Free normal dibands

Free normal dibands

We construct a free normal diband, a free (ℓn,n)-diband, a free (n,rn)-diband and a free (ℓn,rn)-diband. We also describe the structure of free normal dibands and characterize some least congruences on these dibands.

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Datum:2011
1. Verfasser: Zhuchok, A.V.
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Veröffentlicht: Інститут прикладної математики і механіки НАН України 2011
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Zitieren:Free normal dibands / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 112–127. — Бібліогр.: 11 назв. — англ.

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spelling irk-123456789-1548042019-06-17T01:31:11Z Free normal dibands Zhuchok, A.V. We construct a free normal diband, a free (ℓn,n)-diband, a free (n,rn)-diband and a free (ℓn,rn)-diband. We also describe the structure of free normal dibands and characterize some least congruences on these dibands. 2011 Article Free normal dibands / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 112–127. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 Mathematics Subject Classification:08B20, 20M10, 20M50, 17A30,17A32 http://dspace.nbuv.gov.ua/handle/123456789/154804 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 normal diband, a free (ℓn,n)-diband, a free (n,rn)-diband and a free (ℓn,rn)-diband. We also describe the structure of free normal dibands and characterize some least congruences on these dibands.
format Article
author Zhuchok, A.V.
spellingShingle Zhuchok, A.V.
Free normal dibands
Algebra and Discrete Mathematics
author_facet Zhuchok, A.V.
author_sort Zhuchok, A.V.
title Free normal dibands
title_short Free normal dibands
title_full Free normal dibands
title_fullStr Free normal dibands
title_full_unstemmed Free normal dibands
title_sort free normal dibands
publisher Інститут прикладної математики і механіки НАН України
publishDate 2011
url http://dspace.nbuv.gov.ua/handle/123456789/154804
citation_txt Free normal dibands / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 112–127. — Бібліогр.: 11 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT zhuchokav freenormaldibands
first_indexed 2025-07-14T06:53:35Z
last_indexed 2025-07-14T06:53:35Z
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fulltext Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 12 (2011). Number 2. pp. 112 – 127 c© Journal “Algebra and Discrete Mathematics” Free normal dibands Anatolii V. Zhuchok Communicated by V. I. Sushchansky Abstract. We construct a free normal diband, a free (ℓn, n)- diband, a free (n, rn)-diband and a free (ℓn, rn)-diband. We also describe the structure of free normal dibands and characterize some least congruences on these dibands. 1. Introduction and preliminaries The notions of a dialgebra and a dimonoid were introduced by J.-L. Loday [1]. For further details and background see [1], [10]. J.-L. Loday constructed a free dimonoid [1]. Pirashvili [3] introduced the notion of a duplex and constructed a free duplex. Dimonoids in the sense of Loday [1] are examples of duplexes. In [6] a free commutative di- monoid was constructed. Free rectangular dimonoids (rectangular dibands) were construcred in [9]. In this paper the research which was started in [6] and [9] is continued. Here we construct a free normal diband, a free (ℓn, n)-diband, a free (n, rn)-diband and a free (ℓn, rn)-diband. It turns out that the operations of a dimonoid with left (right) normal bands coincide and it is a left (right) normal band. We also describe the structure of free normal dibands and, as a consequence, obtain the description of some least congruences on free normal dibands. We refer to [6] and [9] for the terminology and notations. 