On one class of algebras
In this paper a g-dimonoid which is isomorphic to the free g-dimonoid is given and a free n-nilpotent g-dimonoid is constructed. We also present the least n-nilpotent congruence on a free g-dimonoid and give numerous examples of g-dimonoids.
Saved in:
| Published in: | Algebra and Discrete Mathematics |
|---|---|
| Date: | 2014 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2014
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/153324 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On one class of algebras / Yul.V. Zhuchok // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 2. — С. 306–320. — Бібліогр.: 14 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-153324 |
|---|---|
| record_format |
dspace |
| spelling |
Zhuchok, Yul.V. 2019-06-14T03:20:01Z 2019-06-14T03:20:01Z 2014 On one class of algebras / Yul.V. Zhuchok // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 2. — С. 306–320. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 MSC:08B20, 20M10, 20M50, 17A30, 17A32. https://nasplib.isofts.kiev.ua/handle/123456789/153324 In this paper a g-dimonoid which is isomorphic to the free g-dimonoid is given and a free n-nilpotent g-dimonoid is constructed. We also present the least n-nilpotent congruence on a free g-dimonoid and give numerous examples of g-dimonoids. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On one class of algebras Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On one class of algebras |
| spellingShingle |
On one class of algebras Zhuchok, Yul.V. |
| title_short |
On one class of algebras |
| title_full |
On one class of algebras |
| title_fullStr |
On one class of algebras |
| title_full_unstemmed |
On one class of algebras |
| title_sort |
on one class of algebras |
| author |
Zhuchok, Yul.V. |
| author_facet |
Zhuchok, Yul.V. |
| publishDate |
2014 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
In this paper a g-dimonoid which is isomorphic to the free g-dimonoid is given and a free n-nilpotent g-dimonoid is constructed. We also present the least n-nilpotent congruence on a free g-dimonoid and give numerous examples of g-dimonoids.
|
| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/153324 |
| citation_txt |
On one class of algebras / Yul.V. Zhuchok // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 2. — С. 306–320. — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
AT zhuchokyulv ononeclassofalgebras |
| first_indexed |
2025-11-25T20:37:02Z |
| last_indexed |
2025-11-25T20:37:02Z |
| _version_ |
1850524201712091136 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 18 (2014). Number 2, pp. 306–320
© Journal “Algebra and Discrete Mathematics”
On one class of algebras
Yuliia V. Zhuchok
Communicated by V. I. Sushchansky
Abstract. In this paper a g-dimonoid which is isomorphic
to the free g-dimonoid is given and a free n-nilpotent g-dimonoid is
constructed. We also present the least n-nilpotent congruence on a
free g-dimonoid and give numerous examples of g-dimonoids.
1. Introduction
Recall that a dialgebra (dimonoid) [1] is a vector space (set) with
two binary operations ⊣ and ⊢ satisfying the axioms (x ⊣ y) ⊣ z =
x ⊣ (y ⊣ z) (D1), (x ⊣ y) ⊣ z = x ⊣ (y ⊢ z) (D2), (x ⊢ y) ⊣ z =
x ⊢ (y ⊣ z) (D3), (x ⊣ y) ⊢ z = x ⊢ (y ⊢ z) (D4), (x ⊢ y) ⊢ z =
x ⊢ (y ⊢ z) (D5). In our time dimonoids are standard tool in the theory
of Leibniz algebras. So, for example, free dimonoids were used for con-
structing free dialgebras and for studying a cohomology of dialgebras.
There exist papers devoted to studying structural properties of dimonoids
(see, e.g., [2 – 4]). If in the definition of a dialgebra delete the axioms
(D1), (D3), (D5), then we obtain a 0-dialgebra which was considered
in [5]. Algebras obtained from the definition of a dimonoid by deleting
the axioms (D2) and (D4) were considered in [6]. In the last paper the
free object in the corresponding variety was constructed. Observe that
dimonoids are closely connected with restrictive bisemigroups considered
by B.M. Schein [7]. In [8–11] the notions of interassociativity, respectively,
2010 MSC: 08B20, 20M10, 20M50, 17A30, 17A32.
Key words and phrases: dimonoid, g-dimonoid, free g-dimonoid, free n-nilpotent
g-dimonoid, semigroup, congruence.
Yul. V. Zhuchok 307
strong interassociativity, related semigroups and doppelalgebras which are
naturally connected with dimonoids were considered. Another reason for
interest in dimonoids is their connection with n-tuple semigroups which
were used in [12] for studying properties of n-tuple algebras of associative
type. If in the definition of a dimonoid delete the axiom (D3), then we
obtain an algebraic system which is called a g-dimonoid (see [13, 14]).
In this paper g-dimonoids are studied. In Section 2 we give nume-
rous examples of g-dimonoids. In Section 3 we suggest a new concrete
representation of a free g-dimonoid using the construction of a free g-
dimonoid from [14]. The main result of this section was announced in [13].
In Section 4 the construction of a free n-nilpotent g-dimonoid is given.
Moreover, here we characterize the least n-nilpotent congruence on a free
g-dimonoid.
