Free commutative dimonoids

We construct a free commutative dimonoid and characterize the least idempotent congruence on this dimonoid.

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Дата:	2010
Автор:	Zhuchok, A.V.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154499
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Free commutative dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 109–119. — Бібліогр.: 6 назв. — англ.

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spelling	irk-123456789-1544992019-06-17T01:30:46Z Free commutative dimonoids Zhuchok, A.V. We construct a free commutative dimonoid and characterize the least idempotent congruence on this dimonoid. 2010 Article Free commutative dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 109–119. — Бібліогр.: 6 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:08A05. http://dspace.nbuv.gov.ua/handle/123456789/154499 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We construct a free commutative dimonoid and characterize the least idempotent congruence on this dimonoid.
format	Article
author	Zhuchok, A.V.
spellingShingle	Zhuchok, A.V. Free commutative dimonoids Algebra and Discrete Mathematics
author_facet	Zhuchok, A.V.
author_sort	Zhuchok, A.V.
title	Free commutative dimonoids
title_short	Free commutative dimonoids
title_full	Free commutative dimonoids
title_fullStr	Free commutative dimonoids
title_full_unstemmed	Free commutative dimonoids
title_sort	free commutative dimonoids
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/154499
citation_txt	Free commutative dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 109–119. — Бібліогр.: 6 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT zhuchokav freecommutativedimonoids
first_indexed	2025-07-14T06:35:29Z
last_indexed	2025-07-14T06:35:29Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 9 (2010). Number 1. pp. 109 – 119 c© Journal “Algebra and Discrete Mathematics” Free commutative dimonoids Anatolii V. Zhuchok Communicated by V. I. Sushchansky Dedicated to Professor I.Ya. Subbotin on the occasion of his 60-th birthday Abstract. We construct a free commutative dimonoid and characterize the least idempotent congruence on this dimonoid. 1. Introduction Jean-Louis Loday introduced the notion of a dimonoid [1]. Dimonoids are a tool to study Leibniz algebras. A dimonoid is a set equipped with two binary associative operations satisfying some axioms (see below). If the operations of a dimonoid coincide, then the dimonoid becomes a semigroup. The first result about dimonoids is the description of the free dimonoid generated by a given set [1]. T. Pirashvili [2] introduced the notion of a duplex which generalizes the notion of a dimonoid and constructed a free duplex. In [3] it is proved that every commutative dimonoid is a semilattice of archimedean subdimonoids. In this paper we construct a free commutative dimonoid (Theorem 3), characterize the least idempotent congruence on this dimonoid and the classes of this congruence (Theorem 4). Also we describe the free commutative dimonoids of the small ranks (Propositions 3 and 4). In section 4 we give some properties of commutative dimonoids. 