !( yx) !( xy) ! 6 = (xy) !. 

Corollary 4.12.

If

V

satis es (ef) 2=

ef and P

V



DA

, then

V

also satis es efege=efge.

Proof.

ForS2

V

, the idempotents form a bandE(S). By Lemma 4.11, its dual, and the remarks

preceding it,E(S) must be a regular band, which means precisely thatS satis esefege=efge.

5. Main results

We can now establish the following.

Theorem 5.1.

Let

V

be a pseudovariety of semigroups. Then

PV



DA

if and only if

V

is

contained in at least one of M

K

_

D

,

K

_M

D

and [[exeyxfyf =exyf]].

Proof.

(() follows from Propositions 2.7 and 2.8 forM

K

_

D

, the case of

K

_M

D

being dual.

For the pseudovariety [[exeyxfyf =exyf]], it suces to apply Corollary 4.10.

()) By Proposition 2.6, in view of Lemmas 2.1, 2.2 and 2.3,

V



A

and

V

satis es one of the

pseudoidentities

exeyf =exyf; exfyf =exyf and (exf) 2 =

exf:

In the rst two cases we obtain, respectively,

V

 M

K

_

D

and

V



K

_M

D

by Proposition

2.7. So, we may assume that

V

satis es (exf) 2 =

exf. On the other hand, by Lemma 3.1, I 2=

V

and so, by Propositions 3.4 and 4.4 (writing

V

as the union of equational pseudovarieties [6, Prop. 3.2.4]),

V

must satisfy the pseudoidentityexexf =exf. Dually,

V

must also satisfyexfxf =exf.

By Corollary 4.12,

V

satis es efege = efge. By Lemma 4.6, we deduce that

V

is contained in

[[exeyxfyf =exyf]], as desired.

Actually, the inclusionP

V



DA

is always strict as is shown in the following result.

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Proof.

By Theorem 5.1, if P

V



DA

, then

V

must be contained in one of the pseudovarieties M

K

_

D

,

K

_M

D

and [[exeyxfyf = exyf]]. By Proposition 2.8, its dual, and Lemma 4.9, we

deduce that P

V

satis es, respectively, one of the pseudoidentities

(xy) !( yx) ! e= (xy) ! e; e(xy) !( yx) != e(yx) ! and (( xy) ! x(zx) !) ! = ((xy) !( zx) !) ! :

By [7, Prop. 3.4], each of these pseudoidentities fails in

DA

. Hence P

V

6=

DA

.

For a pseudovariety

V

, let its exponent be the least integer n  0 such that P

n

V

=

S

is

the pseudovariety of all nite semigroups, where we de ne P

0

V

=

V

and, recursively, P

n+1

V

= P(P

n

V

). By [6, Theorem 11.6.8], if

V

is a non-permutative pseudovariety, then

V

has exponent at

most 3. (On the other hand, if

V

is permutative, i.e., if each member of

V

satis es some nontrivial permutation identity, then the exponent of

V

does not exist. A much more precise result may be found in [6, Theorem 11.5.6].) By Lemma 2.1,

DA

has exponent at least 2.

Theorem 5.3.

Let

V

be a non-permutative pseudovariety of aperiodic semigroups. Then: a)

V

does not have exponent 0;

b)

V

has exponent 1 if and only if B 2

2

V

;

c)

V

has exponent 2 if and only if B 2

=

2

V

and at least one of the semigroups Y, Q, I, I ,

3

LeSNB

and (3

LeSNB

)

 belongs to

V

;

d)

V

has exponent 3 if and only if none of the semigroups in (c) belong to

V

.

In particular, the exponent of a decidable non-permutative aperiodic pseudovariety is computable.

Proof.

(a) is trivial and (b) follows from Lemma 2.1.

By (b), Lemmas 2.2, 2.3 and 3.1 and its dual, 4.11 and its dual, if B 2

=

2

V

and at least one of

the semigroups listed in (c) lies in

V

, then

V

has exponent 2. On the other hand, if none of those semigroups lies in

V

then, by the proof of the direct part of Theorem 5.1,

V

is contained in at least one of the pseudovarietiesM

K

_

D

,

K

_M

D

and [[exeyxfyf =exyf]]. Then, by the reverse

part of Theorem 5.1, P

V



DA

, so that P 2

V

6

=

S

by (b). This establishes (c) and (d) since

V

has exponent at most 3.

