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 dened classes (i.e., Birkho's varieties) instead of individual algebraic structures and to dene 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 satised 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 specic 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 pseudovarietyV
is then the numbernof times the power operator needs to be appliedto 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
withV
a pseudovariety of aperiodic monoids were identied (without actuallyThis 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.
\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 undened 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
, PV
denotes thepseudovariety generated by the class consisting of all power semigroupsP(S) withS2
V
.For a pseudovariety
V
and a nonempty setA, anA-ary implicit operation onV
is aV
-indexedfamily ( 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 allA-ary implicit operations on
V
is denotedby A
V
and constitutes a semigroup under pointwise multiplication. For eacha2A, there is an
associated projection (orvariable) adened 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 AV
and is theV
-free semigroup on A, itselements being called A-ary explicit operations on
V
. SinceA
V
is completely determined byV
and jAj, we write
n
V
for AV
withjAj=n.
The most common example of implicit operation which is (usually) not explicit is the unary operation x ! dened 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 AV
! SwithS2
V
, where nite semigroups are taken with the discrete topology. The subsemigroup AV
is dense in A
V
(cf. [18, 6]).A pseudoidentity is a formal equality = where ; 2
n
S
for somen 1,
S
denotingthe 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 consistingof the duals of the semigroups in
V
. Thedual of a word wis the word wobtained 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.
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 S2V
V
(S) = pseudovariety generated bySB
= [[x 2 = x !]] = fnite bandsgA
= [[x !+1 = x !]] =fnite 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 ifV
A
andV
is non-permutative if it contains somenon-permutative semigroup.
Lemma 2.1
[13]. The following are equivalent for an aperiodic pseudovarietyV
of semigroups: i)V
DA
; ii) PV
A
; iii) B 2 = 2V
.Lemma 2.2
[6, Lemma 11.6.4]. B 2 2PV
(Q).Lemma 2.3
[6, Prop. 11.6.7]. B 2 2PV
(Y).Lemma 2.4
[6, Lemma 6.5.14]. A pseudovarietyV
does not contain Q if and only if it satisesat least one of the following pseudoidentities:
exeye=exye or (exe) !+1 =
exe:
Lemma 2.5
[6, Prop. 11.8.1]. A pseudovarietyV
does not contain Y if and only if it satises atleast 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.
LetV
be a pseudovariety contained inA
. Then neither Y nor Q lie inV
ifand only if
V
satises at least one of the following pseudoidentities: (exf)2 =
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 conjunctionexexe=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 identied recently by Azevedo and Zeitoun [8], thereby conrming the guess in [6, Problem 24(b)].
Proposition 2.7
[Azevedo and Zeitoun]. a) MK
_D
= [[exeyf =exyf;x !+1= x !]] = [[ x ! yxzf =x ! yzf]]; a)K
_MD
= [[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 satises allpseudoidentities 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.
LetV
= [[x ! yxzf =x !yzf]]. To show thatP
V
A
, it suces, by Lemma 2.1, to observethat
V
DA
. Indeed, by Proposition 2.7(a), sinceK
_D
DA
, we haveMK
_D
MDA
=DA
.We proceed to show that, if S 2
V
, then P(S) satises 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 satises the pseudoidentityexeye=exye,scan be factorized witht
n as a factor. Using the
pseudoidentity dening
V
, we deduce thatP(S)j=yexye whenever jj x=
1. (2)
Hence, ifjj x =
1 for allx2c(u) =c(v), thenP(S) satises 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
Remark 2.9.
For every T 2DA
, P(T) satises 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 n2 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) satises 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 sideremark 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 allV
[[x!
yxzf =x !
yzf]].
3. The semigroup
Iand its identities
To proceed with the characterization of all pseudovarieties
V
such that PV
DA
, we must stilldetermine necessary and sucient conditions for
V
[[(exf) 2 =exf]] to satisfy P
V
DA
. Werst note that clearly [[(exf) 2=
exf]]
A
. Then, say by [6, Exercise 6.5.13] together with Lemma2.5, [[(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
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.
PV
(I)*DA
.Proof.
We must show that some generator P(II) of the pseudovariety PV
(I) fails theinclusion (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 PV
DA
. Note that I satises 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 satises 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 theleftmost 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
Lemma 3.3.
If the semigroup I satises 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 denition 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 andonly 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 dierent 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.
