On join irreducible J-trivial semigroups
Edmond W. H. Lee (∗) – John Rhodes (∗∗) – Benjamin Steinberg (∗∗∗)
Abstract – A pseudovariety of semigroups is join irreducible if whenever it is contained in the complete join of some pseudovarieties, then it is contained in one of the pseudovarieties. A finite semigroup is join irreducible if it generates a join irreducible pseudovariety. New finite J -trivial semigroups Cn (n ≥ 2) are exhibited I with the property that while each Cn is not join irreducible, the monoid Cn is join I irreducible. The monoids Cn are the first examples of join irreducible J -trivial semigroups that generate pseudovarieties that are not self-dual. Several sufficient conditions are also established under which a finite semigroup is not join irreducible. Based on these results, join irreducible pseudovarieties generated by a J -trivial semigroup of order up to six are completely described. I It turns out that besides known examples and those generated by C2 and its dual monoid, there are no further examples.
Mathematics Subject Classification (2010). 20M07, 08B15.
Keywords. Semigroup, J-trivial, pseudovariety, join irreducible.
(∗) Indirizzo dell’A.: Department of Mathematics, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, FL 33314, USA E-mail: [email protected] (∗∗) Indirizzo dell’A.: Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720, USA E-mail: [email protected] (∗∗∗) Indirizzo dell’A.: Department of Mathematics, City College of New York, NAC 8/133, Convent Avenue at 138th Street, New York, NY 10031, USA E-mail: [email protected] 2 E. W. H. Lee et al.
1. Introduction
1.1 – Join irreducibility
The class S of finite semigroups is closed under the formation of homomor- phic images, subalgebras, and finitary direct products; such a class is called a pseudovariety. Under class inclusion, the subpseudovarieties of S form a lattice, which is denoted by L (S). The lattice L (S) rose to prominence inthe1970safterEilenberg[7]andSchutze¨ nbergerestablishedanisomor- phism between L (S) and the lattice of varieties of regular languages. Since then, several theories have been developed in the study of L (S). Most no- tably,Reiterman[21]characterizedpseudovarietiesasclassesdefinedby pseudoidentities.ThisresultledtothesyntacticapproachofAlmeida[3] that has became a fundamental technique in finite semigroup theory.
The pseudovariety generated by a class {Si | i ∈ I} of finite semigroups is denoted by hhSi | i ∈ Iii. A pseudovariety V is finitely generated if it is generated by a finite class of finite semigroups, that is,
V = hhS1,S2,...,Snii = hhS1 × S2 × · · · × Snii for some S1,S2,...,Sn ∈ S. The compact elements of L (S) are precisely the finitely generated subpseudovarieties of S. The lattice L (S) is complete and join generated by its compact elements, and so is algebraic. Since L (S) is a lattice, it is natural to examine its elements that satisfy important lattice properties, such as those listed in Rhodes and Steinberg[22,Definition6.1.5].Thepresentarticleisprimarilyconcerned with pseudovarieties V that are join irreducible in the sense that for all X ⊆ L (S), the following implication holds: _ V ⊆ X =⇒ V ⊆ X for some X ∈ X . A finite semigroup S is join irreducible if the pseudovariety hhSii is join irreducible. Equivalently, S is join irreducible if and only if the class
Excl(S) = {T ∈ S | S/∈ hhT ii}, called the exclusion class of S, is a pseudovariety; in this case, the pseudo- variety Excl(S) is defined by a single pseudoidentity, called an exclusion pseudoidentity for S. It follows that the pseudovariety hhSii∩ Excl(S) coin- cides with the unique maximal subpseudovariety of hhSii. For more information on join irreducibility and other related properties, seeRhodesandSteinberg[22,Section7.3]. On join irreducible J-trivial semigroups 3
1.2 – Overview
A systematic study of join irreducible pseudovarieties was recently initiated bytheauthors[18].Thissubsectionisanoverviewofsomeresultsrelevant to the present article.
1.2.1 – Examples of join irreducible semigroups
EveryatominthelatticeL(S)isjoinirreducible[22,Table7.2].Inother words, cyclic groups p Zp = hg | g = 1i of prime order p and all semigroups of order two—the nilpotent semigroup
N2 = {0, n}, the semilattice S`2 = {0, 1}, the left zero band L2 = {e, f}, and the right zero band R2={e,f}—arejoinirreducible;seeTable1.
Z2 1 g N2 0 n S`2 0 1 L2 e f R2 e f 1 1 g 0 0 0 0 0 0 e e e e e f g g 1 n 0 0 1 0 1 f f f f e f
Table 1. Multiplication tables of semigroups of order two
n More generally, every cyclic group Zpn of prime power order p and every monogenic nilpotent semigroup n Nn = hn | n = 0i of order n ≥ 1 is join irreducible. Conversely, each join irreducible mono- genic semigroup is isomorphic to some ZpnorNn;seeCorollary4.16. Some examples of join irreducible semigroups generated by two elements include the non-orthodox 0-simple semigroup 2 2 A2 = ha, e | a = 0, aea = a, e = eae = ei oforderfive[9],theBrandtsemigroup 2 2 B2 = ha, b | a = b = 0, aba = a, bab = bi oforderfive[22,Example7.3.4],andtheJ-trivialsemigroups 2 2 n Hn = he, f | e = e, f = f, (ef) = 0i 2 2 n and Kn = he, f | e = e, f = f, (ef) e = 0i oforders4nand4n+2,respectively[14]. 4 E. W. H. Lee et al.
1.2.2 – Constructing new join irreducible semigroups
One way to locate new examples of join irreducible semigroups—and an important problem on its own—is to find operators on S that preserve join irreducibility. Given any semigroup S, let Sop denote the opposite semigroup obtained by reversing the multiplication on S, let SI = S ∪ {I} denote the monoid obtained by adjoining an external identity element I to S, and let S• denote the smallest monoid containing S, that is, ( S if S is a monoid, S• = SI otherwise. Then the operators S 7→ Sop and S 7→ S• on S preserve join irreduci- bility[3,Exercise10.10.1].TheoperatorS7→SIonS,ontheotherhand, does not preserve join irreducibility of every semigroup. For example, the I I semigroup Z2 is join irreducible but Z2 is not, since hhZ2ii = hhZ2ii∨ hhS`2ii. A more substantial operator is the augmentation Sbar of a semigroup S thatisobtainedbyaddingtoSconstantmapsonS•;seeSubsection2.1. Not only does the operator S 7→ Sbar on S preserve join irreducibility, it can also produce join irreducible semigroups from certain non-join irreducible ones. For example, the semigroup n+1 n 2 On = ha, e | a = a e = 0, ea = a, e = ei bar of order 2n + 1 is not join irreducible, while its augmentation On is join irreducible[18,Subsection4.5]. Apart from preserving join irreducibility, the operators S 7→ Sop and S 7→ Sbar on S, when iterated repeatedly, can produce infinite classes of join irreducible semigroups. For instance, by letting [ = op ◦ bar ◦ op, then starting with any join irreducible semigroup S, the pseudovarieties hhSbarii, hh(Sbar)[ii, hh((Sbar)[)barii, hh(((Sbar)[)bar)[ii,... are join irreducible and form a strictly increasing chain in the lattice L (S) [18,Corollary4.11].
1.2.3 – Classification of small join irreducible semigroups
In view of all the results, techniques, and classes of examples discovered so far, the time seems ripe for a classification of join irreducible pseudovarie- ties. Two semigroups are distinct if they are neither isomorphic nor anti- isomorphic. Given that the number ∆n of distinct semigroups of order n On join irreducible J-trivial semigroups 5 increasesveryrapidly[5,Table1],efficientmethodsarecrucialtoany classification attempt.
n 1 2 3 4 5 6 7 8
∆n 1 4 18 126 1,160 15,973 836,021 1,843,120,128
Table 2. Number ∆n of distinct semigroups of order n ≤ 8
An obvious approach is to develop sufficient conditions to determine if one of the following outcomes holds for a finite input semigroup S: • hhSii = hhT ii for some known join irreducible semigroup T ; • S is not join irreducible. A number of these conditions are equational in the sense that they solely require checking if the input semigroup S satisfies and violates certain identities;seeSubsections3.1and4.1.Ingeneral,equationalsufficient conditions are difficult to establish, but the main advantage is that they can be applied by a computer to quickly examine a large number of finite semigroups. Inarecentstudyofjoinirreduciblesemigroups[18],someequational sufficient conditions in the literature, together with a few newly established ones, were used to completely determine the join irreducibility of all pseudo- varieties generated by a semigroup of order up to five.
