Presently, no example of non-finitely based finite semigroup S is known for which the monoid S1 is finitely based. Based on a general result of M. V. Volkov, two methods are established from which examples of such semigroups can be constructed

Lee, Edmond W. H.

Presently, no example of non-finitely based finite semigroup $S$ is known for which the monoid $S^1$ is finitely based. Based on a general result of M. V. Volkov, two methods are established from which examples of such semigroups can be constructed

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Finitely based monoids obtained from non-finitely based semigroups

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UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICAdoi: 10.4467/20843828AM.13.003.2279 FASCICULUS LI, 2013FINITELY BASED MONOIDS OBTAINED FROMNON-FINITELY BASED SEMIGROUPSby Edmond W. H. LeeAbstract. Presently, no example of non-finitely based finite semigroup Sis known for which the monoid S1 is finitely based. Based on a generalresult of M. V. Volkov, two methods are established from which examplesof such semigroups can be constructed.1. Introduction. A semigroup is finitely based if the identities it satisfiesare finitely axiomatizable. Commutative semigroups [9], idempotent semi-groups [3–5], and finite groups [8] are finitely based, but not all semigroupsare finitely based in general. Further, the class FB of finitely based semigroupsis not closed under common operators such as the formation of homomorphicimages, subsemigroups, and direct products. Refer to the survey by Volkov [14]for more information on these operators and the finite basis problem for semi-groups in general. The present article is concerned with the operator thatmaps each semigroup S to the smallest containing monoidS1 ={S if S is a monoid,S ∪ {1} otherwise.The class FB is not closed under this operator; there exist finitely based semi-groups S such that the monoids S1 are non-finitely based. The earliest exam-ple demonstrating this property, published by Perkins [9] in 1969, is a certainsemigroup R24 of order 24; see Section 3. Perkins’s work in fact contains amuch smaller example that he was unaware of at that time: he proved thatthe Brandt monoid B12 is non-finitely based [9], while the Brandt semigroupB2 =〈a, b∣∣ a2 = b2 = 0, aba = a, bab = b〉2000 Mathematics Subject Classification. 20M05.Key words and phrases. Semigroup, monoid, finitely based.46of order five was later shown by Trahtman to be finitely based [12]. Theseexamples led Shneerson [11] to question the existence of semigroups havingthe “opposite” property.Question 1. Do non-finitely based semigroups S exist for which the mo-noids S1 are finitely based?In what follows, it is convenient to call a semigroup S conformable if S isnon-finitely based while S1 is finitely based. Shneerson provided an affirmativeanswer to Question 1 by proving that the semigroupT =〈a, b∣∣ aba = ba〉is conformable [11]. However, unlike the finite examples B2 and R24 thatmotivated Shneerson’s question, the semigroup T is infinite. Apart from T ,no other semigroup has since been found to be conformable. Therefore, therestriction of Question 1 to finite semigroups is of fundamental interest.Question 2. Do finite conformable semigroups exist?Recall that a semigroup S with zero 0 is nilpotent if there exists somen ≥ 1 such that the product of any n elements of S equals 0. Each nilpotentsemigroup satisfies the identityx1x2 · · ·xn = y1y2 · · · ynfor some n ≥ 1 and so is easily shown to be finitely based [9]. It turns outthat by the following general result of Volkov [13], which was established priorto Question 1 being posed by Shneerson, an abundance of finite conformablesemigroups can be constructed from nilpotent semigroups.Lemma 3. Suppose that N is any nilpotent semigroup. Then for any semi-group S, the direct product S × N is finitely based if and only if S is finitelybased.2. Constructing finite conformable semigroups. Recall that the va-riety generated by a semigroup S, denoted by varS, is the smallest class ofsemigroups containing S that is closed under the formation of homomorphicimages, subsemigroups, and arbitrary direct products. A semigroup S satisfiesthe same identities as the variety varS it generates [2].Theorem 4. Suppose that S and N are any semigroups such that(a) S1 is non-finitely based ;(b) N is nilpotent ;(c) S1 ×N1 is finitely based.Then the direct product P = S1 ×N is conformable.47Proof. The semigroup P is non-finitely based by (a), (b), and Lemma 3.Since P is a subsemigroup of S1 ×N1, it belongs to the variety var (S1 ×N1).The inclusion varP 1 ⊆ var (S1 × N1) then follows [1, Lemma 7.1.1]. But themonoids S1 and N1 are embeddable in P 1 so that varP 1 = var (S1 × N1).Therefore, the monoid P 1 is finitely based by (c).Theorem 5. Suppose that S and N are any semigroups such that(a) S1 is non-finitely based ;(b) N is nilpotent ;(c) N1 is finitely based ;(d) varS1 ⊆ varN1.Then the direct product P = S1 ×N is conformable.Proof. Following the proof of Theorem 4, the semigroup P is non-finitelybased with varP 1 = var (S1 × N1). Then (d) implies that varP 1 = varN1,whence the monoid P 1 is finitely based by (c).The following results of Jackson and Sapir [6] now provide the appropri-ate finite semigroups S and N to construct the