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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 16 Выпуск 3 (2015)
УДК 512.57, 512.579
FREE COMMUTATIVE ^-DIMONOIDS
A. V. Zhuchok, Yu. V. Zhuchok
Department of Algebra and System Analysis,
Luhansk Taras Shevchenko National University,
Gogol square, 1, Starobilsk, 92703, Ukraine e-mail: zhuchok_a@mail.ru, yulia.mih@mail.ru
Abstract
A dialgebra is a vector space equipped with two binary operations 4 and b satisfying the following axioms:
(D1) (x 4 y) 4 z = x 4 (y 4 z),
(D2) (x 4 y) 4 z = x 4 (y b z),
(D3) (x b y) 4 z = x b (y 4 z),
(D4) (x 4 y) b z = x b (y b z),
(D5) (x b y) b z = x b (y b z).
This notion was introduced by Loday while studying periodicity phenomena in algebraic K-theory. Leibniz algebras are a non-commutative variation of Lie algebras and dialgebras are a variation of associative algebras. Recall that any associative algebra gives rise to a Lie algebra by [x, y] = xy-yx. Dialgebras are related to Leibniz algebras in a way similar to the relationship between associative algebras and Lie algebras. A dialgebra is just a linear analog of a dimonoid. If operations of a dimonoid coincide, the dimonoid becomes a semigroup. So, dimonoids are a generalization of semigroups.
Pozhidaev and Kolesnikov considered the notion of a 0-dialgebra, that is, a vector space equipped with two binary operations 4 and b satisfying the axioms (D2) and (D4). This notion have relationships with Rota-Baxter algebras, namely, the structure of Rota-Baxter algebras appearing on 0-dialgebras is known.
The notion of an associative 0-dialgebra, that is, a 0-dialgebra with two binary operations 4 and b satisfying the axioms (D1) and (D5), is a linear analog of the notion of a g-dimonoid. In order to obtain a g-dimonoid, we should omit the axiom (D3) of inner associativity in the definition of a dimonoid. Axioms of a dimonoid and of a g-dimonoid appear in defining identities of trialgebras and of trioids introduced by Loday and Ronco.
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The class of all g-dimonoids forms a variety. In the paper of the second author the structure of free g-dimonoids and free n-nilpotent g-dimonoids was given. The class of all commutative g-dimonoids, that is, g-dimonoids with commutative operations, forms a subvariety of the variety of g-dimonoids.
The free dimonoid in the variety of commutative dimonoids was constructed in the paper of the first author.
In this paper we construct a free commutative g-dimonoid and describe the least commutative congruence on a free g-dimonoid.
Keywords: dimonoid, g-dimonoid, commutative g-dimonoid, free commutative g-dimonoid, semigroup, congruence.
Bibliography: 15 titles.
2010 Mathematics Subject Classification: 08B20, 20M10, 20M50, 17A30, 17A32.
СВОБОДНЫЕ КОММУТАТИВНЫЕ g -ДИМОНОИДЫ
Анатолий В. Жучок, Юлия В. Жучок
Кафедра алгебры и системного анализа,
Луганский национальный университет имени Тараса Шевченко, площадь Гоголя, 1, Старобельск, 92703, Украина e-mail: zhuchok_a@mail.ru, yulia.mih@mail.ru
Аннотация
Диалгеброй называется векторное пространство, снабжённое двумя бинарными операциями Ч и Ь, удовлетворяющими следующим аксиомам:
(D1) (х Ч у) Ч z = = х Ч (у Ч z)
(D2) (х Ч у) Ч z = = х Ч (у Ь z)
(D3) (х Ь у) Ч z = = х Ь (у Ч z)
(D4) (х Ч у) Ь z = = х Ь (у Ь z)
(D5) (х Ь у) Ь z = = х Ь (у Ь z)
Это понятие было введено Лодэ во время изучения феномена периодичности в алгебраической K-теории. Алгебры Лейбница являются некоммутативной версией алгебр Ли, а диалгебры - версией ассоциативных алгебр. Напомним, что любая ассоциативная алгебра даёт алгебру Ли, если положить [x,y] = xy — yx. Диалгебры связаны с алгебрами Лейбница аналогично тому как связаны между собой ассоциативные алгебры и алгебры Ли. Диалгебра является линейным аналогом димоноида. Если операции димоноида совпадают, то он превращается в полугруппу. Таким образом, димоноиды обобщают полугруппы.