Theorem1.1([18,Theorem7.1]).Amongallpseudovarietiesgenerated by a nontrivial semigroup of order five or less, precisely 30 are join irreducible:
bar bar op hhZ2ii, hhZ3ii, hhZ4ii, hhZ5ii, hhZ2 ii, hh(Z2 ) ii, bar bar op hhN2ii, hhN3ii, hhN4ii, hhN5ii, hhN2 ii, hh(N2 ) ii, I I I I bar I bar I op hhN1ii, hhN2ii, hhN3ii, hhN4ii, hh(N2 ) ii, hh((N2 ) ) ii, op I I op bar bar op hhL2ii, hhL2 ii, hhL2ii, hh(L2) ii, hhL2 ii, hh(L2 ) ii, I bar bar op hhA2ii, hhB2ii, hhH1ii, hhH1ii, hhO1 ii, hh(O1 ) ii.
bar bar bar bar Information on the augmentation semigroups Z2 , N2 , L2 , and O1 I op canbefoundinSubsection2.1.NotethathhN1ii = hhS`2ii, hhL2 ii = hhR2ii, I op I bar op op bar and hh(L2) ii = hhR2ii. But hh(L2 ) ii6= hh(L2 ) ii because the latter variety bar coincides with hhR2 ii = hhR2ii. 6 E. W. H. Lee et al.
1.3 – Main results
The present article continues the investigation of join irreducible semi- groups but with main emphasis on J -trivial semigroups. The class J of finite J -trivial semigroups coincides with the pseudovariety defined by the pseudoidentities (1.1) xω+1 ≈ xω, (xy)ω ≈ (yx)ω. The pseudovariety J can be generated by several sequences of well-known semigroups, such as the monoid OEn of order-preserving extensive trans- formations on {1, 2, . . . , n}; in other words, _ hhOE2ii⊂ hhOE3ii⊂ · · · ⊂ hhOEnii = J. n≥2 SeeVolkov[23]forothersequencesofgeneratorsforJ. ThepseudovarietyJisnotjoinirreducible;seeAlmeida[2]foranexplicit decomposition of J into the join of two proper subpseudovarieties. It follows that the pseudovariety hhOEnii is not join irreducible for all sufficiently large n. The search for new join irreducible J -trivial semigroups may thus be quite challenging. Up to this point, the known compact join irreducible pseudovarieties of J -trivial semigroups constitute six infinite classes:
I I I (1.2) hhNnii, hhHnii, hhKnii, hhNnii, hhHnii, hhKnii, n ≥ 1. These pseudovarieties are self-dual in the sense that they are closed under the operator S 7→ Sop. InSection2,informationonsomesmallsemigroupsandbackground results are given. I InSection3,themainobjectofstudy—themonoidCn obtained from certain J -trivial semigroup Cn of order 2n + 1—is introduced. A charac- I terization of identities satisfied by Cn is established. Based on this result, a I finite identity basis for Cn is deduced. More crucially, an exclusion pseudo- I I identity for Cn is also obtained, so that Cn is join irreducible; in contrast, the semigroup Cn is not join irreducible. The identity basis and exclusion I pseudoidentity for Cn form an equational sufficient condition to determine I if a finite semigroup generates the pseudovariety hhCnii. Unliketheself-dualpseudovarietiesin(1.2),thejoinirreduciblepseudo- I varieties hhCnii are not self-dual. Therefore all known join irreducible pseudo- varietiesofJ-trivialsemigroupsarethosefrom(1.2)and
I I op (1.3) hhCnii, hh(Cn) ii, n ≥ 2. On join irreducible J-trivial semigroups 7
InviewofTheorem1.1,onehastoexaminesemigroupsoforderatleast six to locate new examples of join irreducible pseudovarieties. Since the I I op join irreducible J -trivial semigroups C2 and (C2) are of order six, an immediate subproblem is to investigate J -trivial semigroups of order six. This is a potentially daunting task since the number of distinct J -trivial semigroups of order six is 6,309 (these semigroups are obtained by checking with a computer which semigroups of order six satisfy the pseudoidenti- ties(1.1)).Incomparison,thetotalnumberofdistinctsemigroupsoforder P5 up to five, J -trivial or otherwise, is i=1∆i=1,309;seeTable2. InSection4,fivenewequationalsufficientconditionsthatdetermine non-joinirreducibilityareestablished.InSection5,thesesufficientcondi- tions, together with some existing ones, are used to settle the join irre- ducibility of 6,306 of the 6,309 distinct J -trivial semigroups of order six; the three remaining semigroups are then individually shown to be non-join irreducible. It turns out that with regard to join irreducible pseudovarieties generated by a J -trivial semigroup of order six, there are no new examples beyondthosefrom(1.2)and(1.3).Specifically,thereareprecisely15join irreducible pseudovarieties generated by a J -trivial semigroup of order six: hhN ii, hhN ii, hhN ii, hhN ii, hhN ii, 2 3 4 5 6 I I I I I (1.4) hhN1ii, hhN2ii, hhN3ii, hhN4ii, hhN5ii, I I I op hhH1ii, hhH1ii, hhK1ii, hhC2ii, hh(C2) ii.
ItiseasilyseenfromTheorem1.1thatthese15pseudovarietiesalsoinclude all join irreducible pseudovarieties generated by a nontrivial J -trivial semigroup of order five or less.
2. Preliminaries
The present section introduces notation and terminology that are used throughout the article. For background information on finite semigroup theoryanduniversalalgebra,refertothemonographsofAlmeida[3]and BurrisandSankappanavar[4],respectively.
2.1 – Augmentation
For any semigroup S, consider the right regular representation (S•,S) of S acting on S• by right multiplication. Then the augmentation of S, denoted by Sbar, is defined by adding to S all constant maps on S•, where 8 E. W. H. Lee et al. multiplication is composition with the variable written on the left. For more information,refertoLeeetal.[18,Section2]. The following are examples of some semigroups and their augmentations:
bar Z2 = {1, g}, Z2 = {1, g, 1, g}; bar N2 = {0, n}, N2 = {0, n, n, i}; bar L2 = {e, f}, L2 = {e, f, e, f, i}; bar O1 = {0, a, e}, O1 = {0, a, e, a, e}; seeTables3and4.TheaugmentationoftherightzerosemigroupR2 is not bar required because hhR2 ii = hhR2ii.
bar bar Z2 1 g 1 g N2 0 n n i 1 1 g 1 g 0 0 0 n i g g 1 1 g n 0 0 n i 1 1 g 1 g n 0 0 n i g g 1 1 g i 0 n n i
bar bar Table 3. Multiplication tables of Z2 and N2
bar bar L2 e f e f i O1 0 a e a e e e e e f i 0 0 0 0 a e f f f e f i a 0 0 0 a e e e e e f i e 0 a e a e f f f e f i a 0 0 0 a e i e f e f i e 0 a e a e
bar bar Table 4. Multiplication tables of L2 and O1
2.2 – Words
Let A + and A ∗ denote the free semigroup and free monoid over a countably infinite alphabet A , respectively. Elements of A are called variables and elements of A ∗ are called words. For any word w and variable x, • the content of w, denoted by con(w), is the set of variables of w; • the number of times x occurs in w is denoted by occ(x, w); On join irreducible J-trivial semigroups 9
• x is simple in w if occ(x, w) = 1, and non-simple in w if occ(x, w) ≥ 2;
• the set of simple variables of w is denoted by sim(w) and the set of non-simple variables of w is denoted by non(w);
• the final part of w, denoted by fin(w), is the word obtained by retaining the last occurrence of each variable in w.
For any h ∈ sim(w) and x ∈ con(w), let occh(x, w) denote the number of times x occurs after h in w; in other words, occh(x, w) = m if and only if w = uhv for some u, v ∈ A ∗ such that h∈ / con(uv) and occ(x, v) = m. A pair (u, v) ∈ A + × A + of words is said to be balanced at a variable x ∈ A if occ(x, u) = occ(x, v), otherwise it is unbalanced at x. A pair of words is fully balanced if it is balanced at every variable.
2.3 – Identities
An identity is a formal expression u ≈ v formed by words u, v ∈ A +. For any semigroup S and identity u ≈ v, write S |= u ≈ v to indicate that S satisfies u ≈ v or u ≈ v is satisfied by S, that is, for any substitution ϕ : A → S, the elements ϕ(u) and ϕ(v) of S coincide; in this case, u ≈ v is also said to be an identity of S.
Lemma 2.1. Let u ≈ v be any identity. Then
I (i) R2 |= u ≈ v if and only if fin(u) = fin(v);
I (ii) Nn |= u ≈ v if and only if for any x ∈ A , either
occ(x, u) = occ(x, v) < n or occ(x, u), occ(x, v) ≥ n.
Proof.(i)SeePetrichandReilly[20,TheoremV.1.9(ix)]. (ii)SeeAlmeida[3,Lemma6.1.4].
A set Σ of identities of a semigroup S is an identity basis for S if every identity of S is deducible from Σ. A semigroup is finitely based if it has some finite identity basis.
