The identities of the free product of a pair of two-element monoids
aa r X i v : . [ m a t h . G R ] J a n THE IDENTITIES OF THE FREE PRODUCTOF A PAIR OF TWO-ELEMENT MONOIDS
M. V. VOLKOV
Abstract.
Up to isomorphism, there exist two non-isomorphic two-el-ement monoids. We show that the identities of the free product of everypair of such monoids admit no finite basis.
Let S stand for the semigroup defined by the semigroup presentation h e, f | e = e, f = f i ; in other words, S is the free product of two one-element semigroups (in the category of semigroups). It is known (and easyto verify) that S is the only free product in the category of semigroups thatsatisfies a nontrivial identity [Shneerson, 1972]. Shneerson and the author[2017] have characterized the identities of S and proved that these identitiesadmit a finite basis.In the present note we address the identities of free products in the cat-egory of monoids considered as algebras of type (2,0). If M and M aretwo monoids, then when constructing their monoidal free product M ∗ M ,one has to amalgamate the identity elements of M and M . Therefore thefree monoidal product of two one-element monoids is again the one-elementmonoid. Moreover, it is evident that the product M ∗ M is isomorphic toone of its factors whenever the other factor is the one-element monoid. Inview of this observation, if we want the operation of free product to producesomething new, we have to assume that both M and M contain at leasttwo elements. On the other hand, if one of the factors M of M contains atleast three elements, M ∗ M cannot satisfy any nontrivial identity. Indeed,let | M | ≥ | M | ≥
3, say. Take a ∈ M \ { } and let b, c ∈ M \ { } besuch that b = c . Then it is easy to see that the elements ab and ac generatea free subsemigroup in the free product M ∗ M .Thus, studying identities of the free product M ∗ M makes sense only ifboth M and M consist of two elements. Up to isomorphism, there exist twonon-isomorphic two-element monoids: one is the two-element idempotentmonoid, which we denote by I , and the other one is the two-element group,which we denote by C . Therefore, up to isomorphism, the free products tobe considered are I ∗ I , C ∗ C , and I ∗ C . These free products can bedefined by the following monoid presentations: I ∗ I = h e, f | e = e, f = f i , (1) C ∗ C = h a, b | a = 1 , b = 1 i , (2) I ∗ C = h e, b | e = e, b = 1 i . (3) The monoid defined by the presentation (1) is denoted J ∞ . Observe thatthe monoid presentation (1) looks exactly as the semigroup presentationused above to define the semigroup S , whence the monoid J ∞ is nothingbut the semigroup S with identity element adjoined. Shneerson and theauthor [2017] have shown that the identities of S with identity elementadjoined are not finitely based.It is easy to see that the presentation (2) defines a group known in theliterature as the infinite dihedral group D ∞ . The group D ∞ is an extensionof the infinite cyclic group generated by the element ba by the two-elementgroup; in particular, D ∞ is a metabeian group. A general result by Sapir[1987] implies that if the semigroup identities of a group G are finitely basedand G is an extension of an abelian subgroup by a group of finite exponent,then G either is abelian or has finite exponent; see [Sapir, 1987, Proposi-tion 6]. Applying this result to D ∞ , we conclude that the semigroup iden-tities of D ∞ are not finitely based. Observe that, in contrast, the group identities of D ∞ are finitely based; this follows from another general result,due to Cohen [1967], who proved that the group identities of any metabeliangroup admit a finite basis.Thus, it remains to analyze the identities of the monoid defined by the pre-sentation (3). This monoid is generated by an idempotent and an involution,and we denote it by K ∞ , having in mind Kuratowski’s closure-complementtheorem . The main result of the present note is the following Theorem 1.
