The identities of the free product of two trivial semigroups

The identities of the free product of two trivial semigroups Semigroup Forum (2017) 95:245–250 DOI 10.1007/s00233-016-9815-8 SHORT NOTE The identities of the free product of two trivial semigroups 1 2 L. M. Shneerson · M. V. Volkov Received: 19 June 2016 / Accepted: 12 July 2016 / Published online: 25 July 2016 © Springer Science+Business Media New York 2016 Abstract We exhibit an example of a finitely presented semigroup S with a minimum number of relations such that the identities of S have a finite basis while the monoid obtained by adjoining 1 to S admits no finite basis for its identities. Our example is the free product of two trivial semigroups. Keywords Semigroup identity · Finitely based semigroup · Nonfinitely based semigroup · Zimin word The finite basis problem, that is, the problem of classifying semigroups according to the finite basability of their identities, has been intensively explored since the 1960s. There exist several powerful methods to attack the finite basis problem for finite semi- groups (see the survey [12] for an overview). The finite basis problem for infinite semigroups is less studied. The reason for this is that infinite semigroups usually arise in mathematics as semigroups of transformations of an infinite set, or semigroups of relations on an infinite domain, or semigroups of matrices over an infinite ring, and as a rule all these semigroups are ‘too big’ to satisfy any nontrivial identity. If all Communicated by Marcel Jackson. B M. V. Volkov L. M. Shneerson Department of Mathematics and Statistics, Hunter College, City University of New York, 695 Park Avenue, New York, NY 10065, USA Institute of Mathematics and Computer Science, Ural Federal University, Lenina 51, Ekaterinburg, Russia 620000 123 246 L. M. Shneerson, M. V. Volkov Fig. 1 The automaton 0|0 generating S as an automaton semigroup 1|1 e f 0|0, 1|0 identities holding in a semigroup S are trivial, S is finitely based in a void way, so to speak. If, however, an infinite semigroup satisfies a nontrivial identity, its finite basis problem may constitute a challenge since ‘finite’ methods are non-applicable in general. Therefore, up to recently, results classifying finitely based and nonfinitely based members within natural series of infinite semigroups satisfying nontrivial iden- tities have been rather sparse. In fact, to the best of our knowledge, before 2015, the problem had been solved for only one natural family of concrete infinite semigroups that contains semigroups with a nontrivial identity, namely, for one-relator semigroups and monoids, see the paper [11] by the first author. This result has been obtained by purely syntactic means. The following observation is a consequence of the result of [11]: if a finitely based semigroup S admits a one-relator presentation, then the monoid S obtained by adjoin- ing1to S also is finitely based. It is known that this observation does not extend to finitely presented semigroups: indeed, an example of a finite (and thus finitely pre- sented) finitely based semigroup S such that the monoid S is nonfinitely based had already appeared in Perkins’s pioneering paper [9]. One can ask what is the least num- ber of relations in a presentation defining a finitely based semigroup S such that the monoid S is nonfinitely based. In this note we show that this number is equal to 2, 2 2 with an example provided by the presentation S =e, f | e = e, f = f , the free product of two trivial semigroups. It turns out that the semigroup S is finitely based while the monoid S is not. We mention in passing that the semigroup S has already arisen in many papers as it has several interesting features. For instance, S is known to be the only (up to isomorphism) infinite semigroup generated by two idempotents [2, Theoreme 3]. Also S is the only free product in which every subsemigroup is finitely generated and finitely presented [4, Sect. 4] and the only free product satisfying a nontrivial identity [10]. Further, S can be presented as an