Remark 2.2. For any finite semigroups S and T , the membership S ∈ hhT ii holds if and only if S satisfies all identities of T ; in other words, if Σ is an identity basis for T , then S ∈ hhT ii if and only if S |= Σ. 10 E. W. H. Lee et al.
2.4 – Pseudoidentities
An n-ary implicit operation is a mapping π that associates each finite semi- group S with an n-ary function πS on S such that for every homomorphism f : S → T between semigroups, π commutes with f in the sense that for any s1, s2, . . . , sn ∈ S, f πS(s1, s2, . . . , sn) = πT f(s1), f(s2), . . . , f(sn) .
Some examples of implicit operations on a finite semigroup S are • the binary operation (x, y) 7→ xy on S; • the unary operation x 7→ xω that maps each element s ∈ S to the unique power sω of s that is an idempotent. A pseudoidentity is a formal expression π ≈ ρ where π and ρ are implicit operations; a finite semigroup satisfies this pseudoidentity if πS = ρS. Recall that a pseudovariety is a class of finite semigroups that is closed under the formation of homomorphic images, subalgebras, and finitary direct products. A class of finite semigroups is a pseudovariety if and only if it coincides with the class of finite semigroups that satisfy some set of pseudoidentities[21].RefertoAlmeida[3]formoreinformationonpseudo- identities and pseudovarieties.
3. A new class of join irreducible semigroups
3.1 – Introduction
Suppose that T is any join irreducible semigroup with exclusion pseudo- identity σ. Then σ defines within hhT ii its unique maximal subpseudovariety hhTii∩Excl(T)[22,Section7.1].ThereforegivenanyfinitesemigroupS,the condition that S ∈ hhT ii and S 6|= σ is sufficient for the equality hhSii = hhT ii—and so also the join irreducibility of S—to hold. Checking for membership S ∈ hhT ii is tedious in general but is straightforward with a computer if the semigroup T is finitely based and afiniteidentitybasisforTisavailable;seeRemark2.2. I This section is devoted to the monoid Cn obtained from the semigroup
n n+1 n Cn = ha, b | ab = 0, ba = a, b = b i On join irreducible J-trivial semigroups 11 of order 2n + 1. The elements of Cn are
a, ab, ab2,..., abn−1, abn = 0, b, b2,..., bn−1, bn.
The semigroup Cn can be represented as a subsemigroup of the monoid POEn+1 of all partial order-preserving extensive transformations of the chain {1, 2, . . . , n + 1}; the generators a and b of Cn can be given as follows:
a b 1 2 3 4 ··· n n+1 b b b
Remark 3.1. (i) The relation ba = a in the presentation of Cn implies 2 n that a = ab a = 0 in Cn. In general, if x1, x 2, . . . , x r are elements i from Cn, two or more of which are of the form ab , then x1x2 · · · xr = 0.
I (ii) In view of part (i), it is easily checked that the monoid Cn satisfies thepseudoidentities(1.1)andsoisJ-trivial.
I ByTheorem1.1,themonoidC1 of order four is not join irreducible. I Therefore when considering the monoid Cn, it suffices to assume that I n≥2.InSubsection3.2,identitiessatisfiedbyCn are characterized and I a finite id entity basis for Cn isgiven.InSubsection3.3,thejoinirre- I ducibility of Cn is estab lished by exhibiting an exclusion pseudoidentity. I The identity basis for Cn and i ts exclusion pseudoidentity thus form an equational sufficient condition that decides if a finite semigroup generates I the pseudovariety hhCnii;seeCondition14inSection5. Incontrast,itisshowninSubsection4.6thatforeachn≥1,the semigroup Cnisnotjoinirreducible;seeCorollary4.15.
I 3.2 – Identity basis for Cn
AsshowninEdmunds[6,Part5ofthefirstproposition],theidenti- ties {x3 ≈ x2, x 2y2 ≈ y2x2, xyx ≈ yx2} constitute an identity basis for the I I monoid C1 of order f our. A finite identity basis for the monoid C2 of order sixwasfoundinLeeandLi[16,Proposition9.1];thefollowingresultisa generalization. 12 E. W. H. Lee et al.
Proposition 3.2. For each n ≥ 2, the following identities form an iden- I tity basis for the monoid Cn: n n Y Y (3.1a) x (Hix) ≈ (Hix), i=1 i=1 ( xyH1xH2y ≈ yxH1xH2y, xH1xyH2y ≈ xH1yxH2y, (3.1b) xH1yH2xy ≈ xH1yH2yx, where Hi ∈ {∅, hi}.
I Proof.ByRemark3.1(i),itiseasilycheckedthatCn satisfies the identities(3.1).TheresultthenfollowsfromLemma3.3below.
Lemma 3.3. For each n ≥ 2, the following statements on any identity u ≈ v are equivalent: I (i) Cn |= u ≈ v; (ii) u and v satisfy the following conditions: (A) for any x ∈ A , either occ(x, u) = occ(x, v) < n or occ(x, u), occ(x, v) ≥ n,
(B) for any h ∈ sim(u) = sim(v) and x ∈ con(u) = con(v), either
occh(x, u) = occh(x, v) < n or occh(x, u), occh(x, v) ≥ n;
(iii)u≈visdeduciblefromtheidentities(3.1).
I Proof.(i)⇒(ii).SupposethatCn |= u ≈ v. Then condition (A) holds 2 n I byLemma2.1(ii)sincethesubmonoid{1,b,b ,...,b }ofCn is isomorphic I to Nn.Chooseanyx∈con(u)=con(v)andh∈sim(u)=sim(v),andlet I e = occh(x,u)andf=occh(x,v).Letϕ:A→Cndenotethesubstitution (x, h, z) 7→ (b, a,I) for all z ∈ A \{x, h}. Then ϕ(u) = abe and ϕ(v) = abf , e f I so that ab = ab in Cn. It is then easily shown that either e = f < n or e, f ≥ n, so that condition (B) holds. (ii) ⇒ (iii). Suppose that u and v satisfy conditions (A) and (B). Then sim(u) = sim(v) and con(u) = con(v) by condition (A). First consider the case when sim(u) = sim(v) = ∅, so that all variables in u and v are non- simple, say non(u) = non(v) = {x1,x2,...,xr}.Thentheidentities(3.1b) can be used to rearrange the variables of u and v to obtain the words
0 e1 e2 er 0 f1 f2 fr u = x1 x2 ··· xr and v = x1 x2 ··· xr , On join irreducible J-trivial semigroups 13 respectively, where ei = occ(xi, u) ≥ 2 and fi = occ(xi, v) ≥ 2 for each i. In otherwords,theidentitiesu≈u0andv≈v0arededuciblefrom(3.1b).
Further, it follows from condition (A) that either ei = fi < n or ei, fi ≥ n. ei fi Therefore the identity xi ≈ xi is either trivial or deducible from the identityxn+1≈xnin(3.1a),whencetheidentityu0≈v0isdeducible from(3.1a).Consequently,theidentityu≈visdeduciblefrom(3.1). It remains to consider the case when sim(u) = sim(v) =6 ∅. Then by condition (B), the order of appearance of the simple variables in u coincides with the order of appearance of the simple variables in v. Hence
m m Y Y u = u0 (hiui) and v = v0 (hivi), i=1 i=1 where sim(u) = sim(v) = {h1, h2, . . . , hm} for some m ≥ 1 and any variable ∗ in ui, vi ∈ A is non-simple. Suppose that the pair (uk, vk) is unbalanced at some variable x, say e = occ(x, uk) and f = occ(x, vk) with e =6 f. Then in thefollowing,itisshownthattheidentities(3.1a)canbeappliedtouandv f e to convert the pair (uk, vk) into the pair (x uk, x vk), which is balanced at x. There are three cases depending on the value of k. Case 1: k = 0. By condition (B), one of the following subcases holds:
(1a) occh1 (x, u) = occh1 (x, v) < n,
(1b) occh1 (x, u), occh1 (x, v) ≥ n. In (1a), since
occ(x, u) = e + occh1 (x, u) =6 f + occh1 (x, v) = occ(x, v), it follows from condition (A) that occ(x, u), occ(x, v) ≥ n. In (1b), it is obvious that occ(x, u), occ(x, v) ≥ n. Therefore occ(x, u), occ(x, v) ≥ n in eithersubcase,whencetheidentities(3.1a)canbeappliedtouandvto f e perform the conversion (u0, v0) → (x u0, x v0):
m (3.1a) m Y f Y u = u0 (hiui) ≈ x u0 (hiui), i=1 i=1 m (3.1a) m Y e Y v = v0 (hivi) ≈ x v0 (hivi). i=1 i=1
Case 2: 0 < k < m. By condition (B), one of the following subcases holds:
(2a) occhk+1 (x, u) = occhk+1 (x, v) < n, 14 E. W. H. Lee et al.