The identities of the monoid K ∞ are not finitely based. We prove Theorem 1 re-using the technique that was applied by Shneersonand the author [2017] to prove the analog of this theorem for the monoid J ∞ .(In fact, the same technique could have been applied to show the absence ofa finite basis for the semigroup identities of the group D ∞ .) The techniquestems from Auinger et al. [2015]; in order to present it, we need to recallthree concepts.The first concept is that of Mal’cev product. The Mal’cev product of twoclasses of semigroups A and B , say, is the class A (cid:13) m B 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 a θ -class forms a subsemigroup of S if and only if the class is anidempotent of the quotient S/θ . We denote by
Com and
Fin the classes ofall commutative semigroups and all finite semigroups, respectively. The classic version of Kuratowski’s closure-complement theorem basically describesthe monoid generated by two operators on subsets of an arbitrary topological space: theoperator of taking the closure of a subset and the operator of forming the complementof a subset; see the excellent survey by Gardner and Jackson [2008] for quite a compre-hensive treatment. Many generalizations have been considered in which the operator oftaking closure has been substituted by various idempotent operators while the operator offorming complement has been substituted by various involutive operators; some of thesegeneralizations are surveyed in [Gardner and Jackson, 2008, Subsection 4.2]. Clearly, allmonoids of operators arising this way are homomorphic images of the monoid K ∞ . HE IDENTITIES OF FREE PRODUCTS OF TWO-ELEMENT MONOIDS 3
The next concept we need is that of a Zimin word. Let x , x , . . . , x n , . . . be a sequence of letters. The sequence { Z n } n =1 , ,... of Zimin words is definedinductively by Z ( x ) := x , Z n +1 ( x , . . . , x n +1 ) := Z n ( x , . . . , x n ) x n +1 Z n ( x , . . . , x n ) . Observe that in the word Z n the letter x i , i = 1 , . . . , n , occurs 2 n − i timesand the length of Z n is 2 n − v is called an isoterm relative to a semigroup S ifthe only word v ′ such that S satisfies the identity v ≏ v ′ is the word v itself.Now we state the main result of Auinger et al. [2015] in a form that itconvenient for the use in the present note. Theorem 2 ([Auinger et al., 2015, Theorem 6]) . The identities of a semi-group S have no finite basis provided that:(i) S lies in the Mal’cev product Com (cid:13) m Fin , and(ii) each Zimin word is an isoterm relative to S .Proof of Theorem 1. From the definition of the free product, it readily fol-lows that each non-identity element of the monoid K ∞ can be uniquelyrepresented as an alternating product of the generators e and b .First we show that K ∞ satisfies the condition (i) of Theorem 2. Considerthe monoid T defined by the following presentation: T = h f, g | f = f gf = f, g = 1 i . It is easy to compute that T consists of 6 elements: 1 , g, f, f g, gf, gf g , allof which except g are idempotents. The map e f, b g extends to amonoid homomorphism K ∞ → T . The kernel θ of this homomorphism isa congruence on the monoid K ∞ with two singleton classes 1 θ = { } and bθ = { b } and four infinite classes: eθ = { ( eb ) k e | k ≥ } , beθ = { ( be ) m | m > } ,ebθ = { ( eb ) ℓ | ℓ > } , bebθ = { ( be ) n b | n > } . The infinite θ -classes and the θ -class 1 θ = { } are subsemigroups in K ∞ .Clearly, the latter subsemigroup is commutative, and a direct computationshows that so are the four other subsemigroups. Thus, the monoid K ∞ liesin the Mal’cev product Com (cid:13) m Fin .Now we aim to verify the condition (i) of Theorem 2, that is, we wantto show that no non-trivial identity of the form Z n ≏ z may hold in K ∞ .This verification repeats the argument used by Shneerson and the author[2017] for the monoid J ∞ , but we reproduce it for the reader’s convenience.We induct on n . Observe that the subsemigroup ebθ = { ( eb ) ℓ | ℓ > } of K ∞ is isomorphic to the additive semigroup of positive integers N . It iswell known that every identity u ≏ v satisfied by N is balanced , that is,every letter occurs the same number of times in u and in v . Therefore ifan identity of the form Z n ≏ z holds in K ∞ , it must be balanced, and thisimmediately implies that for n = 1 any such identity must be trivial. Now M. V. VOLKOV assume that our claim holds for some n and let a word w = w ( x , . . . , x n +1 )be such that the identity Z n +1 ≏ w holds in K ∞ . If we substitute 1 for x in this identity, we conclude that also the identity Z n +1 (1 , x , . . . , x n +1 ) ≏ w (1 , x , . . . , x n +1 )should hold in k ∞ . However, the word Z n +1 (1 , x , . . . , x n +1 ) is nothing butthe Zimin word Z n ( x , . . . , x n +1 ) and by the induction assumption we have w (1 , x , . . . , x n +1 ) = Z n ( x , . . . , x n +1 ). This, together with the fact that theidentity Z n +1 ≏ w must be balanced, means that the word w ( x , . . . , x n +1 ) isobtained from the Zimin word Z n ( x , . . . , x n +1 ) by inserting 2 n occurrencesof the letter x in the latter. If the insertion is made in a way such that theoccurrences of x alternate with 2 n − x , . . . , x n +1 , then w coincides with Z n +1 , and we are done. It remains to verify that any otherway of inserting 2 n occurrences of x in Z n ( x , . . . , x n +1 ) produces a word w such that the identity Z n +1 ≏ w fails in K ∞ . Indeed, substitute e for x and b for all other letters in this identity. The value of the left-handside under this substitution is ( eb ) n − e . On the other hand, since at leasttwo occurrences of x in the word w are adjacent, we are forced to applythe relation e = e at least once to get a representation of the value of theright-hand side as an alternating product of the generators e and b . Hence e occurs less than 2 n times in this representation, and therefore, the valuecannot be equal to ( eb ) n − e .Theorem 1 now follows from Theorem 2. (cid:3) Taking into account the discussion preceding the formulation of Theo-rem 1, we arrive at the following
Corollary 3.