2000 Mathematics Subject Classification: 08A05. Key words and phrases: commutative dimonoid, free commutative dimonoid. Jo u rn al A lg eb ra D is cr et e M at h .110 Free commutative dimonoids 2. Preliminaries A set D equipped with two binary associative operations ≺ and ≻ satis- fying the following axioms: (x ≺ y) ≺ z = x ≺ (y ≻ z), (x ≻ y) ≺ z = x ≻ (y ≺ z), (x ≺ y) ≻ z = x ≻ (y ≻ z) for all x, y, z ∈ D, is called a dimonoid. A map f from a dimonoid D1 to a dimonoid D2 is a homomorphism, if (x ≺ y)f = xf ≺ yf, (x ≻ y)f = xf ≻ yf for all x, y ∈ D1. If f : D1 → D2 is a homomorphism of dimonoids, then corresponding congruence on D1 we denote by ∆f . A subset T of a dimonoid (D, ≺,≻) is called a subdimonoid, if for any a, b ∈ D, a, b ∈ T implies a ≺ b, a ≻ b ∈ T . Now we give the necessary information about varieties of dimonoids. A class H of algebraic systems is called a variety, if there exists such family ℘ of identities of a signature Ω that H consists from that and only that systems of the signature Ω in which all the formulas from ℘ are true. Let H be some class of algebraic systems. We call an arbitrary alge- braic system H ′ a H-system, if H ′ ∈ H. Theorem 1. (Birkhoff [4]) A nonempty class H of algebraic systems is a variety if and only if the following conditions hold: a) the Cartesian product of an arbitrary sequence of H-systems is a H-system, b) any subsystem of an arbitrary H-system is a H-system, c) any homomorphic image of an arbitrary H-system is a H-system. Observe that the class Dim of all dimonoids satisfies the conditions of Birkhoff’s theorem and therefore it is a variety. Indeed, if suppose ℘ = {(x ≺ y) ≺ z = x ≺ (y ≻ z), (x ≻ y) ≺ z = x ≻ (y ≺ z), (x ≺ y) ≻ z = x ≻ (y ≻ z)}, Ω = {≺, ≻}, then H = Dim is a variety. Let U be a dimonoid and let R be some class of dimonoids. A nonempty set X of some elements from U is called independent in U with respect to the class R, if an arbitrary map from X into any R- dimonoid M can be extended to a homomorphism from X̄ into M, where X̄ is a subdimonoid generated by the elements of X in U . Jo u rn al A lg eb ra D is cr et e M at h .A. V. Zhuchok 111 A dimonoid U is called free concerning the class R, if in U there exists a set X of elements which is independent with respect to R and which generates the dimonoid U . The set X satisfying these properties is called a R-free basis of the dimonoid U . The dimonoid U is called a free dimonoid of rank m in the class R, if U ∈ R and in U there exists a R-free basis of cardinality m. The next assertion follows from Malchev’s book [5]. Proposition 1. If in the class R there exist free dimonoids of rank m, then all they are isomorphic and any R-dimonoid having a generating set of cardinality m is a homomorphic image of the free dimonoid of rank m in R. In particular, if in R there exist free dimonoids of an arbitrary rank, then every R-dimonoid U is a homomorphic image of the free dimonoid of rank \|U \| in R. The free dimonoids in the class Dim of all dimonoids are called ab- solutely free. Note that the absolutely free dimonoid was