2010 Mathematics Subject Classification: 08B20, 20M10, 20M50, 17A30, 17A32. Key words and phrases: normal diband, free normal diband, diband of subdi- monoids, dimonoid, semigroup. A. V. Zhuchok 113 Recall that an idempotent semigroup S is called a normal band, if axya = ayxa for all a, x, y ∈ S. It is well-known that a normal band satisfies any identity of the form ax1x2...xnb = ax1πx2π...xnπb, (1) where π is a permutation of {1, 2, ..., n}. A dimonoid (D,⊣,⊢) will be called a normal diband, if both semigroups (D,⊣) and (D,⊢) are normal bands. Lemma 1. ([11], Sect. 3.5, Lemma) Let (D,⊣,⊢) be an arbitrary di- monoid, x, ai ∈ D, 1 ≤ i ≤ n, n ∈ N , n > 1. Then (i) (an ⊣ ... ⊣ ai ⊣ ... ⊣ a1) ⊢ x = an ⊢ ... ⊢ ai ⊢ ... ⊢ a1 ⊢ x; (ii) x ⊣ (a1 ⊢ ... ⊢ ai ⊢ ... ⊢ an) = x ⊣ a1 ⊣ ... ⊣ ai ⊣ ... ⊣ an. Lemma 2. Let (D,⊣,⊢) be an idempotent dimonoid. Then (D,⊣) is a normal band if and only if (D,⊢) is a normal band. Proof. If (D,⊣) is a normal band, a, x, y ∈ D, then a ⊣ x ⊣ y ⊣ a = a ⊣ y ⊣ x ⊣ a. Multiplying both parts of the last equality on the right by a concerning the operation ⊢, we obtain (a ⊣ x ⊣ y ⊣ a) ⊢ a = a ⊢ x ⊢ y ⊢ a ⊢ a = a ⊢ x ⊢ y ⊢ a, (a ⊣ y ⊣ x ⊣ a) ⊢ a = a ⊢ y ⊢ x ⊢ a ⊢ a = a ⊢ y ⊢ x ⊢ a according to Lemma 1 (i) and the idempotent property of the operation ⊢. So, (D,⊢) is a normal band. Conversely, let (D,⊢) be a normal band. Then a ⊢ x ⊢ y ⊢ a = a ⊢ y ⊢ x ⊢ a for all a, x, y ∈ D. Multiplying both parts of the last equality on the left by a concerning the operation ⊣, we obtain a ⊣ (a ⊢ x ⊢ y ⊢ a) = a ⊣ a ⊣ x ⊣ y ⊣ a = a ⊣ x ⊣ y ⊣ a, a ⊣ (a ⊢ y ⊢ x ⊢ a) = a ⊣ a ⊣ y ⊣ x ⊣ a = a ⊣ y ⊣ x ⊣ a according to Lemma 1 (ii) and the idempotent property of the operation ⊣. So, (D,⊣) is a normal band. 114 Free normal dibands For an arbitrary nonempty set X denote the set of all nonempty finite subsets of X by B[X]. Let (D,⊣,⊢) be an arbitrary dimonoid and D be a totally ordered set. For every A = {x1, x2, ..., xn} ∈ B[D] assume −→ A = x1 ⊢ x2 ⊢ ... ⊢ xn, ←− A = x1 ⊣ x2 ⊣ ... ⊣ xn, where x1 < x2 < ... < xn in the total order. Using the identity (1), the idempotent property of the operations of a normal diband and Lemma 1, we can prove the following lemma. Lemma 3. Let (D,⊣,⊢) be a normal diband, D be a totally ordered set and A,B,C ∈ B[D], C ⊆ B, a ∈ A, x, y ∈ D. Then (i) x ⊢ a ⊢ −→ A = x ⊢ −→ A ; (ii) ←− A ⊣ a ⊣ x = ←− A ⊣ x; (iii) −→ A ⊢ a ⊢ x = −→ A ⊢ x = ←− A ⊢ x; (iv) x ⊣ a ⊣ ←− A = x ⊣ ←− A = x ⊣ −→ A ; (v) x ⊢ −−−−→ A ∪B ⊢ y = x ⊢ −→ A ⊢ −→ B ⊢ y = x ⊢ ←−−−− A ∪B ⊢ y; (vi) x ⊣ ←−−−− A ∪B ⊣ y = x ⊣ ←− A ⊣ ←− B ⊣ y = x ⊣ −−−−→ A ∪B ⊣ y; (vii) x ⊢ −→ B ⊢ −→ C ⊢ y = x ⊢ −→ C ⊢ −→ B ⊢ y = x ⊢ −→ B ⊢ y; (viii) x ⊣ ←− B ⊣ ←− C ⊣ y = x ⊣ ←− C ⊣ ←− B ⊣ y = x ⊣ ←− B ⊣ y. Note that the class of normal dibands is a subclass of the variety of all dimonoids which is closed under the taking of homomorphic images, subdimonoids and Cartesian products. Therefore it is a subvariety of the variety of all dimonoids. A dimonoid which is free in the variety of normal dibands will be called a free normal diband. The necessary information about varieties of dimonoids can be found in [6]. Now we consider a free rectangular dimonoid [9]. Let In = {1, 2, ..., n}, n > 1 and let {Xi}i∈In be a family of arbitrary nonempty sets Xi, i ∈ In. Define the operations ⊣ and ⊢ on ∏ i∈In Xi by (x1, ..., xn) ⊣ (y1, ..., yn) = (x1, ..., xn−1, yn), (x1, ..., xn) ⊢ (y1, ..., yn) = (x1, y2, ..., yn) for all (x1, ..., xn), (y1, ..., yn) ∈ ∏ i∈In Xi. Lemma 4. ([9], Lemma 4) For any n > 1, ( ∏ i∈In Xi,⊣,⊢) is a rectan- gular dimonoid. A. V. Zhuchok 115 Obviously, for any n > 1, ( ∏ i∈In Xi,⊣,⊢) is a normal diband. Let X be an arbitrary nonempty set and X3 = X × X × X. We denote the dimonoid (X3,⊣,⊢) by FRct(X). Theorem 1. ([9], Theorem 1) FRct(X) is a free rectangular dimonoid. If f : D1 → D2 is a homomorphism of dimonoids, then the correspond- ing congruence on D1 will be denoted by ∆f . 