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 fSi j i 2 Ig of finite semigroups is denoted by hhSi j i 2 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 2 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 2 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) = fT 2 S j S2 = hhT iig; 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 j g = 1i of prime order p and all semigroups of order two|the nilpotent semigroup N2 = f0; ng, the semilattice S`2 = f0; 1g, the left zero band L2 = fe; fg, and the right zero band R2=fe;fg|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 j 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 j a = 0; aea = a; e = eae = ei oforderfive[9],theBrandtsemigroup 2 2 B2 = ha; b j a = b = 0; aba = a; bab = bi oforderfive[22,Example7.3.4],andtheJ-trivialsemigroups 2 2 n Hn = he; f j e = e; f = f; (ef) = 0i 2 2 n and Kn = he; f j 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 [ fIg 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 j 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) x!+1 ≈ x!; (xy)! ≈ (yx)!: 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 f1; 2; : : : ; ng; in other words, _ hhOE2ii⊂ hhOE3ii⊂ · · · ⊂ hhOEnii = J: n≥2 SeeVolkov[23]forothersequencesofgeneratorsforJ.