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Пожидаев и Колесников рассмотрели понятие 0-диалгебры, то есть векторного пространства, снабжённого двумя бинарными операциями Ч и Ь, удовлетворяющими аксиомам (D2) и (D4). Это понятие имеет связи с алгебрами Рота-Бакстера, а именно известна структура алгебр Рота-Бакстера, возникающих на 0-диалгебрах.
Понятие ассоциативной 0-диалгебры, то есть 0-диалгебры с двумя бинарными операциями Ч и Ь, удовлетворяющими аксиомам (D1) и (D5), является линейным аналогом понятия д-димоноида. Для того, чтобы получить д-димоноид, мы должны опустить аксиому (D3) внутренней ассоциативности в определении димоноида. Аксиомы димоноида и д-димоноида появляются в тождествах триалгебр и триоидов, введенных Лодэ и Ронко.
Класс всех д-димоноидов образует многообразие. Строение свободных д-димоноидов и свободных n-нильпотентных д-димоноидов было описано в статье второго автора. Класс всех коммутативных д-димоноидов, то есть д-димоноидов с коммутативными операциями, образует подмногообразие многообразия д-димоноидов. Свободный димоноид в многообразии коммутативных димоноидов был построен в статье первого автора.
В этой статье мы строим свободный коммутативный д-димоноид, а также описываем наименьшую коммутативную конгруэнцию на свободном д-димоноиде.
Ключевые слова: димоноид, д-димоноид, коммутативный д-димоноид, свободный коммутативный д-димоноид, полугруппа, конгруэнция.
Библиография: 15 названий.
2010 Mathematics Subject Classification: 08B20, 20M10, 20M50, 17A30, 17A32.
1. Introduction and preliminaries
Pozhidaev [1] and Kolesnikov [2] considered the notion of a 0-dialgebra. This notion have relationships with associative dialgebras [3-6] and Rota-Baxter algebras [1]. The notion of an associative 0-dialgebra, that is, a 0-dialgebra with two binary associative operations, is a linear analog of the notion of a g-dimonoid. In order to obtain a g-dimonoid, we should omit the axiom of inner associativity in the definition of a dimonoid [7]. The class of all g-dimonoids forms a variety. Free g-dimonoids and free n-nilpotent g-dimonoids were constructed in [8, 9] and [9], respectively. Axioms of a g-dimonoid also appear in defining identities of trialgebras and of trioids [10-12].
The class of all commutative g-dimonoids, that is, g-dimonoids with commutative operations, forms a subvariety of the variety of g-dimonoids. The free dimonoid in the variety of commutative dimonoids was constructed in [13]. In this paper we construct a free commutative g-dimonoid (Theorem 1) and describe the least commutative congruence on a free g-dimonoid (Theorem 2).
To make the paper almost self-contained, we recall basic definitions that will be used later.
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A nonempty set equipped with two binary operations 4 and h satisfying the axioms (D1)-(D5) is called a dimonoid. For a general introduction and basic theory see [3, 7, 14]. A nonempty set equipped with two binary operations 4 and h satisfying the axioms (D1), (D2), (D4), (D5) is called a generalized dimonoid or simply a g-dimonoid for short. It is obvious that any dimonoid is a g-dimonoid. Other examples of g-dimonoids can be found in [3, 7-9, 13-15]. Independence of axioms of a g-dimonoid follows from independence of axioms of a dimonoid [7].
If f : Di M D2 is a homomorphism of g-dimonoids, then the corresponding congruence on Di will be denoted by А/.
2. The main result
In this section we construct a free commutative g-dimonoid.
A g-dimonoid (D, h) will be called commutative, if both semigroups (D, 4) and
(D, h) are commutative. A g-dimonoid which is free in the variety of commutative g-dimonoids will be called a free commutative g-dimonoid.