2. Free normal dibands In this section we construct a free normal diband. Let {Di}i∈I be a family of arbitrary dimonoids Di, i ∈ I and let∏ i∈IDi be a set of all functions f : I → ⋃ i∈I Di such that if ∈ Di for any i ∈ I. It easy to check that ∏ i∈IDi with multiplications defined by i(f ⊣ g) = if ⊣ ig, i(f ⊢ g) = if ⊢ ig, where i ∈ I, f, g ∈ ∏ i∈IDi, is a dimonoid. It is called the Cartesian product of dimonoids Di, i ∈ I. Observe that if I is finite, then the Cartesian product and the direct product coincide. The Cartesian product of a finite number of dimonoids D1, D2, ..., Dn is denoted by D1 ×D2 × ...×Dn. Let FRct(X) be the free rectangular dimonoid (see Sect. 1), B(X) be the semilattice of all nonempty finite subsets of X with respect to the operation of the set theoretical union and let FND(X) = {((x, y, z), A) ∈ FRct(X)×B(X) |x, y, z ∈ A}. The main result of this section is the following. Theorem 2. FND(X) is a free normal diband. Proof. Clearly, FRct(X) × B(X) is a dimonoid (see above). It is not difficult to see that FND(X) is a subdimonoid of FRct(X)×B(X). It is clear that the operations ⊣ and ⊢ of FND(X) are idempotent. For all ((x, y, z), A), ((a, b, c), B), ((s, c, t), C) ∈ FND(X) we have ((x, y, z), A) ⊣ ((a, b, c), B) ⊣ ((s, c, t), C) ⊣ ((x, y, z), A) = = ((x, y, c), A ∪B) ⊣ ((s, c, t), C) ⊣ ((x, y, z), A) = = ((x, y, t), A ∪B ∪ C) ⊣ ((x, y, z), A) = ((x, y, z), A ∪B ∪ C), ((x, y, z), A) ⊣ ((s, c, t), C) ⊣ ((a, b, c), B) ⊣ ((x, y, z), A) = = ((x, y, t), A ∪ C) ⊣ ((a, b, c), B) ⊣ ((x, y, z), A) = 116 Free normal dibands = ((x, y, c), A ∪ C ∪B) ⊣ ((x, y, z), A) = ((x, y, z), A ∪ C ∪B). Hence FND(X) is a normal band concerning the operation ⊣. By Lemma 2 FND(X) is a normal band concerning the operation ⊢. So, FND(X) is a normal diband. Let us show that FND(X) is free. Let (T,⊣ ′ ,⊢ ′ ) be an arbitrary normal diband, T be a totally ordered set and let γ : X → T be an arbitrary map. For every A = {x1, x2, ..., xn} ∈ B[X] assume Aγ = {xiγ | 1 ≤ i ≤ n} and define a map µ : FND(X)→ (T, ⊣′, ⊢′) : ((x, y, z), A) 7→ ((x, y, z), A)µ, assuming ((x, y, z), A)µ = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ zγ for all ((x, y, z), A) ∈ FND(X). We show that µ is a homomorphism. We will use the axioms of a dimonoid, Lemma 3 and the idempotent property of the operations. For arbitrary elements ((x, y, z), A), ((a, b, c), B) ∈ FND(X) we have ((x, y, z), A)µ = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ zγ, ((a, b, c), B)µ = aγ ⊢′ −→ Bγ ⊢ ′ bγ ⊣′ ←− Bγ ⊣ ′ cγ, (((x, y, z), A) ⊣ ((a, b, c), B))µ = ((x, y, c), A ∪B)µ = = xγ ⊢′ −−−−−→ (A ∪B)γ ⊢ ′ yγ ⊣′ ←−−−−− (A ∪B)γ ⊣ ′ cγ, ((x, y, z), A)µ ⊣′ ((a, b, c), B)µ = = (xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ zγ) ⊣′ (aγ ⊢′ −→ Bγ ⊢ ′ bγ ⊣′ ←− Bγ ⊣ ′ cγ) = = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ zγ ⊣′ aγ ⊣′ −→ Bγ ⊣ ′ bγ ⊣′ ←− Bγ ⊣ ′ cγ = = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ zγ ⊣′ aγ ⊣′ ←− Bγ ⊣ ′ bγ ⊣′ ←− Bγ ⊣ ′ cγ = = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ ←− Bγ ⊣ ′ bγ ⊣′ ←− Bγ ⊣ ′ cγ = = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ ←− Bγ ⊣ ′ ←− Bγ ⊣ ′ cγ = = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ ⊣ ′ ←− Bγ ⊣ ′ cγ = = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ = = (xγ ⊢′ −→ Aγ) ⊢ ′ (yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ) ⊢′ (yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ) = = (xγ ⊢′ −→ Aγ) ⊢ ′ (yγ ⊣′ −−−−−−→ (A ∪B)γ ⊣ ′ cγ) ⊢′ (yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ) = = (xγ ⊢′ −→ Aγ) ⊢ ′ (yγ ⊢′ −−−−−−→ (A ∪B)γ ⊢ ′ cγ) ⊢′ (yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ) = A. V. Zhuchok 117 = xγ ⊢′ −→ Aγ ⊢ ′ −−−−−−→ (A ∪B)γ ⊢ ′ cγ ⊢′ (yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ) = = xγ ⊢′ −−−−−−→ (A ∪B)γ ⊢ ′ cγ ⊢′ (yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ) = = xγ ⊢′ −−−−−−→ (A ∪B)γ ⊢ ′ (yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ) = = xγ ⊢′ −−−−−−→ (A ∪B)γ ⊢ ′ yγ ⊣′ ←−−−−−− (A ∪B)γ ⊣ ′ cγ. Thus, (((x, y, z), A) ⊣ ((a, b, c), B))µ = ((x, y, z), A)µ ⊣′ ((a, b, c), B)µ for all ((x, y, z), A), ((a, b, c), B) ∈ FND(X). Analogously, we can prove that (((x, y, z), A) ⊢ ((a, b, c), B))µ = ((x, y, z), A)µ ⊢′ ((a, b, c), B)µ for all ((x, y, z), A), ((a, b, c), B) ∈ FND(X). This completes the proof of Theorem 2. Obviously, the free normal diband FND(X) generated by a finite set X is finite. Specifically, |FND(X)| = ∑ A∈B[X] |A| 3. 3. Dimonoids and (left, right) normal bands In this section we show that the operations of a dimonoid (D,⊣,⊢) with a left (respectively, right) normal band (D,⊢) (respectively, (D,⊣)) coincide and construct a free (ℓn, n)-diband, a free (n, rn)-diband and a free (ℓn, rn)-diband. Recall that an idempotent semigroup S is called a left normal band, if axy = ayx (2) for all a, x, y ∈ S. If instead of (2) the identity xya = yxa (3) holds, then S is a right normal band. It is well-known that a left normal band satisfies any identity of the form ax1x2...xn = ax1πx2π...xnπ, (4) where π is a permutation of {1, 2, ..., n}. Dually, a right normal band satisfies any identity of the form x1x2...xna = x1πx2π...xnπa, (5) where π is a permutation of {1, 2, ..., n}. 118 Free normal dibands Lemma 5. The operations of a dimonoid (D,⊣,⊢) coincide, if one of the following conditions holds: (i) (D,⊢) is a left normal band; (ii) (D,⊣) is a right normal band. Proof. (i) For all x, y, z ∈ D we have x ⊢ (y ⊣ z) = x ⊢ (y ⊣ z) ⊢ (y ⊣ z) = = x ⊢ (y ⊢ z) ⊢ (y ⊣ z) = x ⊢ (y ⊣ z) ⊢ (y ⊢ z) = = x ⊢ (y ⊢ z) ⊢ (y ⊢ z) = x ⊢ (y ⊢ z) = = (x ⊢ y) ⊢ z = (x ⊢ y) ⊣ z according to the idempotent property of the operation ⊢, the axioms of a dimonoid and the identity (2). Substituting y = x in the last equality and using the idempotent property of the operation ⊢, we obtain x ⊢ z = x ⊣ z. (ii) For all x, y, z ∈ D we have (x ⊢ y) ⊣ z = (x ⊢ y) ⊣ (x ⊢ y) ⊣ z = = (x ⊢ y) ⊣ (x ⊣ y) ⊣ z = (x ⊣ y) ⊣ (x ⊢ y) ⊣ z = = (x ⊣ y) ⊣ (x ⊣ y) ⊣ z = (x ⊣ y) ⊣ z = = x ⊣ (y ⊣ z) = x ⊢ (y ⊣ z) according to the idempotent property of the operation ⊣, the axioms of a dimonoid and the identity (3). Substituting z = y in the last equality and using the idempotent property of the operation ⊣, we obtain x ⊣ y = x ⊢ y. From Lemma 5 (i) (respectively, Lemma 5 (ii)) it follows that a di- monoid (D,⊣,⊢) with left (respectively, right) normal bands (D,⊣) and (D,⊢) is a left (respectively, right) normal band. Consider the semigroups Xℓz, Xrz, Xrb and the dimonoids Xℓz,rz, Xrb,rz, Xℓz,rb which were defined in [9]. It is easy to see that Xℓz, Xrz, Xrb are normal bands and Xℓz,rz, Xrb,rz, Xℓz,rb are normal dibands. Let Brb(X) = {((x, y), A) ∈ Xrb ×B(X) |x, y ∈ A}, Bℓz(X) = {(x,A) ∈ Xℓz ×B(X) |x ∈ A}, Brz(X) = {(x,A) ∈ Xrz ×B(X) |x ∈ A}, Bℓz,rb(X) = {((x, y), A) ∈ Xℓz,rb ×B(X) |x, y ∈ A}, Brb,rz(X) = {((x, y), A) ∈ Xrb,rz ×B(X) |x, y ∈ A}, A. V. Zhuchok 119 Bℓz,rz(X) = {(x,A) ∈ Xℓz,rz ×B(X) |x ∈ A}. It is clear that Brb(X), Bℓz(X), Brz(X) are subsemigroups of Xrb × B(X), Xℓz ×B(X), Xrz ×B(X) respectively, and Bℓz,rb(X), Brb,rz(X), Bℓz,rz(X) are subdimonoids of Xℓz,rb×B(X),Xrb,rz×B(X),Xℓz,rz×B(X) respectively. By [2] Brb(X), Bℓz(X) and Brz(X) are the free normal band, the free left normal band and the free right normal band respectively. A dimonoid (D,⊣,⊢) will be called a (ℓn, n)-diband, if (D,⊣) is a left normal band and (D,⊢) is a normal band. A dimonoid (D,⊣,⊢) will be called a (n, rn)-diband, if (D,⊣) is a normal band and (D,⊢) is a right