2. Examples of g-dimonoids
In this section we give different examples of g-dimonoids.
a) Obviously, any dimonoid is a g-dimonoid.
b) Let X be an arbitrary nonempty set, |X| > 1 and let X∗ be the
set of all finite nonempty words in the alphabet X. Denoting the first
(respectively, the last) letter of a word w ∈ X∗ by w(0) (respectively, by
w(1)), define operations ⊣ and ⊢ on X∗ by w ⊣ u = w(0), w ⊢ u = u(1)
for all w, u ∈ X∗. From the proof of Theorem 2 [2] it follows that (X∗,⊣,⊢)
is a g-dimonoid but not a dimonoid.
c) Let {Di}i∈I be a family of arbitrary g-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 is easy to prove the following lemma.
Lemma 1.
∏
i∈IDi with multiplications defined by
i(f1 ⊣ f2) = if1 ⊣ if2, i(f1 ⊢ f2) = if1 ⊢ if2, (1)
where i ∈ I, f1, f2 ∈
∏
i∈IDi, is a g-dimonoid.
The obtained algebra is called the Cartesian product of g-dimonoids
Di, i ∈ I. If I is finite, then the Cartesian product and the direct product
coincide. The Cartesian product of a finite number of g-dimonoids D1,
D2, ..., Dn is denoted by D1 ×D2 × ...×Dn. In particular, the Cartesian
power of a g-dimonoid can be defined as follows. Let V be an arbitrary
g-dimonoid and X be any nonempty set. Denote by Map(X;V ) the set of
308 On one class of algebras
all maps X → V . Define operations ⊣ and ⊢ on Map(X;V ) by (1) for all
f1, f2 ∈ Map(X;V ) and i ∈ X. Then (Map(X;V ),⊣,⊢) is a g-dimonoid
which is called the Cartesian power of V .
d) As usual, N denotes the set of all positive integers.
Let F [X] be the free semigroup in an alphabet X. We denote the
length of a word w ∈ F [X] by l(w). Fix n ∈ N and define operations ⊣
and ⊢ on F [X] × N by
(w1,m1) ⊣ (w2,m2) = (w1w2, n),
(w1,m1) ⊢ (w2,m2) = (w1w2, l(w1) +m2)
for all (w1,m1), (w2,m2) ∈ F [X]×N. Denote the algebra (F [X]×N,⊣,⊢)
by XNn.
Lemma 2. The algebra XNn is a g-dimonoid but not a dimonoid.
Proof. One can directly verify that XNn is a g-dimonoid. Show that it is
not a dimonoid. For all (w1,m1), (w2,m2), (w3,m3) ∈ XNn obtain
((w1,m1) ⊢ (w2,m2)) ⊣ (w3,m3) = (w1w2, l(w1) +m2) ⊣ (w3,m3) =
= (w1w2w3, n) 6= (w1w2w3, l(w1) + n) = (w1,m1) ⊢ (w2w3, n) =
= (w1,m1) ⊢ ((w2,m2) ⊣ (w3,m3)).
e) Let S be an arbitrary semigroup, a, b ∈ S. By ES denote the set of
all idempotents of S. Define operations ⊣ and ⊢ on S by
x ⊣ y = ax, x ⊢ y = by
for all x, y ∈ S. Denote the algebra (S,⊣,⊢) by S(a, b).
Lemma 3. Let S be an arbitrary right cancellative semigroup, a, b ∈ ES.
(i) If a and b are non-commuting, then S(a, b) is a g-dimonoid but
not a dimonoid.
(ii) If a and b are commuting, then S(a, b) is a dimonoid.
Proof. (i) The axioms (D1), (D2), (D4), (D5) are checked directly. Be-
sides,
(x ⊢ y) ⊣ z = by ⊣ z = aby, x ⊢ (y ⊣ z) = x ⊢ ay = bay
for all x, y, z ∈ S. Suppose that aby = bay. Then, using the right can-
cellability, obtain ab = ba. Thus, we arrive at a contradiction, i.e., the
Yul. V. Zhuchok 309
assumption that aby = bay does not hold. Consequently, S(a, b) is not a
dimonoid.
(ii) If a and b are commuting, then, obviously, all axioms of a dimonoid
hold.
f) Let S be an arbitrary semigroup, a, b ∈ S. Define operations ⊣ and
⊢ on S by
x ⊣ y = xa, x ⊢ y = yb
for all x, y ∈ S. Denote the algebra (S,⊣,⊢) by S[a, b].
Similarly to Lemma 3, the following lemma can be proved.
Lemma 4. Let S be an arbitrary left cancellative semigroup, a, b ∈ ES.
(i) If a and b are non-commuting, then S[a, b] is a g-dimonoid but
not a dimonoid.
(ii) If a and b are commuting, then S[a, b] is a dimonoid.
g) Let S be an arbitrary semigroup, a, b ∈ S. Define operations ⊣ and
⊢ on S by
x ⊣ y = axb, x ⊢ y = ayb
for all x, y ∈ S. Denote the algebra (S,⊣,⊢) by S(a, b].
The following lemma could be proved immediately.