(2b) occhk+1 (x, u), occhk+1 (x, v) ≥ n. In (2a), since
occhk (x, u) = e + occhk+1 (x, u) =6 f + occhk+1 (x, v) = occhk (x, v),
it follows from condition (B) that occhk (x, u), occhk (x, v) ≥ n. In (2b), it is obvious that occhk (x, u), occhk (x, v) ≥ n. Thus occhk (x, u), occhk (x, v) ≥ n ineithersubcase,whencetheidentities(3.1a)canbeappliedtouandvto f e perform the conversion (uk, vk) → (x uk, x vk):
k−1 m Y Y u = u0 (hiui) hkuk (hiui) i=1 i=k+1 (3.1a) k−1 m Y f Y ≈ u0 (hiui) hkx uk (hiui) , i=1 i=k+1 k−1 m Y Y v = v0 (hivi) hkvk (hivi) i=1 i=k+1 (3.1a) k−1 m Y e Y ≈ v0 (hivi) hkx vk (hivi) . i=1 i=k+1
Case 3: k = m. By condition (B), one of the following subcases holds:
(3a) occhm (x, u) = occhm (x, v) < n,
(3b) occhm (x, u), occhm (x, v) ≥ n.
But (3a) cannot hold since occhm (x, u) = e =6 f = occhm (x, v). Therefore only(3b)canhold,whencetheidentities(3.1a)canbeappliedtouandv f e to perform the conversion (um, vm) → (x um, x vm):
m−1 (3.1a) m−1 Y Y f u = u0 (hiui) hmum ≈ u0 (hiui) hmx um, i=1 i=1 m−1 (3.1a) m−1 Y Y e v = v0 (hivi) hmvm ≈ v0 (hivi) hmx vm. i=1 i=1
f e In summary, performing the conversion (ui, vi) → (x ui, x vi) described in Cases 1–3 whenever any pair (ui, vi) is unbalanced at x results in
(3.1a) m (3.1a) m 0 Y 0 0 Y 0 u ≈ u0 (hiui)andv ≈ v0 (hivi), i=1 i=1 On join irreducible J-trivial semigroups 15
0 0 ∗ 0 0 where any variable in ui,vi∈A isnon-simpleandeachpair(ui,vi)isfully 0 0 balanced. Then each identity ui≈viisdeduciblefromtheidentities(3.1b), whencetheidentityu≈visdeduciblefrom(3.1). I (iii) ⇒ (i). This holds because Cnsatisfies(3.1).
I 3.3 – Join irreducibility of Cn
I Theorem 3.4. For each n ≥ 2, the exclusion class Excl(Cn) is the pseudo- variety defined by the pseudoidentity
(3.2) (hωxhω)ωy(hωxhω)ω+n−1 ≈ (hωxhω)ωy(hωxhω)n−1.
I Consequently, the monoid Cn is join irreducible.
I Proof.ItfollowsfromLemma3.3thatthemonoidCn violates the pseudoidentity(3.2).HenceitsufficestoshowthatanypseudovarietyV I ω ω such that Cn∈/ Vsatisfies(3.2).Itisconvenienttowritex=h xh ,so thatthepseudoidentity(3.2)isxωyxω+n−1≈xωyxn−1. I Suppose that T ∈ V. Since Cn∈/ hhTii,thereexistssomeidentityu≈v I that is satisfied by T but violated by Cn.ThereforebyLemma3.3,the words u and v violate conditions (A) or (B). First suppose that u and v I violatecondition(A).ThenbyLemma2.1(ii),themonoidNn violates I I the identity u ≈ v, so that Nn ∈/ hhT ii. Since Nn is join irreducible and the I ω+n−1 n−1 pseudovariety Excl(Nn)isdefinedbythepseudoidentityx ≈x [18, Theorem 5.9], the semigroup T satisfies this pseudoidentity and so also the pseudoidentity(3.2). It remains to assume that u and v satisfy condition (A) but violate con- dition (B). Then there exist h ∈ sim(u) = sim(v) and x ∈ con(u) = con(v) such that e, f ≥ n and e = f < n are both false, where e = occh(x, u) and f = occh(x, v). By symmetry, it suffices to assume that e < n and e < f. Let ϕ denote the substitution (x, h, z) 7→ (x, xωy, hω) for all z ∈ A \{x, h}. Then ϕ(u) = xω+pyxe and ϕ(v) = xω+qyxf for some p, q ≥ 0. Hence T sat- isfies the pseudoidentity xω+pyxe ≈ xω+qyxf and so also the pseudoidentity
(3.3) xωyxn−1 ≈ xω+ryxn−1+s 16 E. W. H. Lee et al. for some r ≥ 0 and s ≥ 1. Since (3.3) xωyxn−1 ≈ xω+ryxn−1+s (3.3) ≈ xω+2ryxn−1+2s . . (3.3) ≈ xω+ωryxn−1+ωs = xωyxn−1+ω, thesemigroupTsatisfiesthepseudoidentity(3.2).
4. Sufficient conditions for non-join irreducibility
4.1 – Introduction
Suppose that V1, V2,..., Vk are two or more pseudovarieties that satisfy the pseudoidentities σ1, σ2, . . . , σk, respectively. Then given any finite semi- group S, the condition that
S ∈ V1 ∨ V2 ∨ · · · ∨ Vk and S 6|= σ1,S 6|= σ2,...,S 6|= σk Wk is sufficient for S to generate a subpseudovariety of i=1 Vi that is not join irreducible. Further, if each pseudovariety Vi is generated by some finite Wk semigroup Si, then the membership S ∈ i=1 Vi is equivalent to
S ∈ hhS1 × S2 × · · · × Skii, and each pseudoidentity σi can be chosen to be an identity of Si. Checking for the above membership is usually done by checking if S satisfies some
finite identity basis Σ for either the direct product S1 × S2 × · · · × Sk or some semigroup in the pseudovariety hhS1 × S2 × · · · × Skii; in both cases,
S |= Σ and S 6|= σ1,S 6|= σ2,...,S 6|= σk form an equational sufficient condition for S to be non-join irreducible. However, locating such equational sufficient conditions is not easy in general, since finding finitely based direct products S1 × S2 × · · · × Sk or finitely based semigroups in hhS1 × S2 × · · · × Skii are highly nontrivial tasks, even when k = 2 and the components Si are small. Nevertheless, a moderate number of such direct products have been found over the years, forexample,inthestudyofcertainvarietiesofsemigroups[3,11,17,24] On join irreducible J-trivial semigroups 17 andvarietiesofmonoids[8,10,13],thesolutiontothefinitebasisproblem forsmallsemigroups[16,19],andearlierworkonjoinirreduciblesemi- groups[18]. This section contains five subsections, each of which exhibits an equa- tional sufficient condition of the aforementioned type that determines non- joinirreducibility.TheseresultsarerequiredinSection5intheclassifica- tion of all join irreducible pseudovarieties generated by a J -trivial semi- group of order six.
I I 4.2 – The pseudovariety hhCn, R2ii The present subsection establishes the non-join irreducibility of some semi- I I groups in hhCn, R2ii.
Proposition 4.1. For each n ≥ 2, the following identities form an iden- I I tity basis for the monoid Cn × R2: n n Y Y (4.1a) x (Hix) ≈ (Hix), i=1 i=1
(4.1b) xyH1xH2y ≈ yxH1xH2y, xH1xyH2y ≈ xH1yxH2y, where Hi ∈ {∅, hi}.
I I Proof.ByRemark3.1(i),itiseasilycheckedthatCn × R2 satisfies the identities(4.1).TheresultthenfollowsfromLemma4.2below.
Lemma 4.2. For each n ≥ 2, the following statements on any identity u ≈ v are equivalent: I I (i) Cn × R2 |= u ≈ v; (ii) u and v satisfy the following conditions: (A) for any x ∈ A , either
occ(x, u) = occ(x, v) < n or occ(x, u), occ(x, v) ≥ n,
(B) for any h ∈ sim(u) = sim(v) and x ∈ con(u) = con(v), either
occh(x, u) = occh(x, v) < n or occh(x, u), occh(x, v) ≥ n,
(C) fin(u) = fin(v); (iii)u≈visdeduciblefromtheidentities(4.1). 18 E. W. H. Lee et al.
Proof.(i)⇒(ii).ThisfollowsfromLemmas2.1(i)and3.3. (ii) ⇒ (iii). Suppose that u and v satisfy conditions (A), (B), and (C). Then sim(u) = sim(v) and con(u) = con(v) by condition (A). First consider the case when sim(u) = sim(v) = ∅, so that all variables in u and v are non- simple, say non(u) = non(v) = {x1, x2, . . . , xr}. In view of condition (C), generality is not lost by assuming that fin(u) = fin(v) = x1x2 ··· xr. Then theidentities(4.1b)canbeusedtorearrangethevariablesofuandvto obtain the words
0 e1 e2 er 0 f1 f2 fr u = x1 x2 ··· xr and v = x1 x2 ··· xr , respectively, where ei = occ(xi, u) ≥ 2 and fi = occ(xi, v) ≥ 2 for all i. In otherwords,theidentitiesu≈u0andv≈v0arededuciblefrom(4.1b).