For each pair of two-element monoids, the identities of theirfree product admit no finite basis.Remark . We have already mentioned in passing that our technique couldhave been used to verify that the semigroup identities of the group D ∞ admit no finite basis. Inspecting the above proof of Theorem 1, the readermay see that this is indeed the case. The containment D ∞ ∈ Com (cid:13) m Fin immediately follows from the fact that D ∞ is an extension of the infinitecyclic group by the two-element group. The inductive proof that all Ziminwords are isoterms relative to K ∞ works also for D ∞ with the followingminimal adjustments: one has to change the generator e of K ∞ to thegenerator a of D ∞ and use the relation a = 1 from the presentation (2)instead of the relation e = e from the presentation (3). Remark . The similarity in the arguments used here and in [Shneerson and Volkov,2017] to establish the absence of finite identity bases for each of the monoids J ∞ , D ∞ , and K ∞ may lead one to the conjecture that some of the threemonoids or perhaps all of them satisfy exactly the same identities. However,this is not the case. Indeed, the results of [Shneerson and Volkov, 2017] eas-ily imply that the monoid J ∞ satisfies the identity x yx ≏ xyx . This HE IDENTITIES OF FREE PRODUCTS OF TWO-ELEMENT MONOIDS 5 identity fails in both D ∞ and K ∞ as one can see by evaluating the variables x and y at the generators of the latter monoids. Further, since the group D ∞ is an extension of the infinite cyclic group by the two-element group,the identity x y ≏ y x holds in D ∞ , but it fails in J ∞ and K ∞ as againis revealed by evaluating the variables at the generators.The observations just made suffice to claim that the sets Id( J ∞ ), Id( D ∞ ),and Id( K ∞ ) of identities holding in respectively J ∞ , D ∞ , and K ∞ are pair-wise distinct; moreover, the sets Id( J ∞ ) and Id( D ∞ ) are incomparable. Infact, one can show that K ∞ satisfies the identity x yx ≏ x yx that fails in D ∞ so that the sets Id( D ∞ ) and Id( K ∞ ) are incomparable as well. One canalso show that Id( J ∞ ) is a proper subset of Id( K ∞ ). These results rely ona characterization of Id( K ∞ ) which will be the subject of a separate paper. Acknowledgements.
The author acknowledges support from the Ministry of Ed-ucation and Science of the Russian Federation, project no. 1.3253.2017, the Com-petitiveness Program of Ural Federal University, and from the Russian Foundationfor Basic Research, project no. 17-01-00551.
References
Auinger, K., Chen, Yuzhu, Hu, Xun, Luo, Yanfeng, and Volkov, M.V. [2015]:
Thefinite basis problem for Kauffman monoids , Algebra Universalis (3–4), 333–350.Cohen, D.E. [1967]: On the laws of a metabelian variety , J. Algebra (3), 267–273.Gardner, B.J., and Jackson, M.G. [2008]: The Kuratowski closure-complement the-orem , New Zealand J. Math. , 9–44.Sapir, M.V. [1987]: Problems of Burnside type and the finite basis property in vari-eties of semigroups , Izv. Akad. Nauk SSSR, Ser. Mat. , 319–340 [in Russian;Engl. translation Math. USSR–Izv. , 295–314].Shneerson, L.M. [1972]: Identities in one-relator semigroups , Uchen. Zap. Ivanov.Gos. Pedag. Inst. (1-2), 139–156 [in Russian].Shneerson, L.M., and Volkov, M.V. [2017]: The identities of the free product of twotrivial semigroups , Semigroup Forum (1), 245–250. Institute of Natural Sciences and Mathematics, Ural Federal University,Lenina 51, 620000 Ekaterinburg, Russia
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