constructed by Loday [1]. It is clear that the variety Dim completely is defined by the absolutely free dimonoids. A variety R is called minimal, if R contains a dimonoid with more than 1 element, and there are not others subvarieties in R, except R and the trivial variety (containing 1-element dimonoids only). For any class R let R̂ be a minimal variety which contains the class R. The next assertion follows from Malchev’s book [5]. Proposition 2. A dimonoid U is free in some class if and only if it has an independent generating set of elements. In this case the dimonoid U is free in the variety Û . A dimonoid (D,≺,≻) will be called a commutative (idempotent) di- monoid, if both semigroups (D,≺) and (D,≻) are commutative (idem- potent). Observe that the class of commutative dimonoids is a subvariety of the variety Dim. A dimonoid which is free in the variety of commutative dimonoids will be called a free commutative dimonoid. We finish this section with the formulations of some results from [3]. Lemma 1. ([3], Lemma 2) In a commutative dimonoid (D,≺,≻) the equalities (x ≺ y) ≺ z = x ≺ (y ≻ z) = = (x ≻ y) ≺ z = x ≻ (y ≺ z) = = (x ≺ y) ≻ z = x ≻ (y ≻ z) hold for all x, y, z ∈ D. Jo u rn al A lg eb ra D is cr et e M at h .112 Free commutative dimonoids From Lemma 1 it follows that the operations ≺ and ≻ of a commu- tative dimonoid (D,≺,≻) are indistinguishable for three and more mul- tipliers and the product of these elements doesn’t depend on the paren- thesizing. A commutative idempotent semigroup will be called a semilattice. If ρ is a congruence on the dimonoid (D,≺,≻) such that (D,≺,≻)/ρ is an idempotent dimonoid, then we say that ρ is an idempotent congruence. Let (D,≺,≻) be a dimonoid with a commutative operation ≺, a, b ∈ D. We say that a ≺-divide b and write a≺\|b, if there exists such element x from (D,≺) with an identity that a ≺ x = b. As usual N denotes the set of positive integers. Let (D,≺,≻) be a dimonoid, a ∈ D, n ∈ N . Denote by an the degree n of an element a concerning the operation ≺. Define a relation η on the dimonoid (D,≺,≻) with a commutative operation ≺ by aηb if and only if there exist positive integers m, n, m 6= 1, n 6= 1 such that a≺\|b m, b≺\|a n. Theorem 2. ([3], Theorem 1) The relation η on the dimonoid (D,≺,≻) with a commutative operation ≺ is the least idempotent congruence, and (D,≺,≻)/η is a commutative idempotent dimonoid which is a semilattice. 3. Constructions In this section we construct a free commutative dimonoid, describe the least idempotent congruence η on this dimonoid and characterize the corresponding classes of the congruence. We also consider separately the free commutative dimonoids of the rank 1 and 2. Let A be an alphabet, F [A] be a free commutative semigroup over A, G be a set of non-ordered pairs (p, q), p, q ∈ A. Define the operations ≺ and ≻ on the set F [A] ⋃ G by a1...am ≺ b1...bn = a1...amb1...bn, a1...am ≻ b1...bn = { a1...amb1...bn, mn > 1, (a1, b1), m = n = 1, a1...am ≺ (p, q) = a1...am ≻ (p, q) = a1...ampq, (p, q) ≺ a1...am = (p, q) ≻ a1...am = pqa1...am, (p, q) ≺ (r, s) = (p, q) ≻ (r, s) = pqrs for all a1...am, b1...bn ∈ F [A], (p, q), (r, s) ∈ G. An immediate verifica- tion shows that axioms of a dimonoid hold concerning the operations ≺,≻ Jo u rn al A lg eb ra D is cr et e M at h .A. V. Zhuchok 113 and thus, (F [A] ⋃ G,≺,≻) is a dimonoid. It is clear that the operations ≺,≻ are commutative. Theorem 3. (F [A] ⋃ G,≺,≻) is a free commutative dimonoid. Proof. Show that (F [A] ⋃ G,≺,≻) is free. Let (T,≺ ′ ,≻ ′ ) be an arbitrary commutative dimonoid, α : A → T an arbitrary map. Define a map θ : (F [A] ⋃ G,≺,≻) → (T,≺ ′ ,≻ ′ ) : w 7→ wθ, assuming wθ = { a1α ≺ ′ ... ≺ ′ amα, w = a1...am, pα ≻ ′ qα, w = (p, q) for all w ∈ F [A] ⋃ G. We show that θ is a homomorphism. For arbitrary elements a1...am, b1...bn ∈ F [A], (p, q), (r, s) ∈ G we obtain (a1...am ≺ b1...bn)θ = (a1...amb1...bn)θ = = a1α ≺ ′ ... ≺ ′ amα ≺ ′ b1α ≺ ′ ... ≺ ′ bnα = = (a1...am)θ ≺ ′ (b1...bn)θ, (a1...am ≺ (p, q))θ = (a1...ampq)θ = = a1α ≺ ′ ... ≺ ′ amα ≺ ′ pα ≺ ′ qα = = (a1...am)θ ≺ ′ pα ≺ ′ qα = = (a1...am)θ ≺ ′ (pα ≻ ′ qα) = = (a1...am)θ ≺ ′ (p, q)θ, ((p, q) ≺ (r, s))θ = (pqrs)θ = = pα ≺ ′ qα ≺ ′ rα ≺ ′ sα = = (pα ≻ ′ qα) ≺ ′ (rα ≻ ′ sα) = = (p, q)θ ≺ ′ (r, s)θ by Lemma 1. So, (w1 ≺ w2)θ = w1θ ≺ ′ w2θ for all w1, w2 ∈ F [A] ⋃ G. If mn > 1, then (a1...am ≻ b1...bn)θ = (a1...amb1...bn)θ = = a1α ≺ ′ ... ≺ ′ amα ≺ ′ b1α ≺ ′ ... ≺ ′ bnα = Jo u rn al A lg eb ra D is cr et e M at h .114 Free commutative dimonoids = (a1...am)θ ≻ ′ (b1...bn)θ by Lemma 1. In the case m = n = 1, (a1 ≻ b1)θ = (a1, b1)θ = = a1α ≻ ′ b1α = a1θ ≻ ′ b1θ. Moreover, (a1...am ≻ (p, q))θ = (a1...ampq)θ = = a1α ≺ ′ ... ≺ ′ amα ≺ ′ pα ≺ ′ qα = = (a1...am)θ ≺ ′ pα ≺ ′ qα = = (a1...am)θ ≻ ′ (pα ≻ ′ qα) = = (a1...am)θ ≻ ′ (p, q)θ, ((p, q) ≻ (r, s))θ = (pqrs)θ = = pα ≺ ′ qα ≺ ′ rα ≺ ′ sα = = pα ≻ ′ qα ≻ ′ rα ≻ ′ sα = = (p, q)θ ≻ ′ (r, s)θ by Lemma 1. So, (w1 ≻ w2)θ = w1θ ≻ ′ w2θ for all w1, w2 ∈ F [A] ⋃ G. Now we describe the least idempotent congruence η (see section 2) on the free commutative dimonoid and characterize the corresponding classes of this congruence. Recall that N denotes the set of positive integers. Define the opera- tions ≺ and ≻ on the set N ⋃ {2̃} by m ≺ n = m+ n, m ≺ 2̃ = 2̃ ≺ m = m ≻ 2̃ = 2̃ ≻ m = m+ 2, m ≻ n = { 2̃, m = n = 1, m+ n otherwise, 2̃ ≺ 2̃ = 2̃ ≻ 2̃ = 4 for all m, n ∈ N . The set N ⋃ {2̃} with the operations ≺ and ≻ is a dimonoid. We denote the dimonoid obtained by N(2̃). Define the operation ≺ on the set N2 ⋃ {1} by (m,n) ≺ (p, l) = (m+ p, n+ l), Jo u rn al A lg eb ra D is cr et e M at h .A. V. Zhuchok 115 (m,n) ≺ 1 = 1 ≺ (m,n) = = (m+ 1, n+ 1), 1 ≺ 1 = (2, 2) for all (m,n), (p, l) ∈ N2. The set N2 ⋃ {1} concerning this operation is a semigroup. We denote by N2 (1) this semigroup. Denote by Nk the Cartesian product of k copies of the additive semi- group of positive integers. For every w ∈ F [A] the set of all elements x ∈ A occurring in w will be denoted by c(w) and assume d(u) = { {p, q}, u = (p, q) ∈ G, c(u), u ∈ F [A] for all u ∈ F [A] ⋃ G. The equivalence w1ηw2 ⇔ d(w1) = d(w2) for all w1, w2 ∈ (F [A] ⋃ G,≺,≻) follows immediately from Theorem 2. Denote by F the dimonoid (F [A] ⋃ G,≺,≻)/η . Theorem 4. The dimonoid F is a semilattice isomorphic to the