Further, it follows from condition (A) that either ei = fi < n or ei, fi ≥ n. ei fi Therefore the identity xi ≈ xi is either trivial or deducible from the identityxn+1≈xnin(4.1a),whencetheidentityu0≈v0isdeducible from(4.1a).Consequently,theidentityu≈visdeduciblefrom(4.1). It remains to consider the case when sim(u) = sim(v) =6 ∅. Then by condition (C), the order of appearance of the simple variables in u coincides with the order of appearance of the simple variables in v. Hence
m m Y Y u = u0 (hiui) and v = v0 (hivi), i=1 i=1 where sim(u) = sim(v) = {h1,h2,...,hm}forsomem≥1andanyvariable ∗ in ui,vi∈A isnon-simple.Sincetheidentities(3.1a)and(4.1a)coincide, theprocedureintheproofoftheimplication(ii)⇒(iii)inLemma3.3can be repeated to convert u and v into the words
m m 0 0 Y 0 0 0 Y 0 u = u0 (hiui) and v = v0 (hivi), i=1 i=1
0 0 ∗ respectively, where any variable in ui, vi ∈ A is non-simple and each pair 0 0 (ui, vi) is fully balanced. Consider any i ∈ {0, 1, . . . , m}. Generality is not lost by assuming that
0 con(ui) = {xi,1, xi,2, . . . , xi,ri , yi,1, yi,2, . . . , yi,si }, where ri, si ≥ 0, such that
0 0 0 0 0 0 (4.2) xi,j ∈/ con(ui+1ui+2 ··· um) and yi,j ∈ con(ui+1ui+2 ··· um) On join irreducible J-trivial semigroups 19 for all j. Note the extreme cases when either ri = 0 or si = 0: ∅ if ri = 0 and si = 0, 0 con(ui) = {xi,1, xi,2, . . . , xi,ri } if ri > 0 and si = 0, {yi,1, yi,2, . . . , yi,si } if ri = 0 and si > 0.
0 0 0 (In particular, if i = m, then con(ui+1ui+2 ··· um) = ∅, so that sm = 0; 0 hence the set con(um) is either {xm,1, xm,2, . . . , xm,rm } or empty.) Now since 0 any variable in uiisnon-simpleinu,theiden tities(4.1b)canbeappliedto 0 gather any non-last occurrence of a variable in ui with its last occurrence 0 in ui.Specifically,theidentities(4.1b)canbeusedtoarrangethevariables 0 of ui into a product p of the powers
e e ei,r f f fi,s x i,1 , x i,2 , . . . , x i , y i,1 , y i,2 , . . . , y i i,1 i,2 i,ri i,1 i,2 i,si
0 0 in some order, where ei,j = occ(xi,j, ui) and fi,j = occ(yi,j, ui). In addition, if si>0,sothatby(4.2),thevariablesyi,1,yi,2,...,yi,si also occur to the right of hi+1,thentheidentities(4.1b)canbeusedtomovethepowers f f fi,s y i,1 , y i,2 , . . . , y i anywhere within p. Therefore i,1 i,2 i,si (4.1b) m 0 Y u ≈ x0y0 (hixiyi), i=1
e e ei,r where each x is a product of the powers x i,1 , x i,2 , . . . , x i in some order i i,1 i,2 i,ri f f fi,s and y = y i,1 y i,2 ··· y i . i i,1 i,2 i,si 0 0 Now since each pair (ui, vi) is fully balanced, when the procedure in the last paragraph is repeated on v0, the deduction
(4.1b) m 0 Y v ≈ xb0y0 (hixbiyi) i=1
e e ei,r is obtained, where each x is a product of the powers x i,1 , x i,2 , . . . , x i in bi i,1 i,2 i,ri someorder.Itisclearthattheidentities(4.1b),whenappliedtoanyword, (C) preserve its final part. Hence fin(u0) = fin(u) = fin(v) = fin(v0). But since m m 0 Y (4.2) Y fin(u ) = fin x0y0 (hixiyi) = fin(x0) hi fin(xi) , i=1 i=1 m m 0 Y (4.2) Y fin(v ) = fin xb0y0 (hixbiyi) = fin(xb0) hi fin(xbi) , i=1 i=1 20 E. W. H. Lee et al. it follows that fin(xi)=fin(xb i)foralli.Thereforexi=xb iforalli,whence theidentityu0≈v0isdeduciblefrom(4.1b).Consequently,theidentity u≈visdeduciblefrom(4.1). I I (iii) ⇒ (i). This holds because Cn×R2satisfies(4.1).
Corollary4.3(cf.Condition20inSection5).Letn≥2befixed. SupposethatSisanyfinitesemigroupthatsatisfiestheidentities(4.1)but violates all of the identities
(4.3) x2 ≈ x, xyhxy ≈ xyhyx.
I I Then hhSii⊆ hhCn, R2ii is not join irreducible.
I I Proof.ByProposition4.1,theinclusionhhSii⊆hhCnii∨ hhR2ii holds. But I I thetwoidentitiesin(4.3)aresatisfiedbyR2 and Cn, respectively. Therefore I I hhSii * CnandhhSii*hhR2ii.
I bar I 4.3 – The pseudovariety hhNn, (L2 ) ii
This subsection establishes the non-join irreducibility of some semigroups I bar I in hhNn, (L2 ) ii. The semigroup
2 Q = a, b, e e = e, ea = a, be = b, eb = ae = ba = 0 = {0, a, b, e, ab} playsacentralrole;seeTable5.
Q 0 ab b a e 0 0 0 0 0 0 ab 0 0 0 0 ab b 0 0 0 0 b a 0 0 ab 0 0 e 0 ab 0 a e
Table 5. Multiplication table of Q
AfiniteidentitybasisforthemonoidQIcanbefoundinLeeandLi[16, I I I I Proposition 4.3]; it is also an identity basis for N2 × Q because N2 ∈ hhQ ii. The following result is a generalization. On join irreducible J-trivial semigroups 21
Proposition 4.4. For each n ≥ 2, the following identities form an iden- I I tity basis for the semigroup Nn × Q :
(4.4a) x2yx ≈ xyx2, n−1 n−1 2 Y Y (4.4b) x (Hix) ≈ x (Hix), i=1 i=1 ( xyH1xH2y ≈ yxH1xH2y, xH1xyH2y ≈ xH1yxH2y, (4.4c) xH1yH2xy ≈ xH1yH2yx, where Hi ∈ {∅, hi}.
I Proof.Itiseasilyshownthattheidentities(4.4)aresatisfiedbyNn I I I and Q , and so also by Nn×Q .Henceitremainstoshowthatanyidentity I I I u ≈ v satisfied by Nn×Q isdeduciblefrom(4.4).SinceNn|=u≈v,it followsfromLemma2.1(ii)that (a) for any x ∈ A , either
occ(x, u) = occ(x, v) < n or occ(x, u), occ(x, v) ≥ n.
In particular, sim(u) = sim(v) and non(u) = non(v), say
sim(u) = sim(v) = {h1, h2, . . . , hm}
and non(u) = non(v) = {x1, x2, . . . , xr}, where m, r ≥ 0 and (m, r) =6 (0, 0). First consider the case when m = 0, that is, all variables in u and v are non-simple.Thentheidentities(4.4c)canbeusedtorearrangethevariables of u and v to obtain the words
0 e1 e2 er 0 f1 f2 fr u = x1 x2 ··· xr and v = x1 x2 ··· xr , respectively, where ei = occ(xi, u) ≥ 2 and fi = occ(xi, v) ≥ 2 for all i. In otherwords,theidentitiesu≈u0andv≈v0arededuciblefrom(4.4c). ei fi By (a), either ei=fi
c1 c2 cr Sm Sm where ui, vi ∈ {x1 x2 ··· xr | cj ≥ 0} with i=0 con(ui) = i=0 con(vi). If con(ui) =6 con(vi) for some i, say there exists x ∈ con(ui)\ con(vi), then by I making the substitution ϕ : A → Q given by (x, hi, hi+1, z) 7→ (e, a, b,I) for all z ∈ A \{x, hi, hi+1}, the contradiction ϕ(u) = 0 =6 ϕ(v) is obtained. Therefore con(ui) = con(vi) for all i. In other words, for each i,
ci1 ci2 cir di1 di2 dir (c) ui = x1 x2 ··· xr and vi = x1 x2 ··· xr for some cij, dij ≥ 0 such that cij = 0 if and only if dij = 0. Itistheneasilyshownby(a)–(c)thattheidentities(4.4a)and(4.4b)can beusedtoconvertuintov,sothatu≈visdeduciblefrom(4.4).