For the computability statement, it suces to observe that in all cases the criterion has been expressed in terms of testing membership in

V

for nitely many semigroups.

Note that, since permutativity of an aperiodic pseudovariety has also been characterized in terms of exclusion of ( ve) speci c semigroups (cf. [6, Theorem 6.5.18]), it is also decidable, for an aperiodic decidable pseudovariety

V

, whether

V

is permutative.

6. Final comments and open problems

The power pseudovariety P

V

(Q) has been computed in [4] andP

LeRegB

was determined in [5].

The calculation of P

V

(Y) amounts to solving the dot-depth two problem [3] which remains open.

Both P

V

(I) andP

LeSNB

also remain unknown.

It would also be interesting to describe all powers of pseudovarieties satisfying the pseudoidentity

exeyxfyf = exyf. Finally, the extension of the determination of power exponents to the

non-aperiodic case, which is left open, should lead to a deeper understanding of the behaviour of the power operator.

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Acknowledgment.

The author wishes to express his thanks to Mario Petrich for his comments on a preliminary version of this paper. For the discovery of the semigroupI and for various calculations

involving it, some software tools were very helpful, namely Amore [14] and the Mathematica [21] package Automata [20].

References

1. J. Almeida, Power pseudovarieties of semigroups II, Semigroup Forum

33

(1986) 375{390. 2. ,On power varieties of semigroups, J. Algebra

120

(1989) 1{17.

3. ,The equation

PX

=

PJ

, in Semigroup and its Related Fields, M. Yamada and H. Tominaga, eds., vol. 1-11, Matsue University, 1990.

4. ,Locally commutative power semigroups and counting factors of words, Theor. Comp. Sci.

108

(1993) 3{16.

5. ,A classi cation of aperiodic power monoids, J. of Algebra

170

(1994) 355{387. 6. ,Finite Semigroups and Universal Algebra, World Scienti c, Singapore, 1995. English

translation.

7. , A syntactical proof of locality of DA, Int. J. Algebra and Computation

6

(1996) 165{177.

8. A. Azevedo and M. Zeitoun,Implicit operations and join computation: some examples, Semi-group forum. To appear.

9. M. N. Bleicher, H. Schneider, and R. L. Wilson,Permanence of identities on algebras, Algebra Universalis

3

(1973) 72{93.

10. J. A. Gerhard and M. Petrich,Varieties of bands revisited, Proc. London Math. Soc.

58

(1989) 323{350.

11. K. Henckell and J. Rhodes, The theorem of Knast, the PG=BG and Type II Conjectures, in Monoids and Semigroups with Applications, J. Rhodes, ed., Singapore, 1991, World Scienti c, 453{463.

12. E. S. Ljapin,Identities valid globally in semigroups, Semigroup Forum

24

(1982) 263{269. 13. S. W. Margolis, On M-varieties generated by power monoids, Semigroup Forum

22

(1981)

339{353.

14. O. Matz, A. Miller, A. Potho , W. Thomas, and E. Valkema,Report on the program AMoRe, Tech. Rep. 9507, Christian Albrechts Universitat, Kiel, 1995.

15. J.-E. Pin,Varieties of Formal Languages, Plenum, London, 1986. English translation.

16. ,BG=PG: A success story, in Semigroups, Formal Languages and Groups, J. Fountain, ed., vol. 466, Dordrecht, 1995, Kluwer, 33{47.

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17. J.-E. Pin and H. Straubing, Monoids of upper triangular matrices, in Semigroups: structure and universal algebraic problems, G. Pollak, ed., Amsterdam, 1985, North-Holland, 259{272. 18. J. Reiterman,The Birkho theorem for nite algebras, Algebra Universalis

14

(1982) 1{10. 19. A. Shafaat,On varieties closed under the construction of power algebras, Bull. Austral. Math.

Soc.

11

(1974) 213{218.

20. K. Suttner,Automata, 1994. Mathematica package, version 1.2.

21. S. Wolfram, Mathematica: a system for doing Mathematics by computer, Addison-Wesley, Reading, Mass., second ed., 1991.

References

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