Lemma 5. If a, b ∈ ES, then S(a, b] is a dimonoid.
h) Let Y be an arbitrary nonempty set, S = SY be some monoid
defined on the set of finite words in the alphabet Y and θ ∈ S be an
empty word which is a unit of S. Denote the operation on S by ∗ and
the length of a word w ∈ S by l(w). By definition l(θ) = 0, u0 = θ for all
u ∈ S. Fix elements a, b ∈ Y , k ∈ N ∪ {0} and define operations ⊣ and ⊢
on S, assuming
u1 ⊣ u2 = u1 ∗ al(u2)+k, u1 ⊢ u2 = u2 ∗ bl(u1)+k
for all u1, u2 ∈ S. The obtained algebra will be denoted by Sb
a(k).
Lemma 6. Let T be the free monoid in the alphabet Y . Then for any
a, b ∈ Y , k ∈ N ∪ {0} the algebra T b
a(k) is a g-dimonoid. If a 6= b, then it
is not a dimonoid.
Proof. Let u1, u2, u3 ∈ T b
a(k). In order to prove that T b
a(k) is a g-dimonoid
we consider the following cases.
310 On one class of algebras
Case 1. Let u1 6= θ, u2 6= θ, u3 6= θ. Then
u1⊣ (u2⊣u3) = u1⊣(u2 ∗ al(u3)+k) =
= u1 ∗ al(u2al(u3)+k)+k = u1 ∗ al(u2)+l(u3)+2k =
= u1 ∗ al(u2)+k ∗ al(u3)+k = (u1 ∗ al(u2)+k)⊣u3 = (u1⊣u2) ⊣u3,
u1⊣ (u2⊢u3) = u1⊣(u3 ∗ bl(u2)+k) =
= u1 ∗ al(u3bl(u2)+k)+k = u1 ∗ al(u3)+l(u2)+2k,
u1⊢ (u2⊢u3) = u1⊢(u3 ∗ bl(u2)+k) =
= u3 ∗ bl(u2)+k ∗ bl(u1)+k = u3 ∗ bl(u2)+l(u1)+2k =
= u3 ∗ bl(u2bl(u1)+k)+k = (u2 ∗ bl(u1)+k)⊢u3 = (u1⊢u2) ⊢u3,
(u1⊣u2)⊢u3 = (u1 ∗ al(u2)+k)⊢u3 =
= u3 ∗ bl(u1al(u2)+k)+k = u3 ∗ bl(u1)+l(u2)+2k.
Case 2. Let u1 = u2 = u3 = θ. Then
θ⊣ (θ⊣θ) = θ⊣(θ ∗ al(θ)+k) = θ⊣ak = θ ∗ al(ak)+k = a2k =
= ak ∗ al(θ)+k = ak⊣θ = (θ ∗ al(θ)+k)⊣θ = (θ⊣θ) ⊣θ,
θ⊣ (θ⊢θ) = θ⊣(θ ∗ bl(θ)+k) = θ⊣bk = θ ∗ al(bk)+k = a2k,
θ⊢ (θ⊢θ) = θ⊢(θ ∗ bl(θ)+k) = θ⊢bk = bk ∗ bl(θ)+k = b2k =
= θ ∗ bl(bk)+k = bk⊢θ = (θ ∗ bl(θ)+k)⊢θ = (θ⊢θ) ⊢θ,
(θ⊣θ) ⊢θ = (θ ∗ al(θ)+k)⊢θ = ak⊢θ = θ ∗ bl(ak)+k = b2k.
Case 3. Let u1 = θ, u2 6= θ, u3 6= θ. Then
θ⊣ (u2⊣u3) = θ⊣(u2 ∗ al(u3)+k) = θ ∗ al(u2al(u3)+k)+k = al(u2)+l(u3)+2k =
= al(u2)+k ∗ al(u3)+k = (θ ∗ al(u2)+k)⊣u3 = (θ⊣u2) ⊣u3,
θ⊣ (u2⊢u3) = θ⊣(u3 ∗ bl(u2)+k) = θ ∗ al(u3bl(u2)+k)+k = al(u2)+l(u3)+2k,
θ⊢ (u2⊢u3) = θ⊢(u3 ∗ bl(u2)+k) = u3 ∗ bl(u2)+k ∗ bl(θ)+k = u3 ∗ bl(u2)+2k =
= u3 ∗ bl(u2∗bl(θ)+k)+k = (u2 ∗ bl(θ)+k)⊢u3 = (θ⊢u2) ⊢u3,
(θ⊣u2) ⊢u3 = (θ ∗ al(u2)+k)⊢u3 = u3 ∗ bl(al(u2)+k)+k = u3 ∗ bl(u2)+2k.