I I bar I Lemma 4.5. The inclusion Q ∈ hhNn, (L2 ) ii holds for any n ≥ 2.
Proof.ItfollowsfromLeeetal.[18,Proposition6.21]thatthepseudo- I bar variety hhN2, L2 ii is defined by the identities
x3 ≈ x2, x2yx2 ≈ xyx, xyhxty ≈ xhxty2, xyhytx ≈ xhy2tx.
I bar It is then easily checked that Q ∈ hhN2, L2 ii, whence the result follows.
Corollary4.6(cf.Condition21inSection5).Letn≥2befixed. SupposethatSisanyfinitesemigroupthatsatisfiestheidentities(4.4)but violates all of the identities
(4.5) x2 ≈ x, xy ≈ yx.
I bar I Then hhSii⊆ hhNn, (L2 ) ii is not join irreducible.
Proof.ByProposition4.4andLemma4.5,theinclusions
I I I bar I hhSii⊆ hhNn, Q ii⊆ hhNnii∨ hh(L2 ) ii
bar I I hold.Butthetwoidentitiesin(4.5)aresatisfiedby(L2 ) and Nn, I bar I respectively. Therefore hhSii * hhNnii and hhSii * hh(L2 ) ii. On join irreducible J-trivial semigroups 23
bar I bar I 4.4 – The pseudovariety hh(N2 ) , (L2 ) ii
The present subsection establishes the non-join irreducibility of some semi- bar I bar I groups in hh(N2 ) , (L2 ) ii. The semigroup W = {a, b, c, d, e} given in Ta- ble6isrequired.
W a b c d e a a a a d e b a a b d e c a a c d e d a a d d e e a d a d e
Table 6. Multiplication table of W
Proposition4.7([16,Proposition15.1]).Thefollowingidentitiesform an identity basis for the monoid WI :
(4.6) x2Hx ≈ xHx, xy2x2 ≈ y2x2, xyHxTy ≈ yxHxTy, where H ∈ {∅, h} and T ∈ {∅, t}.
I bar I bar I Lemma 4.8. The inclusion W ∈ hh(N2 ) , (L2 ) ii holds.
bar I bar Proof. This result holds since the inclusion W ∈ hh(N2 ) , L2 ii holds [18,proofofProposition6.30].
Corollary4.9(cf.Condition22inSection5).SupposethatSis anyfinitesemigroupthatsatisfiestheidentities(4.6)butviolatesallofthe identities
(4.7) x2 ≈ x, xyx2 ≈ yx2.
bar I bar I Then hhSii⊆ hh(N2 ) , (L2 ) ii is not join irreducible.
Proof.ByProposition4.7andLemma4.8,theinclusions
I bar I bar I hhSii⊆ hhW ii⊆ hh(N2 ) ii∨ hh(L2 ) ii
bar I bar I hold.Butthetwoidentitiesin(4.7)aresatisfiedby(L2 ) and (N2 ) , bar I bar I respectively. Therefore hhSii * hh(N2 ) ii and hhSii * hh(L2 ) ii. 24 E. W. H. Lee et al.
I I I I 4.5 – The pseudovariety hhH1, L2, R2, N5ii
The present subsection establishes the non-join irreducibility of some semi- I I I I groups in hhH1, L2, R2, N5ii.
Proposition4.10([18,Proposition6.5]).Thefollowingidentitiesform I I I I an identity basis for the semigroup H1 × L2 × R2 × N5:
(4.8) x6 ≈ x5, x5yx ≈ x4yx, x2yx ≈ xyx2, xyxzx ≈ x2yzx.
Corollary4.11(cf.Condition23inSection5).SupposethatSis anyfinitesemigroupthatsatisfiestheidentities(4.8)butviolatesallofthe identities
(4.9) x3 ≈ x2, xy ≈ yx.
I I I I Then hhSii⊆ hhH1, L2, R2, N5ii is not join irreducible.
Proof.ByProposition4.10,theinclusion
I I I I hhSii⊆ hhH1 × L2 × R2ii∨ hhN5ii
I I I I holds.Butthetwoidentitiesin(4.9)aresatisfiedbyH1 × L2 × R2 and N5, I I I I respectively. Hence hhSii * hhH1 × L2 × R2ii and hhSii * hhN5ii.
4.6 – The pseudovariety Perm
Recall that a semigroup is permutative if it satisfies some permutation identity, that is, an identity of the form
x1x2 ··· xn ≈ x1πx2π ··· xnπ, where π is some nontrivial permutation on {1, 2, . . . , n}. The main aim of the present subsection is to establish the non-join irreducibility of some permutative semigroups.
For each n ≥ 0, let Permn denote the pseudovariety of finite semigroups that satisfy the permutation identity
h1h2 ··· hnxyt1t2 ··· tn ≈ h1h2 ··· hnyxt1t2 ··· tn.
Note that Perm0 ⊂ Perm1 ⊂ Perm2 ⊂ · · · and that Perm0 coincides with the pseudovariety Com of finite commutative semigroups. On join irreducible J-trivial semigroups 25
Lemma4.12([1,Theorem3.9]).Thefollowingstatementsonanyfinite semigroup S are equivalent: (i) S is permutative; (ii) S satisfies the pseudoidentity
(4.10) hωxytω ≈ hωyxtω;
(iii) S ∈ Permn for some n ≥ 0. Consequently, the class Perm of finite permutative semigroups coincides with thepseudovarietydefinedbythepseudoidentity(4.10).
A semigroup S is locally trivial if for any idempotent e ∈ S, the sub- semigroup eSe of S is trivial, that is, eSe = {e}. The class LI of finite locally trivial semigroups coincides with the pseudovariety defined by the pseudoidentity
(4.11) xωyxω ≈ xω.
Let A denote the pseudovariety of finite aperiodic semigroups and G denote the pseudovariety of finite groups.
Lemma4.13([3,Corollary6.4.11andFigure9.1]).
(i) Permn = Com ∨ (Permn ∩ LI). (ii) Com = (Com ∩ A) ∨ (Com ∩ G).
Lemma 4.14. Suppose that S ∈ Perm is such that S/∈ Com and S/∈ LI. Then S is not join irreducible.
Proof.ByLemma4.12,theinclusionhhSii⊆Permn holds for some n ≥ 1. Therefore hhSii⊆ Com ∨ (Permn∩LI)byLemma4.13(i).However, hhSii * Com and hhSii * Permn ∩ LI by assumption, so that S is not join irreducible.
Corollary4.15(cf.Condition24inSection5).SupposethatSisany finitenoncommutativesemigroupthatsatisfiesthepseudoidentity(4.10)but violatesthepseudoidentity(4.11).ThenhhSii⊆Permisnotjoinirreducible.
In particular, the semigroup Cn is not join irreducible for all n ≥ 1.
Corollary 4.16. The class of join irreducible monogenic semigroups consists of monogenic nilpotent semigroups and cyclic groups of prime power order. 26 E. W. H. Lee et al.
Proof. Cyclic groups of prime power order and monogenic nilpotent semigroupsarejoinirreducible[18,Theorems5.3and5.7].Conversely, let S be any join irreducible semigroup generated by some element s. Then since S is commutative, the inclusion hhSii⊆ (Com ∩ A) ∨ (Com ∩ G) holds byLemma4.13(ii).ButSisjoinirreducible,soeitherS∈AorS∈G.If n+1 n ∼ S ∈ A, then s = s in S for some n ≥ 1, whence S = Nn is nilpotent. In ∼ the remaining case when S ∈ G, one has S = Zk for some k ≥ 2. If k is not a prime power, say k = rs for some relatively prime integers r, s ≥ 2, then hhSii = hhZrii∨ hhZsii is contradictorily not join irreducible.
5. J -trivial semigroups of order six
Theorem 5.1. Suppose that S is any J -trivial semigroup of order six thatisjoinirreducible.ThenhhSiiisoneofthe15pseudovarietiesin(1.4).
Proof. There are 6,309 distinct J -trivial semigroups of order six. Withtheaidofacomputer,Conditions1–24inSubsection5.1andtheir dual conditions can be used to settle the join irreducibility of 6,306 of these semigroups: each of the 6,306 semigroups either generates a pseudovari- etyin(1.4)orisnotjoinirreducible.Thethreesemigroupstowhichthe conditions do not apply are U1,U2,andU3,giveninTables7and8.These threesemigroupsareshowntobenon-joinirreducibleinSubsections5.2 and5.3.Consequently,Theorem5.1isestablished.