normal band. A dimonoid (D,⊣,⊢) will be called a (ℓn, rn)-diband, if (D,⊣) is a left normal band and (D,⊢) is a right normal band. Note that every left (right) normal band is normal and the class of (ℓn, n)-dibands ((n, rn)-dibands, (ℓn, rn)-dibands) is a subvariety of the variety of all normal dibands. A dimonoid which is free in the variety of (ℓn, n)-dibands (respectively, (n, rn)-dibands, (ℓn, rn)-dibands) will be called a free (ℓn, n)-diband (respectively, free (n, rn)-diband, free (ℓn, rn)- diband). For the proofs of the following three lemmas we will use the notations from Sect. 1 and from the proof of Theorem 2. Lemma 6. Bℓz,rb(X) is a free (ℓn, n)-diband. Proof. Clearly, Bℓz,rb(X) is a (ℓn, n)-diband. Let us show that Bℓz,rb(X) is free. Let (T,⊣ ′ ,⊢ ′ ) be an arbitrary (ℓn, n)-diband, T be a totally ordered set and let γ : X → T be an arbitrary map. Define the map φℓn,n : Bℓz,rb(X)→ (T, ⊣′, ⊢′) : ((x, y), A) 7→ ((x, y), A)φℓn,n = xγ ⊢′ −→ Aγ ⊢ ′ yγ ⊣′ ←− Aγ . Similarly to the proof of Theorem 2, we can show that φℓn,n is a homo- morphism. For this, we also use (4). Lemma 7. Brb,rz(X) is a free (n, rn)-diband. Proof. Obviously, Brb,rz(X) is a (n, rn)-diband. Show that Brb,rz(X) is free. Let (T,⊣ ′ ,⊢ ′ ) be an arbitrary (n, rn)-diband, T be a totally ordered set and let γ : X → T be an arbitrary map. Define the map φn,rn : Brb,rz(X)→ (T, ⊣′, ⊢′) : ((x, y), A) 7→ ((x, y), A)φn,rn = −→ Aγ ⊢ ′ xγ ⊣′ ←− Aγ ⊣ ′ yγ. Analysis similar to that in the proof of Theorem 2 shows that φn,rn is a homomorphism. Our proof also uses (5). 120 Free normal dibands Lemma 8. Bℓz,rz(X) is a free (ℓn, rn)-diband. Proof. It is evident that Bℓz,rz(X) is a (ℓn, rn)-diband. Let (T,⊣ ′ ,⊢ ′ ) be an arbitrary (ℓn, rn)-diband, T be a totally ordered set and let γ : X → T be an arbitrary map. Define the map φℓn,rn : Bℓz,rz(X)→ (T, ⊣′, ⊢′) : (x,A) 7→ (x,A)φℓn,rn = −→ Aγ ⊢ ′ xγ ⊣′ ←− Aγ . Similarly to the proof of Theorem 2, the fact that φℓn,rn is a homomorphism can be proved. To do this, also use (4) and (5). 4. Decompositions of FND(X) In this section we describe the structure of free normal dibands and characterize some least congruences on these dibands. Let B(i,j,k)(X) = {A ∈ B(X) | i, j, k ∈ A}, B (i) rb (X) = {((x, y), A) ∈ Brb(X) | i ∈ A}, B (i,j) ℓz (X) = {(x,A) ∈ Bℓz(X) | i, j ∈ A}, B(i,j) rz (X) = {(x,A) ∈ Brz(X) | i, j ∈ A}, B (i) ℓz,rb(X) = {((x, y), A) ∈ Bℓz,rb(X) | i ∈ A}, B (i) rb,rz(X) = {((x, y), A) ∈ Brb,rz(X) | i ∈ A}, B (i,j) ℓz,rz(X) = {(x,A) ∈ Bℓz,rz(X) | i, j ∈ A} for all i, j, k ∈ X. It is evident thatB(i,j,k)(X),B (i) rb (X),B (i,j) ℓz (X),B (i,j) rz (X) are subsemigroups of B(X), Brb(X), Bℓz(X), Brz(X) respectively, and B (i) ℓz,rb(X), B (i) rb,rz(X), B (i,j) ℓz,rz(X) are subdimonoids of Bℓz,rb(X), Brb,rz(X), Bℓz,rz(X) respectively. For all i, j, k ∈ X put M(i,j,k) = {((x, y, z), A) ∈ FND(X) | (x, y, z) = (i, j, k)}, M(i,j) = {((x, y, z), A) ∈ FND(X) | (x, y) = (i, j)}, M(i,j] = {((x, y, z), A) ∈ FND(X) | (y, z) = (i, j)}, M[i,j] = {((x, y, z), A) ∈ FND(X) | (x, z) = (i, j)}, M(i) = {((x, y, z), A) ∈ FND(X) |x = i}, A. V. Zhuchok 121 M(i] = {((x, y, z), A) ∈ FND(X) | y = i}, M[i] = {((x, y, z), A) ∈ FND(X) | z = i}; for all i, j ∈ X, Y ∈ B(X) such that i, j ∈ Y put MY (i,j) = {((x, y, z), A) ∈ FND(X) | ((x, y), A) = ((i, j), Y )}, MY (i,j] = {((x, y, z), A) ∈ FND(X) | ((y, z), A) = ((i, j), Y )}, MY [i,j] = {((x, y, z), A) ∈ FND(X) | ((x, z), A) = ((i, j), Y )}; for all i ∈ X, Y ∈ B(X) such that i ∈ Y put MY (i) = {((x, y, z), A) ∈ FND(X) | (x,A) = (i, Y )}, MY (i] = {((x, y, z), A) ∈ FND(X) | (y,A) = (i, Y )}, MY [i] = {((x, y, z), A) ∈ FND(X) | (z,A) = (i, Y )}; for all Y ∈ B(X) put MY = {((x, y, z), A) ∈ FND(X) |A = Y }. The notion of a diband of subdimonoids was introduced in [4] and investigated in [5] (see also [9]). Subsequently, we will deal with diband decompositions and band decompositions of free normal dibands. The following structure theorem gives decompositions of free normal dibands into dibands of subsemigroups. Theorem 3. Let FND(X) be the free normal diband. Then (i) FND(X) is a rectangular diband FRct(X) of subsemigroups M(i,j,k), (i, j, k) ∈ FRct(X) such that M(i,j,k) ∼= B(i,j,k)(X) for every (i, j, k) ∈ FRct(X); (ii) FND(X) is a diband Xℓz,rb of subsemigroups M(i,j), (i, j) ∈ Xℓz,rb such that M(i,j) ∼= B (i,j) rz (X) for every (i, j) ∈ Xℓz,rb; (iii) FND(X) is a diband Xrb,rz of subsemigroups M(i,j], (i, j) ∈ Xrb,rz such that M(i,j] ∼= B (i,j) ℓz (X) for every (i, j) ∈ Xrb,rz; (iv) FND(X) is a left and right diband Xℓz,rz of subsemigroups M(i], i ∈ Xℓz,rz such that M(i] ∼= B (i) rb (X) for every i ∈ Xℓz,rz; (v) FND(X) is a diband Bℓz,rb(X) of subsemigroups MY (i,j), ((i, j), Y )∈ Bℓz,rb(X) such that MY (i,j) ∼= Yrz for every ((i, j), Y ) ∈ Bℓz,rb(X); (vi) FND(X) is a diband Brb,rz(X) of subsemigroups MY (i,j], ((i, j), Y )∈ Brb,rz(X) such that MY (i,j] ∼= Yℓz for every ((i, j), Y ) ∈ Brb,rz(X); (vii) FND(X) is a diband Bℓz,rz(X) of subsemigroups MY (i], (i, Y )∈ Bℓz,rz(X) such that MY (i] ∼= Yrb for every (i, Y ) ∈ Bℓz,rz(X). 122 Free normal dibands Proof. (i) By Theorem 2 the map µFRct : FND(X)→ FRct(X) : ((x, y, z), A) 7→ ((x, y, z), A)µFRct = (x, y, z) is a homomorphism. It is clear that M(i,j,k), (i, j, k) ∈ FRct(X) is a class of ∆µFRct which is a subdimonoid of FND(X). If ((x, y, z), A), ((a, b, c), B) ∈M(i,j,k), then x = a = i, y = b = j, z = c = k and ((x, y, z), A) ⊣ ((a, b, c), B) = ((x, y, c), A ∪B) = ((i, j, k), A ∪B), ((x, y, z), A) ⊢ ((a, b, c), B) = ((x, b, c), A ∪B) = ((i, j, k), A ∪B). Hence the operations of M(i,j,k) coincide and so, it is a semigroup. It is not difficult to show that for every (i, j, k) ∈ FRct(X) the map M(i,j,k) → B(i,j,k)(X) : ((i, j, k), A) 7→ A is an isomorphism. (ii) By Theorem 2 the map µℓz,rb : FND(X)→ Xℓz,rb : ((x, y, z), A) 7→ ((x, y, z), A)µℓz,rb = (x, y) is a homomorphism. It is evident that M(i,j), (i, j) ∈ Xℓz,rb is a class of ∆µℓz,rb which is a subdimonoid of FND(X). If ((x, y, z), A), ((a, b, c), B) ∈ M(i,j), then x = a = i, y = b = j. Similarly to (i), the operations of M(i,j) coincide and so, it is a semigroup. It is easy to check that for every (i, j) ∈ Xℓz,rb the map M(i,j) → B(i,j) rz (X) : ((i, j, z), A) 7→ (z,A) is an isomorphism. (iii) By Theorem 2 the map µrb,rz : FND(X)→ Xrb,rz : ((x, y, z), A) 7→ ((x, y, z), A)µrb,rz = (y, z) is a homomorphism. Similarly to (ii), M(i,j], (i, j) ∈ Xrb,rz is a class of ∆µrb,rz which is a semigroup isomorphic to B (i,j) ℓz (X). (iv) By Theorem 2 the map µℓz,rz : FND(X)→ Xℓz,rz : ((x, y, z), A) 7→ ((x, y, z), A)µℓz,rz = y is a homomorphism. Then M(i], i ∈ Xℓz,rz is a class of ∆µℓz,rz which is a subdimonoid of FND(X). If ((x, y, z), A), ((a, b, c), B) ∈M(i], then A. V. Zhuchok 123 y = b = i. Similarly to (i), the operations of M(i] coincide and so, it is a semigroup. It is easily seen that for every i ∈ Xℓz,rz the map M(i] → B (i) rb (X) : ((x, i, z), A) 7→ ((x, z), A) is an isomorphism. (v) By Theorem 2 the map µ∗ ℓz,rb : FND(X)→ Bℓz,rb(X) : ((x, y, z), A) 7→ ((x, y, z), A)µ∗ ℓz,rb = ((x, y), A) is a homomorphism. Then MY (i,j), ((i, j), Y ) ∈ Bℓz,rb(X) is a class of ∆µ∗ ℓz,rb which is a subdimonoid of FND(X). If ((x, y, z), A), ((a, b, c), B) ∈MY (i,j), then x = a = i, y = b = j, A = B = Y . Similarly to (i), the operations of MY (i,j) coincide and so, it is a semigroup. It is immediate to check that for every ((i, j), Y ) ∈ Bℓz,rb(X) the map MY (i,j) → Yrz : ((i, j, z), Y ) 7→ z is an isomorphism. (vi) By Theorem 2 the map µ∗ rb,rz : FND(X)→ Brb,rz(X) : ((x, y, z), A) 7→ ((x, y, z), A)µ∗ rb,rz = ((y, z), A) is a