Yul. V. Zhuchok 311
Case 4. Let u1 = θ, u2 6= θ, u3 = θ. Then
θ⊣ (u2⊣θ) = θ⊣(u2 ∗ al(θ)+k) = θ⊣(u2 ∗ ak) = θ ∗ al(u2∗ak)+k = al(u2)+2k =
= al(u2)+k ∗ al(θ)+k = (θ ∗ al(u2)+k)⊣θ = (θ⊣u2) ⊣θ,
θ⊣ (u2⊢θ) = θ⊣(θ ∗ b
l(u2)+k
) = θ ∗ al(b
l(u2)+k
)+k = al(u2)+2k,
θ⊢ (u2⊢θ) = θ⊢(θ ∗ bl(u2)+k) = bl(u2)+k ∗ bl(θ)+k = bl(u2)+2k =
= θ ∗ bl(u2∗bk)+k = (u2 ∗ bl(θ)+k)⊢θ = (θ⊢u2) ⊢θ,
(θ⊣u2) ⊢θ = (θ ∗ al(u2)+k)⊢θ = θ ∗ bl(al(u2)+k)+k = bl(u2)+2k.
The cases u1 6= θ, u2 = θ, u3 6= θ; u1 6= θ, u2 6= θ, u3 = θ; u1 = u2 = θ,
u3 6= θ; u1 6= θ, u2 = u3 = θ are considered in a similar way.
Thus, T b
a(k) is a g-dimonoid.
Finally, show that T b
a(k) is not a dimonoid when a 6= b. For u1 6= θ,
u2 6= θ and u3 6= θ we have
(u1⊢u2) ⊣u3 = (u2 ∗ bl(u1)+k)⊣u3 = u2 ∗ bl(u1)+k ∗ al(u3)+k =
= u2b
l(u1)+kal(u3)+k 6= u2a
l(u3)+kbl(u1)+k =
= u2 ∗ al(u3)+k ∗ bl(u1)+k = u1⊢(u2 ∗ al(u3)+k) = u1⊢ (u2⊣u3)
and so, the axiom (D3) of a dimonoid does not hold.
The following lemma gives an answer on the question when Sb
a(k) is a
dimonoid.
Lemma 7. Let M be the free commutative monoid in the alphabet Y .
For any a, b ∈ Y , k ∈ N ∪ {0} algebras M b
a(k) and T a
a (k) are dimonoids.
Proof. From Lemma 6 it follows that M b
a(k) satisfies the axioms (D1),
(D2), (D4), (D5). Show that the axiom (D3) also holds.
Let u1, u2, u3 ∈ M b
a(k). Consider the following eight cases.
Case 1. Let u1 6= θ, u2 6= θ, u3 6= θ. Then
(u1⊢u2) ⊣u3 = (u2 ∗ bl(u1)+k)⊣u3 = u2 ∗ bl(u1)+k ∗ al(u3)+k =
= u2 ∗ al(u3)+k ∗ bl(u1)+k = u1⊢(u2 ∗ al(u3)+k) = u1⊢ (u2⊣u3) .
Case 2. Let u1 = u2 = u3 = θ. Then
(θ⊢θ) ⊣θ = (θ ∗ bl(θ)+k)⊣θ = bk⊣θ = bk ∗ al(θ)+k = bk ∗ ak =
312 On one class of algebras
= ak ∗ bk = ak ∗ bl(θ)+k = θ⊢ak = θ⊢(θ ∗ al(θ)+k) = θ⊢ (θ⊣θ) .
Case 3. Let u1 = θ, u2 6= θ, u3 6= θ. Then
(θ⊢u2) ⊣u3 = (u2 ∗bl(θ)+k)⊣u3 = u2 ∗bl(θ)+k ∗al(u3)+k = u2 ∗bk ∗al(u3)+k =
= u2∗al(u3)+k∗bk = u2∗al(u3)+k∗bl(θ)+k = θ⊢(u2∗al(u3)+k) = θ⊢ (u2⊣u3) .
Case 4. Let u1 6= θ, u2 = θ, u3 6= θ. Then
(u1⊢θ) ⊣u3 = (θ ∗ bl(u1)+k)⊣u3 = bl(u1)+k ∗ al(u3)+k =
= al(u3)+k ∗ bl(u1)+k = u1⊢(θ ∗ al(u3)+k) = u1⊢ (θ⊣u3) .
Case 5. Let u1 6= θ, u2 6= θ, u3 = θ. Then
(u1⊢u2) ⊣θ = (u2 ∗bl(u1)+k)⊣θ = u2 ∗bl(u1)+k ∗al(θ)+k = u2 ∗bl(u1)+k ∗ak =
= u2 ∗ ak ∗ bl(u1)+k = u1⊢(u2 ∗ al(θ)+k) = u1⊢ (u2⊣θ) .
Case 6. Let u1 = u2 = θ, u3 6= θ. Then
(θ⊢θ) ⊣u3 = (θ ∗ bl(θ)+k)⊣u3 = bk⊣u3 = bk ∗ al(u3)+k =
= al(u3)+k ∗ bk = al(u3)+k ∗ bl(θ)+k = θ⊢(θ ∗ al(u3)+k) = θ⊢ (θ⊣u3) .
Case 7. Let u1 6= θ, u2 = u3 = θ. Then
(u1⊢θ) ⊣θ = (θ ∗ bl(u1)+k)⊣θ = bl(u1)+k ∗ al(θ)+k = bl(u1)+k ∗ ak =
= ak ∗ bl(u1)+k = u1⊢ak = u1⊢(θ ∗ al(θ)+k) = u1⊢ (θ⊣θ) .