semi- lattice Ω(A) of nonempty finite subsets of the set A with respect to the operation of a union. Let Fw be a class of the congruence η on the di- monoid (F [A] ⋃ G,≺,≻) with a representative w ∈ Fw. Then 1) if \|d(w)\| = 1, then Fw ∼= N(2̃), 2) if \|d(w)\| = 2, then Fw ∼= N2 (1) , 3) if \|d(w)\| = k ≥ 3, then Fw ∼= Nk. Proof. It is immediate to cheek that the map d : F → Ω(A) : Fw 7→ d(w) is an isomrphism. Let \|d(w)\| = 1 and let d(w) = {x}. It is easy to cheek that the map α1 : Fw → N(2̃) : u 7→ uα1 = { s, u = xs, 2̃, u = (x, x) is an isomorphism. If \|d(w)\| = 2, d(w) = {x, y} and u = xs1ys2 is the canonical form of a word u ∈ Fw\{(x, y)}, s1, s2 ∈ N , then we can show that the map α2 : Fw → N2 (1) : v 7→ vα2 = { (s, t), v = xsyt, 1, v = (x, y) Jo u rn al A lg eb ra D is cr et e M at h .116 Free commutative dimonoids is an isomorphism. Finally, let \|d(w)\| = k ≥ 3 and let u = x p1 1 x p2 2 ...x pk k be the canonical form of a word u ∈ Fw, xi ∈ A, pi ∈ N, 1 ≤ i ≤ k. It is not difficult to show that the map α3 : Fw → Nk : u = x p1 1 x p2 2 ...x pk k 7→ uα3 = (p1, p2, ..., pk) is an isomorphism. Now we consider the free commutative dimonoids of the small ranks. Proposition 3. If \|A\| = 1, then (F [A] ⋃ G,≺,≻) ∼= N(2̃). Proof. Let A = {a}. Define a map µ : (F [A] ⋃ G,≺,≻) → N(2̃) : w 7→ wµ, where wµ = { k, w = ak ∈ F [A], 2̃, w = (a, a). An immediate verification shows that µ is an isomorphism. Let Ñ = (N0 × N0)\{(0, 0)}, where N0 is the additive semigroup of positive integers with a zero, and let {1, 2̃, 2̄} be an arbitrary three- element set. Define the operations ≺ and ≻ on the set Ñ ⋃ {1, 2̃, 2̄} by (m,n) ∗ 1 = 1 ∗ (m,n) = (m+ 1, n+ 1), (m,n) ∗ 2̃ = 2̃ ∗ (m,n) = (m+ 2, n), (m,n) ∗ 2̄ = 2̄ ∗ (m,n) = (m, n+ 2), 1 ∗ 1 = (2, 2), 2̃ ∗ 2̃ = (4, 0), 2̄ ∗ 2̄ = (0, 4), 1 ∗ 2̃ = 2̃ ∗ 1 = (3, 1), 2̃ ∗ 2̄ = 2̄ ∗ 2̃ = (2, 2), 1 ∗ 2̄ = 2̄ ∗ 1 = (1, 3), where ∗ =≺ or ≻, and (m,n) ≺ (p, l) = (m+ p, n+ l), (m,n) ≻ (p, l) =            1, if (m,n) = (1, 0), (p, l) = (0, 1), 1, if (m,n) = (0, 1), (p, l) = (1, 0), 2̃, if (m,n) = (p, l) = (1, 0), 2̄, if (m,n) = (p, l) = (0, 1), (m+ p, n+ l) otherwise for all (m,n), (p, l) ∈ Ñ . A long verification shows that (Ñ ⋃ {1, 2̃, 2̄},≺ ,≻) is a dimonoid. We denote this dimonoid by N(1,2̃,2̄). Jo u rn al A lg eb ra D is cr et e M at h .A. V. Zhuchok 117 Proposition 4. If \|A\| = 2, then (F [A] ⋃ G,≺,≻) ∼= N(1,2̃,2̄). Proof. Let A = {a, b} and let u = ambn be the canonical form of a word u ∈ F [A],m, n ∈ N0 (m and n are not equal to a zero concurrently). Define a map τ : (F [A] ⋃ G,≺,≻) → N(1,2̃,2̄) : w 7→ wτ, where wτ =        (m,n), w = ambn ∈ F [A], 1, w = (a, b) ∈ G, 2̃, w = (a, a) ∈ G, 2̄, w = (b, b) ∈ G. An immediate verification shows that τ is an isomorphism. 