automaton semigroup in the sense of [3], namely, S is generated by the Mealy automaton in Fig. 1,see [3, Example 4.5]. In [13] S occurs in the context of the Stone–Cech compactifications of countably infinite abelian groups considered as discrete semigroups. The main results of the present note were found by the first author at the beginning of the 1990s but remained unpublished. The proof was of the same syntactic flavor as the proofs in [11]. Recently, Auinger, Chen, Hu, Luo, and the second author [1] have found a new condition under which a semigroup admits no finite identity basis; it turns out that this condition readily applies to the monoid S , thus providing a new 1 1 In contrast, if a one-relator semigroup S is nonfinitely based, the monoid S may happen to be finitely based, see [11] for an example. 123 The identities of the free product of two trivial semigroups 247 calculation-free proof that S is nonfinitely based. In order to present this new proof, we first recall some notions involved in the formulation of the condition from [1]. Let A and B be classes of semigroups. The Mal’cev product A  B of A and B is the class of all semigroups S for which there exists a congruence θ such that the quotient semigroup S/θ lies in B while all θ-classes that are subsemigroups in S belong to A. Notice that for a congruence θ on a semigroup S,a θ-class forms a subsemigroup of S if and only if the class is an idempotent of the quotient S/θ. We denote by Com the variety of all commutative semigroups and by Fin the class of all finite semigroups. Let x , x ,..., x ,... be a sequence of letters. The sequence { Z } of Zimin 1 2 n n n=1,2,... words is defined inductively by Z (x ) := x , Z (x ,..., x ) := Z (x ,..., x )x Z (x ,..., x ). 1 1 1 n+1 1 n+1 n 1 n n+1 n 1 n n−i Observe that in the word Z the letter x , i = 1,..., n, occurs 2 times and the n i length of Z is 2 − 1. We say that a word v is an isoterm for a semigroup S if the only word v such that S satisfies the identity v  v is the word v itself. Now we state the main result of [1] in a form that it convenient for the use in the present note. Theorem 1 ([1, Theorem 6]) A semigroup S is nonfinitely based provided that: (1) S lies in the Mal’cev product Com  Fin, and (2) Each Zimin word is an isoterm relative to S. 2 2 We are ready to prove our first result. Clearly, in S =e, f | e = e, f = f each element is uniquely represented as an alternating product of the generators e and f , and different such products represent different elements of S . Theorem 2 The monoid S is nonfinitely based. 2 2 Proof Consider the semigroup R =a, b | a = a = aba, b = b = bab.Obvi- ously, R consists of 4 idempotents: a, b, ab, and ba.The map e → a, f → b extends to a homomorphism S → R which in turn can be extended to a homomorphism 2 2 1 1 S → R by sending 1 to 1. The kernel θ of this homomorphism is a congruence on 2 2 S with one singleton class 1θ ={1} and four infinite classes: k m eθ ={(ef ) e | k ≥ 0}, feθ ={( fe) | m > 0}, ef θ ={(ef ) | > 0}, f θ ={( fe) f | n ≥ 0}. Since each element of the monoid R is an idempotent, each θ-class is a subsemigroup of S , and a direct computation shows that all these subsemigroups are commutative. We see that S lies in the Mal’cev product Com  Fin, and thus, it satisfies the condition (1) of Theorem 1. It remains to verify that S satisfies no non-trivial identity of the form Z  z.We induct on n. Observe that the subsemigroup ef θ ={(ef ) | > 0} of S is isomorphic to the additive semigroup of positive integers N. It is well known that every identity u  v satisfied by N is balanced, that is, every letter occurs the same number of 123 248 L. M. Shneerson, M. V. Volkov times in u and in v. Therefore if an identity of the form Z  z holds in S ,itmustbe balanced, and this immediately implies that for n = 1 any such identity must be trivial. Now assume that our claim holds for some n and let a word w = w(x ,..., x ) be 1 n+1 such that the identity Z  w holds in S . If we substitute 1 for x in this identity, n+1 1 we conclude that also the identity Z (1, x ,..., x )  w(1, x ,..., x ) n+1 2 n+1 2 n+1 should hold in S . However, it is easy to see that the word Z (1, x ,..., x ) is n+1 2 n+1 nothing but the Zimin word Z (x ,..., x ) and by the induction assumption we have n 2 n+1 w(1, x ,..., x ) = Z (x ,..., x ). This means that the word w(x ,..., x ) 2 n+1 n 2 n+1 1 n+1 is obtained from the Zimin word Z (x ,..., x ) by inserting in the latter 2 occur- n 2 n+1 rences of the letter x . If the insertion is made in a way such that the occurrences of x 1 1 alternate with 2 − 1 occurrences of x ,..., x , then w coincides with Z , and 2 n+1 n+1 we are done. It remains to verify that any other way of inserting 2 occurrences of x in Z (x ,..., x ) produces a word w such that the identity Z  w fails in S . n 2 n+1 n+1 Indeed, substitute e for x and f for all other letters in this identity. The value of the 2 −1 left-hand side under this substitution is (ef ) e. On the other hand, since at least two occurrences of x in the word w are adjacent, we are forced to apply the relation e = e at least once to get a representation of the value of the right-hand side as an alternating product of the generators e and f . Hence e occurs less than 2 times in 2 −1 this representation, and therefore, the value cannot be equal to (ef ) e. The syntactic proof of Theorem 2, which we do not include here, is much longer but provides an explicit infinite sequence {σ } of identities of S such that σ n n=4,5... n does not follow from the set of all identities holding in S and involving at most n letters. The identity σ looks as follows: x x ... x x yx x ... x x yx yx ... yx yx x ... x 2 3 n−1 n n−1 n−2 2 1 1 2 n 1 2 n−1 x x ... x x yx yx ... yx yx x ... x x yx x ... x . 2 3 n−1 n 1 2 n n−1 n−2 2 1 1 2 n−1 In contrast, we show that the semigroup S is finitely based and exhibit an explicit finite basis of this semigroup. This is an easy consequence of a result by Kim and Roush [6] and the following description of the identities holding in S . Theorem 3 An identity u  v holds in the semigroup S if and only if the words u and v satisfy the following three conditions: (a) the first letter of u is the same as the first letter of v; (b) the last letter of u is the same as the last letter of v; (c) for each word of length 2, the number of its occurrences as a factor is the same in u and v. Proof The claim admits rather a transparent syntactic proof, but we present a shorter argument that uses the Rees matrix construction (cf. [5, Chapter 3]). Let Z stand for the additive group of integers, set I := {e, f }, and define the matrix P := ( p ) ij i, j ∈ I 123 The identities of the free product of two trivial semigroups 249 over Z by setting p = p = p := 0 and p := 1. Consider the Rees matrix ee fe ff ef semigroup M( I, Z, I ; P) and define a map ψ : S → M( I, Z, I ; P) as follows: k k (ef ) e = e(ef ) e → (e, k, e) for each k ≥ 0, (ef ) = e(ef ) f → (e,, f ) for each > 0, m m−1 ( fe) = f (ef ) e → ( f, m − 1, e) for each m > 0, n n ( fe) f = f (ef ) f → ( f, n, f ) for each n ≥ 0. Clearly, ψ is 1-1, and a straightforward verification shows that ψ is a homomorphism. Thus, S is isomorphic to a subsemigroup of M( I, Z, I ; P). It is known (see, e.g., [6, Theorem 9]) and easy to verify that every identity u  v with u and v satisfying (a)–(c) holds in each Rees matrix semigroup over an abelian group. Hence, every such identity holds in M( I, Z, I ; P), and thus, in S . This proves the ‘if’ part of the theorem. To prove the ‘only if’ part, we consider the series of semigroups 2 p 2 p R =a, b | a = a = (ab) a, b = b = (ba) b 2, p for all primes p. For each p = 2, 3, 5,... ,the map e → a, f → b extends to an onto homomorphism S → R whence every identity u  v that holds in S must hold 2 2, p 2 in all semigroups R . The semigroup R is easily seen to be the union of 4 cyclic 2, p 2, p subgroups of order p and can be identified with the Rees matrix semigroup over the additive group of integers modulo p with the sandwich-matrix . As shown by Mashevitzky [8], an identity u  v holds in the latter semigroup if and only if the