Case 8. Let u1 = θ, u2 6= θ, u3 = θ. Then
(θ⊢u2) ⊣θ = (u2 ∗ bl(θ)+k)⊣θ = u2 ∗ bk ∗ al(θ)+k = u2 ∗ bk ∗ ak =
= u2 ∗ ak ∗ bk = u2 ∗ ak ∗ bl(θ)+k = θ⊢(u2 ∗ al(θ)+k) = θ⊢ (u2⊣θ) .
Thus, M b
a(k) is a dimonoid.
A proof is the same for T a
a (k).
Note that independence of axioms of a g-dimonoid follows from inde-
pendence of axioms of a dimonoid (see [2], Theorem 2).
Yul. V. Zhuchok 313
3. Free g-dimonoids
In this section we construct a g-dimonoid which is isomorphic to
the free g-dimonoid of an arbitrary rank and consider separately free
g-dimonoids of rank 1.
A nonempty subset A of a g-dimonoid (D, ⊣,⊢) is called a g-subdimo-
noid, if for any a, b ∈ D, a, b ∈ A implies a ⊣ b, a ⊢ b ∈ A.
Note that the class of all g-dimonoids is a variety as it is closed under
taking of homomorphic images, g-subdimonoids and Cartesian products.
A g-dimonoid which is free in the variety of all g-dimonoids is called a
free g-dimonoid.
In order to prove the main result of this section we need the construc-
tion of a free g-dimonoid from [14].
Let e be an arbitrary symbol. Consider the following sets:
I1 = {e}, In = {(ε1, . . . , εn−1) | εk ∈ {0, 1}, 1 6 k 6 n− 1}, n > 1,
I =
⋃
n>1
In.
If l = 0, we will regard the sequence ε1, . . . , εl without brackets as
empty, and the sequence (ε1, . . . , εl) with brackets as e. Define operations
⊣ and ⊢ on I by
(ε1, . . . , εn−1) ⊣ (θ1, . . . , θm−1) = (ε1, . . . , εn−1, 1, 1, . . . , 1︸ ︷︷ ︸
m
),
(ε1, . . . , εn−1) ⊢ (θ1, . . . , θm−1) = (θ1, . . . , θm−1, 0, 0, . . . , 0︸ ︷︷ ︸
n
).
By Lemma 3 from [14] (I,⊣,⊢) is a g-dimonoid. Observe that e⊣e=(1),
e ⊢ e = (0) and (I,⊣,⊢) is not a dimonoid.
Let X be an arbitrary nonempty set and F [X] be the free semigroup
in the alphabet X. Define operations ⊣ and ⊢ on FG = {(w, ε) |w ∈
F [X], ε ∈ I l(w)} by
(w1, ε) ⊣ (w2, ξ) = (w1w2, ε ⊣ ξ),
(w1, ε) ⊢ (w2, ξ) = (w1w2, ε ⊢ ξ)
for all (w1, ε), (w2, ξ) ∈ FG. The algebra (FG,⊣,⊢) is denoted by FG[X].
By Theorem 4 from [14] FG[X] is the free g-dimonoid.
Using notations from Section 2, introduce the set
XT b
a(k) = {(w, u) ∈ F [X] × T b
a(k) | l(w) − l(u) = 1}.
314 On one class of algebras
If s = 1, we will regard the sequence y1y2...ys−1 ∈ T b
a(k) as θ.
The main result of this section is the following.
Theorem 1. The g-dimonoid XT b
a(1) is free if |Y | = 2 and a 6= b.
Proof. By Lemma 1 F [X] × T b
a(k) is a g-dimonoid. It is not difficult to
check that XT b
a(1) is a g-subdimonoid of F [X] × T b
a(1).
Let |Y | = 2 and a 6= b. Let us show that XT b
a(1) is free. Take
(x1x2...xs, y1y2...ys−1) ∈ XT b
a(1), where xi ∈ X, 1 6 i 6 s, yj ∈ Y ,
1 6 j 6 s− 1, and define a map
π : XT b
a(1) → FG[X] :
(x1x2...xs, y1y2...ys−1) 7→ (x1x2...xs, y1y2...ys−1)π,
assuming
(x1x2...xs, y1y2...ys−1)π = (x1x2...xs, (ỹ1, ỹ2, ..., ỹs−1)),
where
ỹi =
{
1, yi = a,
0, yi = b
for all 1 6 i 6 s− 1, s 6= 1, and (ỹ1, ỹ2, ..., ỹs−1) is e for s = 1. Show that
π is an isomorphism.