Now we give a new example of a g-dimonoid. Let A be an arbitrary nonempty set and A = {x | x E A}. For every x E A assume x = x and introduce a map a = aA : A U A M A by the following rule:
ya
Г У, У E A, [ У, У E A.
Let further S be an arbitrary semigroup. Define operations 4 and h on S U S by a 4 b = (aaS)(baS), a h b = (aaS)(baS) for all a,b E S U S. Denote (S U S, h) by S.
Lemma 1. Sis a g-dimonoid but not a dimonoid.
Proof. The proof follows by a routine verification. □
Evidently, if S is commutative, then Sis a commutative g-dimonoid. If X is a generating set for a semigroup S, then, obviously, S(A\X is a g-subdimonoid of Sgenerated by X. Denote by FCgD(X) the g-dimonoid S(a')\X in which S is the free commutative semigroup on X.
Theorem 1. FCgD(X) is the free commutative g-dimonoid.
Proof. Show that FCgD(X) is free in the variety of commutative g-dimonoids.
Let (G, 4 , h ) be an arbitrary commutative g-dimonoid, ф : X ^ G be an arbitrary map and xi, yj E X, i E {1, 2, ...,m},j E {1, 2,..., n}. Define a map
£ : FCgD(X) ^ (G, 4 , h ) : w w£, assuming
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w£
Х\ф 4 ... 4 хтф, w = xi...xm,m > 1,
x-дф h ... h хтф, w = ХГТХт, m > 1.
Further prove that £ is a homomorphism.
Let w,u E FCgD(X). In the case w = x\...xm, u = yi...yn obtain
(w 4 u)£ = xpf 4 ... 4 xmф 4 Угф 4 ... 4 УиФ =
= (xiФ 4 ... 4 Хmф) 4 (угф h' ... h' упф) =
= (У1Ф h ... h' упф) 4 (x^ 4 ... 4 xm^) =
= (у1ф h ... h Упф) 4 (x^ h ... h' xmf) =
= (xiф h ... h' xmф) 4 (угф h' ... h' Упф) =
= xi...xm£ 4 yi...yn£ = w£ 4 u£.
For w = xi...xm, u = yi...yn get
(w 4 u)£ = Хlф 4 ... 4 xmф 4 У1ф 4 ... 4 Упф =
= (у1ф 4 ... 4 Упф) 4 (xiif 4 ... 4 xmf) =
= (угф 4 ... 4 Упф) 4 (x^ 4 ... 4 xm4) =
= (xiф h ... h xm4) 4 (у1ф 4 ... 4 упф) =
= xl...xm£ 4 (yi...yn)£ = w£ 4 u£.
The remaining two cases are considered in a similar way. So, (w 4 u)£ = w£ 4 u£ for all w,u E FCgD(X).
Similarly, one can check that (w h u)£ = w£ h u£ for all w,u E FCgD(X).
Consequently, £ is a homomorphism and FCgD(X) is the free commutative g-dimonoid. □
If N+ is the additive semigroup of all positive integers, obviously, N+a)\{1} is the free commutative g-dimonoid of rank 1.
It is not difficult to see that the automorphism group of the free commutative g-dimonoid FCgD(X) is isomorphic to the symmetric group on X and semigroups of FCgD(X) are isomorphic.
We conclude this section with some additional property of g-dimonoids.
Lemma 2. Operations of a g-dimonoid (D, 4, h) with a commutative idempotent operation 4 (respectively, h) coincide.
Proof. For all x,y,z E D we have
x h у = (x h y) 4 (x h y)
(x h y) 4 (x 4 y) =
= (x 4 y) 4 (x h y) = (x 4 y) 4 (x 4 y) = x 4 у
according to the idempotency, the commutativity of 4 and the axioms (D1), (D2) of a g-dimonoid. The case with the operation h is proved similarly. □
From Lemma 2 it follows that there do not exist commutative g-dimonoids with different idempotent operations.
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3. The least commutative congruence on a free g-dimonoid
In this section we present the least commutative congruence on a free g-dimonoid.
If p is a congruence on a g-dimonoid (D, H, h) such that (D, H, h)/p is a commutative g-dimonoid, we say that p is a commutative congruence.
In our next result we need the following construction.