homomorphism. Similarly to (v), MY (i,j], ((i, j), Y ) ∈ Brb,rz(X) is a class of ∆µ∗ rb,rz which is a semigroup isomorphic to Yℓz. (vii) By Theorem 2 the map µ∗ ℓz,rz : FND(X)→ Bℓz,rz(X) : ((x, y, z), A) 7→ ((x, y, z), A)µ∗ ℓz,rz = (y,A) is a homomorphism. Similarly to (iv), MY (i], (i, Y ) ∈ Bℓz,rz(X) is a class of ∆µ∗ ℓz,rz which is a semigroup isomorphic to Yrb. If ρ is a congruence on a dimonoid (D,⊣,⊢) such that (D,⊣,⊢)/ρ is a (ℓn, n)-diband (respectively, (n, rn)-diband, (ℓn, rn)-diband), then we say that ρ is a (ℓn, n)-congruence (respectively, (n, rn)-congruence, (ℓn, rn)-congruence). Using the terminology of [9], from Theorem 3 we obtain 124 Free normal dibands Corollary 1. Let FND(X) be the free normal diband. Then (i) ∆µFRct is the least rectangular diband congruence on FND(X); (ii) ∆µℓz,rb is the least (ℓz, rb)-congruence on FND(X); (iii) ∆µrb,rz is the least (rb, rz)-congruence on FND(X); (iv) ∆µℓz,rz is the least left zero and right zero congruence on FND(X); (v) ∆µ∗ ℓz,rb is the least (ℓn, n)-congruence on FND(X); (vi) ∆µ∗ rb,rz is the least (n, rn)-congruence on FND(X); (vii) ∆µ∗ ℓz,rz is the least (ℓn, rn)-congruence on FND(X). Proof. (i) By Theorem 1 FRct(X) is the free rectangular dimonoid. Ac- cording to Theorem 3 (i) we obtain (i). (ii) By Lemma 7 from [9] Xℓz,rb is the free (ℓz, rb)-dimonoid. According to Theorem 3 (ii) we obtain (ii). The proof of (iii) is similar. (iv) By Lemma 5 from [9] Xℓz,rz is the free left zero and right zero dimonoid. According to Theorem 3 (iv) we obtain (iv). (v) By Lemma 6 Bℓz,rb(X) is the free (ℓn, n)-diband. According to Theorem 3 (v) we obtain (v). The proof of (vi) is similar. (vii) By Lemma 8 Bℓz,rz(X) is the free (ℓn, rn)-diband. According to Theorem 3 (vii) we obtain (vii). The following structure theorem gives decompositions of free normal dibands into bands of subdimonoids. Theorem 4. Let FND(X) be the free normal diband. Then (i) FND(X) is a rectangular band Xrb of subdimonoids M[i,j], (i, j) ∈ Xrb such that M[i,j] ∼= B (i,j) ℓz,rz(X) for every (i, j) ∈ Xrb; (ii) FND(X) is a left band Xℓz of subdimonoids M(i), i ∈ Xℓz such that M(i) ∼= B (i) rb,rz(X) for every i ∈ Xℓz; (iii) FND(X) is a right band Xrz of subdimonoids M[i], i ∈ Xrz such that M[i] ∼= B (i) ℓz,rb(X) for every i ∈ Xrz; (iv) FND(X) is a normal band Brb(X) of subdimonoids MY [i,j], ((i, j), Y ) ∈ Brb(X) such that MY [i,j] ∼= Yℓz,rz for every ((i, j), Y ) ∈ Brb(X); (v) FND(X) is a left normal band Bℓz(X) of subdimonoids MY (i), (i, Y )∈ Bℓz(X) such that MY (i) ∼= Yrb,rz for every (i, Y ) ∈ Bℓz(X); (vi) FND(X) is a right normal band Brz(X) of subdimonoids MY [i], (i, Y )∈ Brz(X) such that MY [i] ∼= Yℓz,rb for every (i, Y ) ∈ Brz(X); (vii) FND(X) is a semilattice B(X) of subdimonoids MY , Y ∈ B(X) such that MY ∼= FRct(Y ) for every Y ∈ B(X). A. V. Zhuchok 125 Proof. (i) By Theorem 2 the map µrb : FND(X)→ Xrb : ((x, y, z), A) 7→ ((x, y, z), A)µrb = (x, z) is a homomorphism. It is clear that M[i,j], (i, j) ∈ Xrb is a class of ∆µrb which is a subdimonoid of FND(X). It can be shown that for every (i, j) ∈ Xrb the map M[i,j] → B (i,j) ℓz,rz(X) : ((i, y, j), A) 7→ (y,A) is an isomorphism. (ii) By Theorem 2 the map µℓz : FND(X)→ Xℓz : ((x, y, z), A) 7→ ((x, y, z), A)µℓz = x is a homomorphism. It is evident that M(i), i ∈ Xℓz is a class of ∆µℓz which is a subdimonoid of FND(X). It is easy to check that for every i ∈ Xℓz the map M(i) → B (i) rb,rz(X) : ((i, y, z), A) 7→ ((y, z), A) is an isomorphism. (iii) By Theorem 2 the map µrz : FND(X)→ Xrz : ((x, y, z), A) 7→ ((x, y, z), A)µrz = z is a homomorphism. Similarly to (ii), M[i], i ∈ Xrz is a class of ∆µrz which is a dimonoid isomorphic to B (i) ℓz,rb(X). (iv) By Theorem 2 the map µ∗ rb : FND(X)→ Brb(X) : ((x, y, z), A) 7→ ((x, y, z), A)µ∗ rb = ((x, z), A) is a homomorphism. Similarly to (i), MY [i,j], ((i, j), Y ) ∈ Brb(X) is a class of ∆µ∗ rb which is a dimonoid isomorphic to Yℓz,rz. (v) By Theorem 2 the map µ∗ ℓz : FND(X)→ Bℓz(X) : ((x, y, z), A) 7→ ((x, y, z), A)µ∗ ℓz = (x,A) is a homomorphism. It is clear that MY (i), (i, Y ) ∈ Bℓz(X) is a class of ∆µ∗ ℓz which is a subdimonoid of FND(X). It can be shown that for every (i, Y ) ∈ Bℓz(X) the map MY (i) → Yrb,rz : ((i, y, z), Y ) 7→ (y, z) is an isomorphism. 