For all
(x1x2 . . . xs, y1y2 . . . ys−1) , (a1a2 . . . am, b1b2 . . . bm−1) ∈ XT b
a(1),
where ai ∈ X, 1 6 i 6 m, bj ∈ Y, 1 6 j 6 m− 1, obtain
((x1x2 . . . xs, y1y2 . . . ys−1) ⊣ (a1a2 . . . am, b1b2 . . . bm−1))π =
= (x1x2 . . . xsa1a2 . . . am, y1y2 . . . ys−1 ∗ am)π =
=
x1x2 . . . xsa1a2 . . . am, (ỹ1, ỹ2, . . . , ỹs−1, ã, ã, . . . , ã︸ ︷︷ ︸
m
)
=
=
x1x2 . . . xsa1a2 . . . am, (ỹ1, ỹ2, . . . , ỹs−1, 1, 1, . . . , 1︸ ︷︷ ︸
m
)
=
= (x1x2 . . . xs, (ỹ1, ỹ2, . . . , ỹs−1)) ⊣
(
a1a2 . . . am, (b̃1, b̃2, . . . , b̃m−1)
)
=
= (x1x2 . . . xs, y1y2 . . . ys−1)π ⊣ (a1a2 . . . am, b1b2 . . . bm−1)π,
Yul. V. Zhuchok 315
((x1x2 . . . xs, y1y2 . . . ys−1) ⊢ (a1a2 . . . am, b1b2 . . . bm−1))π =
= (x1x2 . . . xsa1a2 . . . am, b1b2 . . . bm−1 ∗ bs)π =
=
x1x2 . . . xsa1a2 . . . am, (b̃1, b̃2, . . . , b̃m−1, b̃, b̃, . . . , b̃︸ ︷︷ ︸
s
)
=
=
x1x2 . . . xsa1a2 . . . am, (b̃1, b̃2, . . . , b̃m−1, 0, 0, . . . , 0︸ ︷︷ ︸
s
)
=
= (x1x2 . . . xs, (ỹ1, ỹ2, . . . , ỹs−1)) ⊢
(
a1a2 . . . am, (b̃1, b̃2, . . . , b̃m−1)
)
=
= (x1x2 . . . xs, y1y2 . . . ys−1)π ⊢ (a1a2 . . . am, b1b2 . . . bm−1)π.
So, π is a homomorphism. Obviously, π is a bijection and thus, π is an
isomorphism. Hence we obtain that XT b
a(1) is the free g-dimonoid.
The following lemma gives one property of Sb
a(k).
Lemma 8. If Sb
a(k) is a dimonoid, then ak and bk are commuting in S.
Proof. Let Sb
a(k) be a dimonoid. Then
(θ⊢θ) ⊣θ = (θ ∗ bl(θ)+k)⊣θ = bk⊣θ = bk ∗ al(θ)+k = bk ∗ ak,
θ⊢ (θ⊣θ) = θ⊢(θ ∗ al(θ)+k) = θ⊢ak = ak ∗ bl(θ)+k = ak ∗ bk
and, using the axiom (D3), obtain bk ∗ ak = ak ∗ bk.
Now we construct a g-dimonoid which is isomorphic to the free g-
dimonoid of rank 1.
Let |Y | = 2, a 6= b. Define operations ⊣ and ⊢ on
ÑT
b
a(1) = {(m,u) ∈ N × T b
a(1) |m− l(u) = 1}
by
(m1, u1)⊣(m2, u2) = (m1 +m2, u1 ∗ al(u2)+1),
(m1, u1)⊢(m2, u2) = (m1 +m2, u2 ∗ bl(u1)+1)
for all (m1, u1) , (m2, u2) ∈ ÑT
b
a(1). By Lemma 1 (N,+) × T b
a(1) is a
g-dimonoid. An immediate verification shows that operations ⊣ and ⊢ are
well-defined. Thus, (ÑT
b
a(1),⊣,⊢) is a g-subdimonoid of (N,+) × T b
a(1).
Denote it by NT b
a(1).
316 On one class of algebras
Lemma 9. The free g-dimonoid of rank 1 is isomorphic to the g-dimonoid
NT b
a(1).
Proof. Let X = {r}. An easy verification shows that a map
ξ : XT b
a(1) → NT b
a(1),
defined by ωξ = (k, u) ⇔ ω = (rk, u), is an isomorphism.
4. Free n-nilpotent g-dimonoids
In this section we construct a free n-nilpotent g-dimonoid of an arbi-
trary rank and consider separately free n-nilpotent g-dimonoids of rank 1.
We also characterize the least n-nilpotent congruence on a free g-dimonoid.
An element 0 of a g-dimonoid (D,⊣,⊢) will be called zero, if x ⋆ 0 =
0 = 0 ⋆ x for all x ∈ D and ⋆ ∈ {⊣,⊢}.
A g-dimonoid (D,⊣,⊢) with zero will be called nilpotent, if for some
n ∈ N and any xi ∈ D, 1 6 i 6 n + 1, and ∗j ∈ {⊣,⊢}, 1 6 j 6 n, any
parenthesizing of
x1 ∗1 x2 ∗2 . . . ∗n xn+1 (2)
gives 0 ∈ D. The least such n we shall call the nilpotency index of (D,⊣,⊢).
For k ∈ N a nilpotent g-dimonoid of nilpotency index 6 k is said to be
k-nilpotent.
Note that from (2) it follows that operations of any 1-nilpotent g-
dimonoid coincide and it is a zero semigroup.
It is not difficult to see that the class of all n-nilpotent g-dimonoids is
a subvariety of the variety of all g-dimonoids. A g-dimonoid which is free
in the variety of n-nilpotent g-dimonoids will be called a free n-nilpotent
g-dimonoid.