Let X be an arbitrary nonempty set and let w be an arbitrary word in the alphabet X. The length of w will be denoted by l(w). Let further T be the free monoid on the two-element set {a,b}, 9 E T be an empty word and * denotes the operation on T. By definition, l(9) = 0. For every u E T\{9} denote the last letter of u by u(1). Define operations H and h on T, assuming
ui H u2 = u1 * al(u2')+1, u1 h u2 = u2 * bl(ui')+1
for all u1,u2 E T. The obtained algebra is denoted by T^(1).
Let F [X] be the free semigroup on X and
XTba(1) = {(w, u) E F[X] x Tb(1) | l(w) - l(u) = 1}.
By Theorem 1 from [9] XTf(1) is the free g-dimonoid.
Theorem 2. Let XTf(1) be the free g-dimonoid and FCgD(X) be the free commutative g-dimonoid. A map
в : XTb(1) ^ FCgD(X) :
, , , ,Q \ w, u(1) = b,
(w, u) (w,u)p = < ,,
[ w otherwise
is an epimorphism inducing the least commutative congruence on XT%(1). Proof. Take arbitrary elements (w1,u1), (w2,u2) E XT%(1). We have ((w1,u1) H (w2,u2))в = (wpw2,u1 * al(u2)+1)e =
= ww = (w1,uf)e H (w2,u2)e,
((w1,u1) h (w2,u2))e = (w1w2,u2 * + =
= ww = (w1,ui)e h (w2,m)p.
Thus, в is a homomorphism.
Let FC[X] be the free commutative semigroup on X and w,x E FC[X], where l(w) > 1 and l(x) = 1. For elements w,w,x E FCgD(X) there exist elements (w,ua), (w,ub), (x,9) E XTf(1), where u E T, such that
(w, ua)e
w
(w,ub)e = w, (х,9)в = x.
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So, в is surjective. By Theorem 1 FCgD(X) is the free commutative g-dimonoid. Then A^ is the least commutative congruence on XT^(1). □
Let a be an arbitrary fixed congruence on F[X]. Define a relation a1 on XT^(1) by
(w1,u1)a1 (w2,u2) ^ w1 aw2
for all (wi,ui), (w2,u2) E XTf(1).
It is not hard to prove the following lemma.
Lemma 3. The relation a' is a congruence on the free g-dimonoid XTf(T). Besides, operations of XTf(1)/a1 coincide.
From Lemma 3 we obtain
Corollary 1. If a is a diagonal of F [X ], then XTf (1)/a' is the free semigroup.
4. Conclusions
In this paper we consider g-dimonoids which are sets with two binary associative operations satisfying additional axioms. Dimonoids in the sense of Loday are examples of g-dimonoids. The main result of this paper is the construction of a free commutative g-dimonoid. We also present the least commutative congruence on a free g-dimonoid.
REFERENCES
1. Pozhidaev, A. P. 2009, “0-dialgebras with bar-unity and nonassociative Rota-Baxter algebras”, Sib. Math. J., vol. 50, no. 6, pp. 1070-1080.
2. Kolesnikov, P. S. 2008, “Varieties of dialgebras and conformal algebras”, Sib. Math. J., vol. 49, no. 2, pp. 257-272.
3. Loday, J.-L. 2001, “Dialgebras. In: Dialgebras and related operads”, Lecture Notes in Math., Berlin: Springer-Verlag, vol. 1763, pp. 7-66.
4. Frabetti, A. 2001, “Dialgebra (co)homology with coefficients. In: Dialgebras and related operads”, Lecture Notes in Math., Berlin: Springer-Verlag, vol. 1763, pp. 67-103.
5. Bokut, L. A., Chen, Y. & Liu, C. 2010, “Grobner-Shirshov bases for dialgebras”, Int. J. Algebra Comput., vol. 20, no. 3, pp. 391-415.
6. Kolesnikov, P. S. & Voronin, V. Yu. 2013, “On the special identities for dialgebras”, Linear and Multilinear Algebra, vol. 61, no. 3, pp. 377-391.
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7. Zhuchok, A. V. 2011, “Dimonoids”, Algebra and Logic, vol. 50, no. 4, pp. 323340.