U1 0 a b c d e U2 0 a b c d e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 a a 0 0 0 0 0 a b 0 0 0 0 0 b b 0 0 0 0 0 b c 0 0 a 0 a 0 c 0 0 a 0 a a d 0 0 a 0 a b d 0 0 a 0 a b e 0 a 0 c c e e 0 a a c c e
Table 7. Multiplication tables of U1 and U2
5.1–SufficientconditionsrequiredforTheorem5.1
TheproofofTheorem5.1requiresthefollowingsufficientconditionsthat determine if a finite semigroup S is join irreducible. On join irreducible J-trivial semigroups 27
U3 0 a b c d e 0 0 0 0 0 0 0 a 0 0 0 0 0 a b 0 0 0 0 0 b c 0 0 a 0 c 0 d 0 0 b 0 d 0 e 0 a a c c e
Table 8. Multiplication table of U3
Condition1([18,Subsection5.4]).Supposethat 2 • S |= x ≈ y1y2, • S 6|= x ≈ y.
Then hhSii = hhN2ii is join irreducible.
Condition2([18,Subsection5.4]).Supposethat 3 • S |= {xy ≈ yx, x ≈ y1y2y3}, • S 6|= x3 ≈ x2;
Then hhSii = hhN3ii is join irreducible.
Condition3([18,Subsection5.4]).Supposethat 2 2 4 • S |= {xy ≈ yx, x y ≈ xy , x ≈ y1y2y3y4}, • S 6|= x4 ≈ x3;
Then hhSii = hhN4ii is join irreducible.
Condition4([18,Subsection5.4]).Supposethat 2 2 5 • S |= {xy ≈ yx, x yz ≈ xy z, x ≈ y1y2y3y4y5}, • S 6|= x5 ≈ x4;
Then hhSii = hhN5ii is join irreducible.
Condition5([18,Subsection5.4]).Supposethat 2 2 6 • S |= {xy ≈ yx, x yzt ≈ xy zt, x ≈ y1y2y3y4y5y6}, • S 6|= x6 ≈ x5.
Then hhSii = hhN6ii is join irreducible. 28 E. W. H. Lee et al.
Condition6([18,Subsection5.5]).Supposethat • S |= {x2 ≈ x, xy ≈ yx}, • S 6|= x ≈ y; I Then hhSii = hhN1ii is join irreducible.
Condition7([18,Subsection5.5]).Supposethat • S |= {x3 ≈ x2, xy ≈ yx}, • S 6|= x2y ≈ xy2; I Then hhSii = hhN2ii is join irreducible.
Condition8([18,Subsection5.5]).Supposethat • S |= {x4 ≈ x3, xy ≈ yx}, • S 6|= x3y2 ≈ x2y3; I Then hhSii = hhN3ii is join irreducible.
Condition9([18,Subsection5.5]).Supposethat • S |= {x5 ≈ x4, xy ≈ yx}, • S 6|= x4y3 ≈ x3y4; I Then hhSii = hhN4ii is join irreducible.
Condition10([18,Subsection5.5]).Supposethat • S |= {x6 ≈ x5, xy ≈ yx}, • S 6|= x5y4 ≈ x4y5. I Then hhSii = hhN5ii is join irreducible.
Condition11([18,Subsections5.11]).Supposethat • S |= {x3 ≈ x2, xyx ≈ xyxy, xyx ≈ yxyx}, • S 6|= x2y2 ≈ y2x2.
Then hhSii = hhH1ii is join irreducible.
Condition12([18,Subsections5.12]).Supposethat • S |= {x3 ≈ x2, x2yx2 ≈ xyx, xyxy ≈ yxyx, xyxzx ≈ xyzx}, • S 6|= xy2z2x ≈ xz2y2x. I Then hhSii = hhH1ii is join irreducible. On join irreducible J-trivial semigroups 29
Condition13([12,Lemma15],[15,Theorem2]).Supposethat ( 3 2 2 2 ) x ≈ x , x yx ≈ xyx, xh1yh2xh3y ≈ yh1xh2yh3x, • S |= Q|S| Q1 , x i=1(yihiyi) x ≈ x i=|S|(yihiyi) x • S 6|= xy2x ≈ yx2y.
Then hhSii = hhK1ii is join irreducible.
Condition14(Proposition3.2andTheorem3.4).Supposethat xHxTx ≈ HxTx, xyHxTy ≈ yxHxTy, H ∈ {∅, h}, • S |= , xHxyTy ≈ xHyxTy, xHyTxy ≈ xHyTyx T ∈ {∅, t} • S 6|= xyxh2 ≈ yx2h2. I Then hhSii = hhC2ii is join irreducible.
Condition15([18,Corollary6.13]).Supposethat 6 • S |= x ≈ x1x2x3x4x5x6, • S 6|= xy ≈ yx. Then hhSii is not join irreducible.
Condition16([18,Corollary6.15]).Supposethat • S |= {xy ≈ yx, x3yz ≈ x2yz}, • S 6|= x3 ≈ x2, S 6|= x2y ≈ xy2. I Then hhSii⊆ hhN4, N2ii is not join irreducible.
Condition17([18,Corollary6.16]).Supposethat 2 2 2 • S |= {xy ≈ yx, x yz ≈ xy z, x y1y2y3y4 ≈ xy1y2y3y4}, • S 6|= x2 ≈ x, S 6|= x5 ≈ y5. I Then hhSii⊆ hhN5, N1ii is not join irreducible.
Condition18([18,Corollary6.24]).Supposethat • S |= {x3 ≈ x2, xyx ≈ x2y2, xy2z ≈ xyz}, • S 6|= x2y ≈ xy, S 6|= xy2 ≈ xy. I op Then hhSii⊆ hhL2, O1, O1 ii is not join irreducible.
Condition19([18,Corollary6.26]).Supposethat x3 ≈ x2, x2yx2 ≈ xyx, xyxy ≈ yxyx, • S |= , xyxzx ≈ xyzx, xy2z2x ≈ xz2y2x 30 E. W. H. Lee et al.
• S 6|= xyx ≈ yxy, S 6|= xyx ≈ x2y, S 6|= xyx ≈ yx2.
I op I Then hhSii⊆ hhH1, O1, (O1 ) ii is not join irreducible.
Condition20(Corollary4.3).Supposethat xHxTx ≈ HxTx, xyHxTy ≈ yxHxTy, H ∈ {∅, h}, • S |= , xHxyTy ≈ xHyxTy T ∈ {∅, t} • S 6|= x2 ≈ x, S 6|= xyhxy ≈ xyhyx.
I I Then hhSii⊆ hhC2, R2ii is not join irreducible.
Condition21(Corollary4.6).Supposethat 2 2 2 x yx ≈ xyx , x HxTx ≈ xHxTx, H ∈ {∅, h}, • S |= xyHxTy ≈ yxHxTy, xHxyTy ≈ xHyxTy, , T ∈ { , t} xHyTxy ≈ xHyTyx ∅ • S 6|= x2 ≈ x, S 6|= xy ≈ yx.
I bar I Then hhSii⊆ hhN3, (L2 ) ii is not join irreducible.
Condition22(Corollary4.9).Supposethat 2 2 2 2 2 x Hx ≈ xHx, xy x ≈ y x , H ∈ {∅, h}, • S |= , xyHxTy ≈ yxHxTy T ∈ {∅, t} • S 6|= x2 ≈ x, S 6|= xyx2 ≈ yx2.
bar I bar I Then hhSii⊆ hh(N2 ) , (L2 ) ii is not join irreducible.
Condition23(Corollary4.11).Supposethat • S |= {x6 ≈ x5, x5yx ≈ x4yx, x2yx ≈ xyx2, xyxzx ≈ x2yzx},
• S 6|= x3 ≈ x2, S 6|= xy ≈ yx.
I I I I Then hhSii⊆ hhH1, L2, R2, N5ii is not join irreducible.