Fix n ∈ N and, using notations from Section 3, assume
Gn = {(w, u) ∈ XT b
a(1) | l(w) 6 n} ∪ {0} (|Y | = 2, a 6= b).
Define operations ≺ and ≻ on Gn by
(w1, u1) ≺ (w2, u2) =
{ (
w1w2, u1 ∗ al(u2)+1
)
, l(w1w2) 6 n,
0, l(w1w2) > n,
(w1, u1) ≻ (w2, u2) =
{ (
w1w2, u2 ∗ bl(u1)+1
)
, l(w1w2) 6 n,
0, l(w1w2) > n,
Yul. V. Zhuchok 317
(w1, u1) ⋆ 0 = 0 ⋆ (w1, u1) = 0 ⋆ 0 = 0
for all (w1, u1) , (w2, u2) ∈ Gn\{0} and ⋆ ∈ {≺,≻}. The algebra (Gn,≺,≻)
will be denoted by Gn(X).
Theorem 2. Gn(X) is the free n-nilpotent g-dimonoid.
Proof. Prove that Gn(X) is a g-dimonoid. Let (w1, u1), (w2, u2) ,
(w3, u3) ∈ Gn\{0}. If l(w1w2) > n or l(w2w3) > n, then the proof
is straightforward. The fact that axioms of a g-dimonoid hold when
l(w1w2w3) 6 n follows from Theorem 1. In the case l(w1w2) 6 n,
l(w2w3) 6 n and l(w1w2w3) > n we have
((w1, u1) ∗1 (w2, u2)) ∗2 (w3, u3) = 0 = (w1, u1) ∗1 ((w2, u2) ∗2 (w3, u3))
for ∗1, ∗2 ∈ {≺,≻}. The proofs of the remaining cases are obvious. Thus,
Gn(X) is a g-dimonoid.
For any (wi, ui) ∈ Gn\{0}, 1 6 i 6 n+ 1, and ∗j ∈ {≺,≻}, 1 6 j 6 n,
any parenthesizing of
(w1, u1) ∗1 (w2, u2) ∗2 . . . ∗n (wn+1, un+1)
gives 0, hence Gn(X) is nilpotent. Moreover, for any (xi, θ) ∈ Gn\{0},
where xi ∈ X, 1 6 i 6 n,
(x1, θ) ≺ (x2, θ) ≺ . . . ≺ (xn, θ) = (x1x2 . . . xn, a
n−1) 6= 0.
It means that Gn(X) has nilpotency index n.
Let us show that Gn(X) is free in the variety of n-nilpotent g-dimo-
noids.
The g-dimonoid (G(X),⊣,⊢) which is isomorphic to FG[X] from
Section 3 was constructed in [14]. The corresponding isomorphism
(G(X),⊣,⊢) → FG[X] is denoted by σ (see [14], Theorem 4). In the
last paper for an arbitrary g-dimonoid (D,⊣,⊢) the homomorphism ψ0
from (G(X),⊣,⊢) to (D,⊣,⊢) was given. We will call ψ0 as a canonical
homomorphism. Observe that ψ0 sends an arbitrary term with elements
x1, ..., xn to the product of some n elements from D.
Let
(
P,⊣′,⊢′
)
be an arbitrary n-nilpotent g-dimonoid, α be the ca-
nonical homomorphism from (G(X),⊣,⊢) to (P,⊣
′
,⊢
′
) and µ = πσ−1α
(see Section 3). Obviously, µ is a homomorphism from XT b
a(1), where
|Y | = 2, a 6= b, to (P,⊣
′
,⊢
′
). Define a map
δ : Gn(X) → (P,⊣′,⊢′) : ω 7→ ωδ,
318 On one class of algebras
assuming
ωδ =
{
ωµ, ω ∈ Gn\{0},
0, ω = 0.
Show that δ is a homomorphism.
Let ω1 = (x1x2...xs, y1y2...ys−1), ω2 = (a1a2 . . . am, b1b2 . . . bm−1) ∈
Gn\{0}, where xi ∈ X, 1 6 i 6 s, yj ∈ Y , 1 6 j 6 s−1, ai ∈ X, 1 6 i 6 m,
bj ∈ Y , 1 6 j 6 m− 1. Assume s+m 6 n. As ω1 ≺ ω2 ∈ Gn\{0}, then
(ω1 ≺ ω2)δ = (ω1 ≺ ω2)µ = (ω1 ⊣ ω2)µ = ω1µ ⊣′ ω2µ = ω1δ ⊣′ ω2δ.
Analogously, (ω1 ≻ ω2)δ = ω1δ ⊢′ ω2δ. Taking into account the previous
arguments, in the remaining cases the equalities
(ω1 ≺ ω2)δ = (ω1 ≻ ω2)δ = 0 = ω1δ⊢
′ω2δ = ω1δ⊣
′ω2δ
hold. Thus, δ is a homomorhism.
The proof is complete.
Now we construct a g-dimonoid which is isomorphic to the free n-
nilpotent g-dimonoid of rank 1.
Assume |Y | = 2, a 6= b. For any n ∈ N let
L̃n = {(m,u) ∈ N × T b
a(1) |m− l(u) = 1,m 6 n} ∪ {0}.