8. Movsisyan, Y., Davidov, S. & Safaryan, Mh. 2014, “Construction of free
g-dimonoids”, Algebra and Discrete Math., vol. 18, no. 1, pp. 138-148.
9. Zhuchok, Yul. V. 2014, “On one class of algebras”, Algebra and Discrete Math., vol. 18, no. 2, pp. 306-320.
10. Loday, J.-L. & Ronco, M. O. 2004, “Trialgebras and families of polytopes”, Contemp. Math., vol. 346, pp. 369-398.
11. Casas, J. M. 2006, “Trialgebras and Leibniz 3-algebras”, Boletin de la Sociedad Matematica Mexicana, vol. 12, no. 2, pp. 165-178.
12. Zhuchok, A. V. 2014, “Semiretractions of trioids”, Ukr. Math. J., vol. 66, no. 2, pp. 218-231.
13. Zhuchok, A. V. 2010, “Free commutative dimonoids”, Algebra and Discrete Math., vol. 9, no. 1, pp. 109-119.
14. Zhuchok, A. V. 2014, “Elements of dimonoid theory”, Mathematics and its Applications. Proceedings of Institute of Mathematics of NAS of Ukraine, Kiev, vol. 98, 304 p. (in Ukrainian).
15. Zhuchok, A. V. 2011, “Semilattices of subdimonoids”, Asian-Eur. J. Math., vol. 4, no. 2, pp. 359-371.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Pozhidaev A. P. 0-dialgebras with bar-unity and nonassociative Rota-Baxter algebras // Sib. Math. J. 2009. Vol. 50, no. 6. P. 1070-1080.
2. Kolesnikov P. S. Varieties of dialgebras and conformal algebras // Sib. Math. J. 2008. Vol. 49, no. 2. P. 257-272.
3. Loday J.-L. Dialgebras. In: Dialgebras and related operads // Lecture Notes in Math. Berlin: Springer-Verlag. 2001. Vol. 1763. P. 7-66.
4. Frabetti A. Dialgebra (co)homology with coefficients. In: Dialgebras and related operads // Lecture Notes in Math. Berlin: Springer-Verlag. 2001. Vol. 1763. P. 67-103.
5. Bokut L. A., Chen Y., Liu C. Grobner-Shirshov bases for dialgebras // Int. J. Algebra Comput. 2010. Vol. 20, no. 3. P. 391-415.
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6. Kolesnikov P. S., Voronin V. Yu. On the special identities for dialgebras // Linear and Multilinear Algebra. 2013. Vol. 61, no. 3. P. 377-391.
7. Zhuchok A. V. Dimonoids // Algebra and Logic. 2011. Vol. 50, no. 4. P. 323-340.
8. Movsisyan Y., Davidov S., Safaryan Mh. Construction of free g-dimonoids // Algebra and Discrete Math. 2014. Vol. 18, no. 1. P. 138-148.
9. Zhuchok Yul. V. On one class of algebras // Algebra and Discrete Math. 2014. Vol. 18, no. 2. P. 306-320.
10. Loday J.-L., Ronco M. O. Trialgebras and families of polytopes // Contemp. Math. 2004. Vol. 346. P. 369-398.
11. Casas J. M. Trialgebras and Leibniz 3-algebras // Boletin de la Sociedad Matematica Mexicana. 2006. Vol. 12, no. 2. P. 165-178.
12. Zhuchok A. V. Semiretractions of trioids // Ukr. Math. J. 2014. Vol. 66, no. 2. P. 218-231.
13. Zhuchok A. V. Free commutative dimonoids // Algebra and Discrete Math. 2010. Vol. 9, no. 1. P. 109-119.
14. Жучок А. В. Елементи теорп дiмоноi'дiв // Математика та ii застосування. Пращ 1нституту математики НАН Укра'ши, Кшв. 2014. Т. 98. 304 с.
15. Zhuchok A. V. Semilattices of subdimonoids // Asian-Eur. J. Math. 2011. Vol. 4, no. 2. P. 359-371.
Луганский национальный университет имени Тараса Шевченко, Украина. Получено 01.07.2015