4. Some properties In this section we describe some properties of commutative dimonoids. Recall the definitions of Green’s relations on a semigroup S. Green’s relations on S are called the binary relations: L = {(x; y) ∈ S × S ∣ ∣S1 x = S1y}, ℜ = {(x; y) ∈ S × S \|xS 1 = yS1}, ℑ = {(x; y) ∈ S × S ∣ ∣S1xS1 = S1yS1}, H = L ⋂ ℜ, D = L ◦ ℜ, where S1 is a semigroup with an identity, L ◦ ℜ is the composition of binary relations. Let (D,≺,≻) be a dimonoid and let K be one of Green’s relations on (D,≺). Then we will call K a Green’s relation on the dimonoid (D,≺,≻). Lemma 2. In a commutative dimonoid (D,≺,≻) all Green’s relations coincide and are congruences. Proof. An equality of Green’s relations on (D,≺,≻) follows from the equality of Green’s relations on the commutative semigroup (D,≺) (see [6]). It is well-known also that the relation L is a congruence on the semi- group (D,≺) (see [6]). We show that L is compatible with the operation ≻. Jo u rn al A lg eb ra D is cr et e M at h .118 Free commutative dimonoids Let xLy, x, y, c ∈ D. Then y = t1 ≺ x, x = t2 ≺ y for some t1, t2 ∈ D. Hence, c ≻ y = c ≻ (t1 ≺ x) = (c ≻ t1) ≺ x = = (t1 ≻ c) ≺ x = t1 ≺ (c ≻ x), c ≻ x = c ≻ (t2 ≺ y) = (c ≻ t2) ≺ y = = (t2 ≻ c) ≺ y = t2 ≺ (c ≻ y) according to Lemma 1. So, c ≻ xL c ≻ y. From the commutativity of the operation ≻ it follows that x ≻ cL y ≻ c. Thus, L is a congruence on (D,≺,≻). Corollary 1. Green’s relations on the free commutative dimonoid (F [A] ⋃ G, ≺,≻) are equal to the diagonal of F [A] ⋃ G. Let (D,≺,≻) be a commutative dimonoid, n ∈ N, n > 1. Recall that we denote by an the degree n of an element a ∈ D concerning the operation ≺. Lemma 3. The map β : x 7→ xn is an endomorphism of the commuta- tive dimonoid (D,≺,≻) and the operations of the dimonoid (D,≺,≻)/∆β coincide. Proof. If a, b ∈ D, then (a ≺ b)β = (a ≺ b)n = = an ≺ bn = aβ ≺ bβ, (a ≻ b)β = (a ≻ b)n = an ≺ bn = = an ≻ bn = aβ ≻ bβ according to Lemma 1. As, in a commutative dimonoid, (a ≺ b)n = (a ≻ b)n for all a, b ∈ D, n ∈ N, n > 1, then the operations of (D,≺,≻)/∆β coincide. Corollary 2. The map β is a homomorphism from the commutative dimonoid (D,≺,≻) to the semigroup (D,≺). For all h = (x, y) ∈ G let [h] be an element xy ∈ F [A]. Jo u rn al A lg eb ra D is cr et e M at h .A. V. Zhuchok 119 Corollary 3. Let (F [A] ⋃ G,≺,≻) be a free commutative dimonoid. For all w, u ∈ (F [A] ⋃ G,≺,≻) we have w∆βu if and only if one of the fol- lowing statements holds: (i) if w, u ∈ F [A], then w = u, (ii) if w ∈ G, then u = [w] or u = w, (iii) if u ∈ G, then w = [u] or w = u. References [1] J.-L. Loday, Dialgebras, In: Dialgebras and related operads, Lecture Notes in Math. 1763, Springer-Verlag, Berlin, 2001, pp.7-66. [2] T. Pirashvili, Sets with two associative operations, Central European Journal of Mathematics, 2, 2003, pp.169-183. [3] A.V. Zhuchok, Commutative dimonoids, Algebra and Discrete Mathematics, N.2, 2009, pp.116-127. [4] G. Birkhoff, On the structure of abstract algebras, Proc. Cambr. Phil. Soc., 31, 1935, pp.433-454. [5] A. Malchev, Algebraic systems, Publish. «Nauka», Moscow, 1970, 392 p. (In Rus- sian). [6] A.H. Clifford, G.B. Preston, The algebraic theory of semigroups, vol.1, 2, American Mathematical Society, 1964, 1967. Contact information A. V. Zhuchok Department of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska str., 64, 01033 Kyiv, Ukraine E-Mail: zhuchok_a@mail.ru Received by the editors: 15.09.2009 and in final form 30.04.2010.

Free commutative dimonoids

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