Define operations ⊣ and ⊢ on L̃n by the rule
(m1, u1) ⊣ (m2, u2) =
{ (
m1 +m2, u1 ∗ al(u2)+1
)
, m1 +m2 6 n,
0, m1 +m2 > n,
(m1, u1) ⊢ (m2, u2) =
{ (
m1 +m2, u2 ∗ bl(u1)+1
)
, m1 +m2 6 n,
0, m1 +m2 > n,
(m1, u1) ⋆ 0 = 0 ⋆ (m1, u1) = 0 ⋆ 0 = 0
for all (m1, u1) , (m2, u2) ∈ L̃n\{0} and ⋆ ∈ {⊣,⊢}. An immediate verifi-
cation shows that axioms of a g-dimonoid hold concerning operations ⊣
and ⊢. So, (L̃n,⊣,⊢) is a g-dimonoid. Denote it by Ln.
Lemma 10. If |X| = 1, then Gn(X) ∼= Ln.
Proof. Let X = {r}. An easy verification shows that a map ̺ : Gn(X) →
Ln, defined by
ω̺ =
{
(k, u), ω = (rk, u),
0, ω = 0,
is an isomorphism.
Yul. V. Zhuchok 319
We finish this section with the description of the least n-nilpotent
congruence on a free g-dimonoid.
If f : D1 → D2 is a homomorphism of g-dimonoids, then the corre-
sponding congruence on D1 will be denoted by ∆f . If ρ is a congruence on
a g-dimonoid (D,⊣,⊢) such that (D,⊣,⊢) /ρ is an n-nilpotent g-dimonoid,
then we say that ρ is an n-nilpotent congruence.
Let XT b
a(1) be the free g-dimonoid (|Y | = 2, a 6= b) (see Section 3).
Fix n ∈ N and define a relation κ(n) on XT b
a(1) by
(w1, u1)κ(n)(w2, u2) if and only if
(w1, u1) = (w2, u2) or l(w1) > n, l(w2) > n.
Theorem 3. The relation κ(n) on the free g-dimonoid XT b
a(1) is the
least n-nilpotent congruence.
Proof. Define a map τ : XT b
a(1) → Gn(X) by
(w, u) τ =
{
(w, u) , l(w)6n,
0, l(w) > n,
(w, u) ∈ XT b
a(1).
Similarly to the proof of Theorem 4 from [4], the facts that τ is a
surjective homomorphism and ∆τ = κ(n) can be proved.
References
[1] J.-L. Loday, Dialgebras, In: Dialgebras and related operads, Lect. Notes Math.
1763, Springer-Verlag, Berlin (2001), 7–66.
[2] A.V. Zhuchok, Dimonoids, Algebra and Logic 50 (2011), no. 4, 323–340.
[3] A.V. Zhuchok, Free dimonoids, Ukr. Math. J. 63 (2011), no. 2, 196–208.
[4] A.V. Zhuchok, Free n-nilpotent dimonoids, Algebra and Discrete Math. 16 (2013),
no. 2, 299–310.
[5] A.P. Pozhidaev, 0-dialgebras with bar-unity and nonassociative Rota-Baxter alge-
bras, Sib. Math. J. 50 (2009) no. 6, 1070–1080.
[6] T. Pirashvili, Sets with two associative operations, Cent. Eur. J. Math. 2 (2003),
169–183.
[7] B.M. Schein, Restrictive bisemigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (44)
(1965), 168–179 (in Russian).
[8] M. Gould, K.A. Linton, A.W. Nelson, Interassociates of monogenic semigroups,
Semigroup Forum 68 (2004), 186–201.
[9] M. Gould, R.E. Richardson, Translational hulls of polynomially related semigroups,
Czechoslovak Math. J. 33 (1983), no. 1, 95–100.
[10] E. Hewitt, H.S. Zuckerman, Ternary operations and semigroups, Semigroups, Proc.
Sympos. Detroit, Michigan 1968. (1969), 95–100.
320 On one class of algebras
[11] B. Richter, Dialgebren, Doppelalgebren und ihre Homologie, Diplomarbeit, Univer-
sitat Bonn. (1997). Available at http://www.math.uni-bonn.de/people/richter/.
[12] N.A. Koreshkov, n-tuple algebras of associative type, Izv. Vyssh. Uchebn. Zaved.
Mat. 12 (2008), 34–42 (in Russian).
[13] Yul. V. Zhuchok, On one class of algebras, International Algebraic Conference
dedicated to the 100th anniversary of L.A. Kaluzhnin: Abstracts, Kyiv, Ukraine
(2014), p. 91.
[14] Y. Movsisyan, S. Davidov and Mh. Safaryan, Construction of free g-dimonoids,
Algebra and Discrete Math. 18 (2014), no. 1, 138–148.
Contact information
Yul. V. Zhuchok Department of Algebra and System Analysis,
Luhansk Taras Shevchenko National University,
Gogol square, 1, Starobilsk, 92700, Ukraine
E-Mail(s): yulia.mih@mail.ru
Received by the editors: 10.11.2014
and in final form 12.01.2015.
|