Proof.
Taking x = xf in exexf = exf gives exfexf = exf. On the other hand, assuming thepseudoidentity 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 everyother 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 variablesxandysuchthat 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 rightside, 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.
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 pseudoidentitiesfexexf =exf =exfxf;efege=efgeg
is equivalent to the single pseudoidentity exeyxfyf =exyf.
Proof.
Assume rst the pseudoidentities in the set. Thenexyf = 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
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]]satises 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 satisesexexyf = exyf (8)
exyfyf = exyf: (9)
We rst show that P(S) satises 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 :
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) satises the inclusion (10).
Using (10), we nd that P(S) satises
(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 satises (7).
Lemma 4.9.
The power pseudovariety P[[exeyxfyf = exyf]] satises all pseudoidentities of theform (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 ! :
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
=MK
1 MD
1 =RiRegB
[[x 2 = x;xyz=xyxz]]From the description of the lattice of pseudovarieties of bands, it follows that, if a pseudovariety
V
of bands is not contained in the pseudovarietyRegB
= [[x 2 =x;efege=efge]]
of all niteregular bands, then
V
contains either the pseudovarietyLeSNB
= [[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 prex 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.
PLeSNB
*DA
.Proof.
LetS be theLeSNB
-free semigroup over the setfa;b;cg. Evaluatex=fag,y=fb;cgtoobtain 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.
IfV
satises (ef) 2=ef and P
V
DA
, thenV
also satises efege=efge.Proof.
ForS2V
, the idempotents form a bandE(S). By Lemma 4.11, its dual, and the remarkspreceding it,E(S) must be a regular band, which means precisely thatS satisesefege=efge.
5. Main results
We can now establish the following.
Theorem 5.1.
LetV
be a pseudovariety of semigroups. ThenPV
DA
if and only ifV
iscontained in at least one of M
K
_D
,K
_MD
and [[exeyxfyf =exyf]].Proof.
(() follows from Propositions 2.7 and 2.8 forMK
_D
, the case ofK
_MD
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
andV
satises one of thepseudoidentities
exeyf =exyf; exfyf =exyf and (exf) 2 =
exf:
In the rst two cases we obtain, respectively,
V
MK
_D
andV
K
_MD
by Proposition2.7. So, we may assume that
V
satises (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
satises efege = efge. By Lemma 4.6, we deduce thatV
is contained in[[exeyxfyf =exyf]], as desired.
Actually, the inclusionP
V
DA
is always strict as is shown in the following result.Proof.
By Theorem 5.1, if PV
DA
, thenV
must be contained in one of the pseudovarieties MK
_D
,K
_MD
and [[exeyxfyf = exyf]]. By Proposition 2.8, its dual, and Lemma 4.9, wededuce that P
V
satises, 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 PV
6=DA
.For a pseudovariety
V
, let its exponent be the least integer n 0 such that Pn
V
=S
isthe pseudovariety of all nite semigroups, where we dene P
0
V
=V
and, recursively, Pn+1
V
= P(Pn
V
). By [6, Theorem 11.6.8], ifV
is a non-permutative pseudovariety, thenV
has exponent atmost 3. (On the other hand, if
V
is permutative, i.e., if each member ofV
satises some nontrivial permutation identity, then the exponent ofV
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.
LetV
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 22
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 (3LeSNB
)belongs to
V
;d)
V
has exponent 3 if and only if none of the semigroups in (c) belong toV
.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 ofthe semigroups listed in (c) lies in
V
, thenV
has exponent 2. On the other hand, if none of those semigroups lies inV
then, by the proof of the direct part of Theorem 5.1,V
is contained in at least one of the pseudovarietiesMK
_D
,K
_MD
and [[exeyxfyf =exyf]]. Then, by the reversepart of Theorem 5.1, P
V
DA
, so that P 2V
6
=
S
by (b). This establishes (c) and (d) sinceV
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) specic semigroups (cf. [6, Theorem 6.5.18]), it is also decidable, for an aperiodic decidable pseudovariety
V
, whetherV
is permutative.6. Final comments and open problems
The power pseudovariety P
V
(Q) has been computed in [4] andPLeRegB
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) andPLeSNB
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.
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 calculationsinvolving it, some software tools were very helpful, namely Amore [14] and the Mathematica [21] package Automata [20].
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