126 Free normal dibands (vi) By Theorem 2 the map µ∗ rz : FND(X)→ Brz(X) : ((x, y, z), A) 7→ ((x, y, z), A)µ∗ rz = (z,A) is a homomorphism. Similarly to (v), MY [i], (i, Y ) ∈ Brz(X) is a class of ∆µ∗ rz which is a dimonoid isomorphic to Yℓz,rb. (vii) By Theorem 2 the map µ∗ : FND(X)→ B(X) : ((x, y, z), A) 7→ ((x, y, z), A)µ∗ = A is a homomorphism. Clearly, MY , Y ∈ B(X) is a class of ∆µ∗ which is a subdimonoid of FND(X). One can show that for every Y ∈ B(X) the map MY → FRct(Y ) : ((x, y, z), Y ) 7→ (x, y, z) is an isomorphism. If ρ is a congruence on a dimonoid (D,⊣,⊢) such that the operations of (D,⊣,⊢)/ρ coincide and it is a (left, right) normal band, then we say that ρ is a (left, right) normal band congruence. Using the terminology of [9], from Theorem 4 we obtain Corollary 2. Let FND(X) be the free normal diband. Then (i) ∆µrb is the least rectangular band congruence on FND(X); (ii) ∆µℓz is the least left zero congruence on FND(X); (iii) ∆µrz is the least right zero congruence on FND(X); (iv) ∆µ∗ rb is the least normal band congruence on FND(X); (v) ∆µ∗ ℓz is the least left normal band congruence on FND(X); (vi) ∆µ∗ rz is the least right normal band congruence on FND(X); (vii) ∆µ∗ is the least semilattice congruence on FND(X). Proof. (i) Xrb is the free rectangular band (see Sect. 3 of [9]). By Theorem 4 (i) we obtain (i). (ii) It is well-known that Xℓz is the free left zero semigroup. By Theorem 4 (ii) we obtain (ii). The proof of (iii) is similar. (iv) Brb(X) is the free normal band (see Sect. 3). By Theorem 4 (iv) we obtain (iv). (v) Bℓz(X) is the free left normal band (see Sect. 3). By Theorem 4 (v) we obtain (v). The proof of (vi) is similar. (vii) It is well-known that B(X) is the free semilattice. By Theorem 4 (vii) we obtain (vii). A. V. Zhuchok 127 Note that the least congruences on dimonoids and the corresponding decompositions of these dimonoids were also described in [4] and [6–9]. References [1] J.-L. Loday, Dialgebras, In: Dialgebras and related operads, Lect. Notes Math. 1763, Springer-Verlag, Berlin, 2001, 7–66. [2] M. Petrich, P.V. Silva, Structure of relatively free bands, Commun. Algebra 30 (2002), no. 9, 4165–4187. [3] T. Pirashvili, Sets with two associative operations, Cent. Eur. J. Math. 2 (2003), 169–183. [4] A.V. Zhuchok, Commutative dimonoids, Algebra and Discrete Math. 2 (2009), 116–127. [5] A.V. Zhuchok, Dibands of subdimonoids, Mat. Stud. 33 (2010), no. 2, 120–124. [6] A.V. Zhuchok, Free commutative dimonoids, Algebra and Discrete Math. 9 (2010), no. 1, 109–119. [7] A.V. Zhuchok, Free dimonoids, Ukr. Math. J. 63 (2011), no. 2, 165–175 (in Ukrainian). [8] A.V. Zhuchok, Semilattices of subdimonoids, Asian-Eur. J. Math. 4 (2011), no. 2, 359–371. [9] A.V. Zhuchok, Free rectangular dibands and free dimonoids, Algebra and Discrete Math. 11 (2011), no. 2, 92–111. [10] A.V. Zhuchok, Dimonoids, Algebra i Logika 50 (2011), no. 4, 471–496 (in Russian). [11] A.V. Zhuchok, Dimonoids with an idempotent operation, Proc. Inst. Applied Math. and Mech. 22 (2011), 99–107 (in Ukrainian). Contact information A. V. Zhuchok Department of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska str., 64, Kyiv, 01033, Ukraine E-Mail: zhuchok_a@mail.ru Received by the editors: 25.10.2011 and in final form 05.12.2011.

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