words u and v satisfy (a), (b) and the following condition: (c) for each word of length 2, the number of its occurrences as a factor in u and the number of its occurrences as a factor in v are congruent modulo p. Therefore, if u  v holds in S ,the words u and v must satisfy (a), (b), and (c) for 2 p every prime p. The conjunction of all the conditions (c) implies (c) since two integers congruent modulo every prime must be equal. Remark 1 Kizimenko [7, Sect. 2] considered a map similar to our homomorphism ψ from the proof of Theorem 3 but his homomorphism was only defined on the subsemigroup S \{e, f } of S , and therefore, could not be used for our purposes. 2 2 Combining Theorem 3 with [6, Proposition 7] and [6, Theorem 9], we obtain an explicit finite identity basis for the semigroup S : Theorem 4 The following identities form an identity basis for the semigroup S : xz yz xz y  xz yz xz y, xz yx z y  xz yx z y, xz yz xy  xyz xz y, 1 2 3 3 2 1 1 3 3 1 1 2 2 1 2 2 xz yx y  xyx z y, xz xz x  xz xz x , xz x  x z x . 1 1 1 2 2 1 1 1 Remark 2 Comparing Theorem 3 and [6, Theorem 9], we see that the semigroup S and the Rees matrix semigroup M := M( I, Z, I ; P) are equationally equivalent, i.e., 123 250 L. M. Shneerson, M. V. Volkov 1 1 they satisfy the same identities. This readily implies that the monoids S and M are equationally equivalent, and hence, by Theorem 2 the latter monoid is nonfinitely based. The fact that M is nonfinitely based also follows from [1, Theorem 4.6], and therefore, Theorem 2 could have been deduced from [1, Theorem 4.6], Theorem 3 and [6, Theorem 9]. We have preferred the proof presented above as it is short, direct, and does not appeal to any information about the identities of S . Acknowledgements The work of the first author was partially supported by the CUNY Collaborative Incentive Research Grant Program (Project #94592-0001). The second author acknowledges support from the Russian Foundation for Basic Research, Project No. 14-01-00524, the Ministry of Education and Science of the Russian Federation, Project No. 1.1999.2014/K, and the Competitiveness Program of Ural Federal University. The paper was written during the second author’s stay at Hunter College of the City University of New York as Ada Peluso Visiting Professor of Mathematics and Statistics with a generous support from the Ada Peluso Endowment. The authors would like to thank the anonymous reviewer for his/her comments and suggestions. References 1. Auinger, K., Chen, Yuzhu, Hu, Xun, Luo, Yanfeng, Volkov, M.V.: The finite basis problem for Kauffman monoids. Algebra Univ. 74(3–4), 333–350 (2015) 2. Benzaken, C., Mayr, H.C.: Notion de demi-bande: demi-bandes de type deux. Semigroup Forum 10(2), 115–128 (1975) 3. Cain, A.J.: Automaton semigroups. Theoret. Comput. Sci. 410(47–49), 5022–5038 (2009) 4. Campbell, C.M., Robertson, E.F., Ruškuc, N., Thomas, R.M.: On subsemigroups and ideals in free products of semigroups. Int. J. Algebra Comput. 6(5), 571–591 (1996) 5. Clifford, A.H., Preston, G.B.: The algebraic theory of semigroups, vol. I. Am. Math. Soc. (1961) 6. Kim, K.H., Roush, F.: The semigroup of adjacency patterns of words. In: Algebraic theory of semi- groups. Colloq. Math. Soc. János Bolyai, North-Holland. 20, 281–297 (1979) 7. Kizimenko, O.M.: On free products of one-element semigroups. Visn. Kiïv. Univ. Ser. Fiz.-Mat. Nauki. 1, 37–41 (2000) (in Ukraine) 8. Mashevitzky, G.I.: On identities in varieties of completely simple semigroups over abelian groups. In: Contemporary algebra, 81–89. Leningr. Gos. Ped. Inst. (1978) (in Russian) 9. Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969) 10. Shneerson, L.M.: Identities in one-relator semigroups. Uchen. Zap. Ivanov. Gos. Ped. Inst. 1(1–2), 139–156 (1972) (in Russian) 11. Shneerson, L.M.: On the axiomatic rank of varieties generated by a semigroup or monoid with one defining relation. Semigroup Forum 39, 17–38 (1989) 12. Volkov, M.V.: The finite basis problem for finite semigroups. Sci. Math. Jpn. 53(1), 171–199 (2001) 13. 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