conformable semigroups P inTheorems 4 and 5.Lemma 6. There exist finite nilpotent semigroups S and N such that S1and N1 are non-finitely based while S1 ×N1 is finitely based.Lemma 7. There exist finite nilpotent semigroups S and N such that S1is non-finitely based, N1 is finitely based, and varS1 ⊆ varN1.Jackson and Sapir in fact presented methods for locating as many of the semi-groups in Lemmas 6 and 7 as desired [6, Corollaries 3.1 and 5.2].3. Explicit examples of finite conformable semigroups. Let A+denote the free semigroup over a countably infinite alphabet A. Elementsof A+ are called words. For any finite set W = {w1, . . . , wk} of words, letR(w1, . . . , wk) denote the Rees quotient of A+ over the ideal of all words thatare not factors of any word in W. Equivalently, R(w1, . . . , wk) can be treatedas the semigroup that consists of every nonempty factor of every word in W,together with a zero element 0, with binary operation · given byu · v ={uv if uv is a factor of some word in W,0 otherwise.It is easily seen that the semigroup R(w1, . . . , wk) is nilpotent. The semigroupR24 of Perkins introduced in Section 1 is R(xyzyx, xzyxy, xyxy, xxz).Consider the semigroupsR8 = R(xyxy), R12 = R(xxyy, xyyx), and R15 = R(xyxy, xxyy, xyyx)where |R8| = 8, |R12| = 12, and |R15| = 15. Then48• R18 is non-finitely based [6, Example 4.2];• R112 is non-finitely based [6, proof of Corollary 5.1];• R115 is finitely based [6, Corollary 3.2 and proof of Corollary 5.1];• var (R18 ×R112) = varR115 [6, Lemma 5.1].It follows that the pairs (S,N) = (R8, R12) and (S,N) = (R8, R15) satisfyLemmas 6 and 7, respectively. Therefore, by Theorems 4 and 5, the semigroupsR18 ×R12 and R18 ×R15 are conformable.Now since the conformable semigroup P = S1 × N in Theorems 4 and 5is a direct product, its order |S1||N | can be quite large in general. But itturns out that the semigroup P contains a proper subsemigroup that is alsoconformable. DefineP∗ = S1∗ ∪N∗where S1∗ ={(a, 0)∣∣ a ∈ S1} and N∗ = {(0, b) ∣∣ b ∈ N}. Then it is easily seenthat S1∗ , N∗, and P∗ are subsemigroups of P .Proposition 8. The semigroup P∗ is conformable.Proof. The isomorphic relations S1 ∼= S1∗ and N ∼= N∗ clearly hold.Therefore,varP = var (S1 ×N) = var (S1∗ ×N∗) ⊆ varP∗ ⊆ varP,whence the semigroups P and P∗ generate the same variety and so satisfy thesame identities. The result thus follows.The semigroup P∗ has order |S1| + |N | − 1 and so is often much smallerthan the semigroup P with order |S1||N |. For instance,(|P∗|, |P |) ={(20, 108) if (S,N) = (R8, R12),(23, 135) if (S,N) = (R8, R15).On the other hand, the semigroup P∗ is still quite large; the order of anynon-finitely based monoid of the form R(w1, . . . , wk)1 is at least nine [6, The-orem 4.3] so that |P∗| ≥ 9 + 2− 1 = 10.In view of the small semigroup B2 that motivated Question 1, it is naturalto pose the following question:Question 9. What is the smallest possible order of a conformable semi-group?Based on results of Lee et al. [7], Sapir [10], and Zhang [15], the order of anyconformable semigroup is at least seven.Acknowledgment. The author would like to thank the anonymous re-viewer for a number of helpful, constructive comments.49References1. Almeida J., Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994.2. Birkhoff G., On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31(1935), 433–454.3. Birjukov A. P., Varieties of idempotent semigroups, Algebra and Logic, 9 (1970), 153–164; translation of Algebra i Logika, 9 (1970), 255–273.4. Fennemore C. F., All varieties of bands. I, II, Math. Nachr., 48 (1971), 237–252; ibid.48 (1971), 253–262.5. Gerhard J. A., The lattice of equational classes of idempotent semigroups, J. Algebra, 15(1970), 195–224.6. Jackson M., Sapir O., Finitely based, finite sets of words, Internat. J. Algebra Comput.,10 (2000), 683–708.7. Lee E. W. H., Li J. R., Zhang W. T., Minimal non-finitely based semigroups, SemigroupForum, 85 (2012), 577–580.8. Oates S., Powell M. B., Identical relations in finite groups, J. Algebra, 1 (1964), 11–39.9. Perkins P., Bases for equational theories of semigroups, J. Algebra, 11 (1969), 298–314.10. Sapir M. V., Problems of Burnside type and the finite basis property in varieties of semi-groups, Math. USSR-Izv., 30 (1988), 295–314; translation of Izv. Akad. Nauk SSSR Ser.Mat., 51 (1987), 319–340.11. Shneerson L. M., On the axiomatic rank of varieties generated by a semigroup or monoidwith one defining relation, Semigroup Forum, 39 (1989), 17–38.12. Trahtman A. N., A basis of identities of the five-element Brandt semigroup, Ural. Gos.Univ. Mat. Zap., 12, No. 3 (1981), 147–149 (in Russian).13. Volkov M. V., On the join of varieties, Simon Stevin, 58 (1984), 311–317.14. Volkov M. V., The finite basis problem for finite semigroups, Sci. Math. Jpn., 53 (2001),171–199.15. Zhang W. T., Existence of a new limit variety of aperiodic monoids, Semigroup Forum,86 (2013), 212–220.Received September 30, 2013Division of Math, Science, and TechnologyNova Southeastern University3301 College AvenueFort LauderdaleFlorida 33314, USAe-mail : edmond.lee@nova.edu

Finitely Based Monoids Obtained From Non-Finitely Based Semigroups

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Finitely Based Monoids Obtained From Non-Finitely Based Semigroups

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