Condition24(Corollary4.15).Supposethat • S |= {x4 ≈ x3, h3xyt3 ≈ h3yxt3},
• S 6|= xy ≈ yx, S 6|= x3yx3 ≈ x3. Then hhSii⊆ Perm is not join irreducible. On join irreducible J-trivial semigroups 31
5.2 – The semigroups U1 and U2
The main goal of the present subsection is to show that the semigroups U1 and U2,giveninTable7,arenotjoinirreducible.SinceU1andU2generate thesamepseudovariety[19,Proposition17.1],itsufficestoconsideronlyU1. The semigroups required in this subsection are
I 3 2 N3 = hn | n = 0i ∪ {I} = {0, n, n ,I}, 2 2 K1 = he, f | e = e, f = f, efe = 0i = {0, e, f, ef, fe, fef}, and the syntactic semigroup of {a, b}∗bab{a, b}∗ given by
3 2 2 2 2 2 SYN = ha, b | a = a ba = a , b = ba b = b, bab = 0i = {0, a, b, a2, ab, ba, a2b, aba, ba2, a2ba, aba2}; seeTables9and10. I Let Y be the subsemigroup of N3 × K1 × SYN generated by the elements a = (n, e, a) and b = (I, f, b). Then Y can be given by the presentation
4 3 2 2 a = a , b = b, abab = baba, ababa = a bab, Y = a, b a3ba = a2ba = a2ba2ba, aba3 = aba2 = aba2ba2 with the following 25 elements:
a = (n, e, a), b = (I, f, b), c = (n2, e, a2) = a2, d = (n, ef, ab) = ab, e = (n, fe, ba) = ba, f = (0, e, a2) = a3, g = (n2, ef, a2b) = a2b, h = (n2, 0, aba) = aba, i = (n2, fe, ba2) = ba2, j = (n, fef, 0) = bab, k = (0, ef, a2b) = a3b, l = (0, 0, a2ba) = a2ba, m = (0, 0, aba2) = aba2, n = (0, fe, ba2) = ba3, o = (0, 0, 0) = ababa, p = (n2, 0, 0) = abab, q = (n2, fef, b) = ba2b, r = (0, 0, a2) = a2ba2, s = (0, 0, ab), = aba2b t = (0, 0, ba) = ba2ba, u = (0, fef, b) = ba3b, v = (0, 0, a2b) = a2ba2b, w = (0, 0, aba) = aba2ba, x = (0, 0, ba2) = ba2ba2, y = (0, 0, b) = ba2ba2b; seeTable11.Notethato=ababa=a2bab=baba2isthezeroelementofY. 32 E. W. H. Lee et al.
I 2 N3 0 n n I K1 0 fef ef fe e f 0 0 0 0 0 0 0 0 0 0 0 0 n2 0 0 0 n2 fef 0 0 0 0 0 fef n 0 0 n2 n ef 0 0 0 0 0 ef I 0 n2 n I fe 0 0 fef 0 fe fef e 0 0 ef 0 e ef f 0 fef fef fe fe f
I Table 9. Multiplication tables of N3 and K1
2 2 2 2 2 SYN 0 ab aba aba b a b ba a ba a ba a 0 00000000000 ab 0 0 0 0 ab ab aba aba aba aba2 aba2 aba 0 ab aba aba2 0 ab 0 aba aba2 0 aba2 aba2 0 ab aba aba2 ab ab aba aba aba2 aba2 aba2 b 0 0 0 0 b b ba ba ba ba2 ba2 a2b 0 0 0 0 a2b a2b a2ba a2ba a2ba a2 a2 ba 0 b ba ba2 0 b 0 ba ba2 0 ba2 a2ba 0 a2b a2ba a2 0 a2b 0 a2ba a2 0 a2 a 0 a2b a2ba a2 ab a2b aba a2ba a2 aba2 a2 ba2 0 b ba ba2 b b ba ba ba2 ba2 ba2 a2 0 a2b a2ba a2 a2b a2b a2ba a2ba a2 a2 a2
Table 10. Multiplication table of SYN On join irreducible J-trivial semigroups 33
Y abcdefghijklmnopqrstuvwxy a cdfghfklmpklrmoosrvwsvlms b ebijenqpijutonopqxotuyoxy c fgfklfklroklrroovrvlvvlrv d hdmphmsompswomoosmowssoms e ijnqpnutoputxooooxyooytoo f fkfklfklroklrroovrvlvvlrv g lgrolrvorovloroovrolvvorv h mpmsomswooswmoooomsooswoo i nqnutnutxoutxxooyxytyytxy j pjoppoooopooooooooooooooo k lkrolrvorovloroovrolvvorv l rorvorvloovlroooorvoovloo m msmswmswmoswmmoosmswsswms n nunutnutxoutxxooyxytyytxy o ooooooooooooooooooooooooo p opooooooooooooooooooooooo q tqxotxyoxoytoxooyxotyyoxy r rvrvlrvlrovlrroovrvlvvlrv s wsmowmsomoswomoosmowssoms t xoxyoxytooytxooooxyooytoo u tuxotxyoxoytoxooyxotyyoxy v lvrolrvorovloroovrolvvorv w momsomswooswmoooomsooswoo x xyxytxytxoytxxooyxytyytxy y tyxotxyoxoytoxooyxotyyoxy
Table 11. Multiplication table of Y
The partition ∼ on Y given by
{a}, {b}, {c, g, i, q}, {d}, {e}, {f, h, j, k, l, m, n, o, p, r, s, t, u, v, w, x, y} is a congruence on Y. The quotient semigroup Y/ ∼ is isomorphic to U1; see Table12.Hencetheinclusion
I (5.1) hhU1ii⊆ hhSYN × K1ii∨ hhN3ii 34 E. W. H. Lee et al.
3 2 holds. But the identity x ≈ x of SYN × K1 and the identity xy ≈ yx I I of N3 are violated by U1. Therefore hhU1ii * hhSYN × K1ii and hhU1ii * hhN3ii, whence U1 is not join irreducible.
Y/ ∼ f c d e a b U1 0 a b c d e f f f f f f f 0 0 0 0 0 0 0 c f f f f f c a 0 0 0 0 0 a d f f f f f d b 0 0 0 0 0 b e f f c f c f c 0 0 a 0 a 0 a f f c f c d d 0 0 a 0 a b b f c f e e b e 0 a 0 c c e
Table 12. Multiplication tables of Y/ ∼ and U1
Remark 5.2. The intuition behind the proof in this subsection is that U1 is the syntactic semigroup of the language of words mapping to 0 in it. Expressing that language as a Boolean combination of languages then leads tothejoindecomposition(5.1)forthepseudovarietyhhU1ii.
I Corollary 5.3. The monoid U1 is not join irreducible.
I I I Proof. The inclusion hhU1ii⊆hhSYN ×K1ii∨hhN3iifollowsfrom(5.1). 3 2 I I I But the identity x ≈ x of SYN × K1 and the identity xy ≈ yx of N3 are I I I I I I violated by U1. Therefore hhU1ii * hhSYN × K1ii and hhU1ii * hhN3ii.
5.3 – The semigroup U3
The main goal of the present subsection is to show that the semigroup U3, giveninTable8,isnotjoinirreducible.Thesemigroups
2 2 A2 = ha, e | a = 0, aea = a, e = eae = ei = {0, a, e, ae, ea} I 2 and N2 = hn | n = 0i ∪ {I} = {0, n,I} arerequiredforthispurpose;seeTable13. I The semigroup A2 × N2 consists of the 15 elements
0 = (0, 0), 1 = (0, n), 2 = (0, 1), 3 = (a, 0), 4 = (a, n), 5 = (a, 1), 6 = (ae, 0), 7 = (ae, n), 8 = (ae, 1), 9 = (ea, 0), s = (ea, n), t = (ea, 1), x = (e, 0), y = (e, n), z = (e, 1). On join irreducible J-trivial semigroups 35
I It is straightforwardly checked that S14 = (A2 × N2)\{z} is a subsemigroup I of A2×N2oforder14withidealI={0,1,2,3,4,6,9,s,x}andthatthequo- tient semigroup S14/I={0,5,7,8,t,y}isisomorphictoU3;seeTables14 I and15.HencehhU3ii⊆ hhA2ii∨ hhN2ii. But the identity xyxyx ≈ xyx of A2 I and the identity xy ≈ yx of N2 are violated by U3. Therefore hhU3ii * hhA2ii I and hhU3ii * hhN2ii, whence U3 is not join irreducible.
I A2 0 a ae ea e N2 0 n I 0 0 0 0 0 0 0 0 0 0 a 0 0 0 a ae n 0 0 n ae 0 a ae a ae I 0 n I ea 0 0 0 ea e e 0 ea e ea e
I Table 13. Multiplication tables of A2 and N2
S14 0123456789stxy 0 00000000000000 1 00100100100100 2 01201201201201 3 00000000033366 4 00100100133466 5 01201201234567 6 00033366633366 7 00133466733466 8 01234567834567 9 000000000999xx s 00100100199sxx t 0120120129stxy x 000999xxx999xx y 00199sxxy99sxx
Table 14. Multiplication table of S14 36 E. W. H. Lee et al.
S14/I 0 7 y 5 t 8 U3 0 a b c d e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 7 a 0 0 0 0 0 a y 0 0 0 0 0 y b 0 0 0 0 0 b 5 0 0 7 0 5 0 c 0 0 a 0 c 0 t 0 0 y 0 t 0 d 0 0 b 0 d 0 8 0 7 7 5 5 8 e 0 a a c c e
Table 15. Multiplication tables of S14/I and U3
Acknowledgments. The authors thank the reviewer for several sugges- tions that improved the article and for bringing to their attention the rep- resentation of the semigroup Cn as a subsemigroup of the monoid POEn+1; seeSubsection3.1.TheyarealsogratefultoWendyWongforcheckingthe sufficientconditionsinSection5,withacomputer,againstallJ-trivial semigroups of order six. John Rhodes was supported by Simons Foundation Collaboration Grants for Mathematicians #313548. Benjamin Steinberg was supported by PSC- CUNY.
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