On Join Irreducible J-Trivial Semigroups

On Join Irreducible J-Trivial Semigroups

Rend. Sem. Mat. Univ. Padova, DRAFT, 1{37 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 in the 1970s after Eilenberg [7] and Sch¨utzenberger established an isomor- 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] characterized pseudovarieties as classes defined by pseudoidentities. This result led to the syntactic approach of Almeida [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, Definition 6.1.5]. The present article is primarily concerned 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, see Rhodes and Steinberg [22, Section 7.3]. On join irreducible J-trivial semigroups 3 1.2 { Overview A systematic study of join irreducible pseudovarieties was recently initiated by the authors [18]. This subsection is an overview of some results relevant to the present article. 1.2.1 { Examples of join irreducible semigroups Every atom in the lattice L (S) is join irreducible [22, Table 7.2]. In other 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|are join irreducible; see Table1. 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 Zpn or Nn; see Corollary 4.16. 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 of order five [9], the Brandt semigroup 2 2 B2 = ha; b j a = b = 0; aba = a; bab = bi of order five [22, Example 7.3.4], and the J -trivial semigroups 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 of orders 4n and 4n + 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, Exercise 10.10.1]. The operator S 7! SI on S, on the other hand, 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 that is obtained by adding to S constant maps on S•; see Subsection 2.1. 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, Subsection 4.5]. 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, Corollary 4.11]. 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 increases very rapidly [5, Table 1], efficient methods are crucial to any 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; see Subsections 3.1 and 4.1. In general, equational sufficient 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. In a recent study of join irreducible semigroups [18], some equational 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. Theorem 1.1 ([18, Theorem 7.1]). Among all pseudovarieties generated 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 can be found in Subsection 2.1.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    37 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us