Commutative dimonoids
We present some congruence on the dimonoid with a commutative operation and use it to obtain a decomposition of a commutative dimonoid.
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1546132019-06-16T01:30:22Z Commutative dimonoids Zhuchok, A.V. We present some congruence on the dimonoid with a commutative operation and use it to obtain a decomposition of a commutative dimonoid. 2009 Article Commutative dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 116–127. — Бібліогр.: 4 назв. — англ. 2000 Mathematics Subject Classification:08A05. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/154613 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We present some congruence on the dimonoid with a commutative operation and use it to obtain a decomposition of a commutative dimonoid. |
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Zhuchok, A.V. Commutative dimonoids Algebra and Discrete Mathematics |
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Zhuchok, A.V. |
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Zhuchok, A.V. |
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Commutative dimonoids |
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Commutative dimonoids |
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Commutative dimonoids |
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Commutative dimonoids |
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Commutative dimonoids |
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commutative dimonoids |
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Інститут прикладної математики і механіки НАН України |
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Commutative dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 116–127. — Бібліогр.: 4 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT zhuchokav commutativedimonoids |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2009). pp. 116 – 127
c⃝ Journal “Algebra and Discrete Mathematics”
Commutative dimonoids
Anatolii V. Zhuchok
Communicated by V. I. Sushchansky
Abstract. We present some congruence on the dimonoid
with a commutative operation and use it to obtain a decomposition
of a commutative dimonoid.
1. Introduction
Jean-Louis Loday introduced the notion of dimonoid [1]. Dimonoid is a
set equipped with two associative operations satisfying some axioms (see,
below). If the operations of a dimonoid coincide, then the dimonoid be-
comes a semigroup. The first result about dimonoids is the description of
free dimonoid generated by a given set [1]. Other notion which belongs to
the theory of semigroups is a notion of band of semigroups. The decompo-
sitions of semigroups are effectively described according to the construc-
tion of a band of semigroups. Tamura and Kimura [2] proved, in partic-
ular, that every commutative semigroup is a semilattice of archimedean
semigroups. The decompositions of some semigroups were studied in the
terms of bands in the papers [3-4].
In this work we introduce the notion of diband of dimonoids to de-
scribe the decompositions of dimonoids. In section 2 we give the necessary
definitions, some properties of dimonoids with one and two commutative
operations (Lemmas 1-4) and one example of a dimonoid. In section 3 we
first present the least idempotent congruence on an arbitrary dimonoid
with a commutative operation (Theorem 1). The main result of this pa-
per is a generalization of the theorem by Tamura and Kimura (Theorem
2000 Mathematics Subject Classification: 08A05.
Key words and phrases: dimonoid with a commutative operation, commutative
dimonoid, semilattice of dimonoids.
A. V. Zhuchok 117
2): every commutative dimonoid is a semilattice of archimedean subdi-
monoids. In section 4 we construct different examples of dimonoids.
2. Preliminaries
A set D equipped with two associative operations ≺ and ≻ satisfying 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 dimonoid D1 to dimonoid D2 is homomorphism, if
(x ≺ y)f = xf ≺ yf , (x ≻ y)f = xf ≻ yf for all x, y ∈ D1.
Define the notion of diband of dimonoids.
A dimonoid (D,≺,≻) will be called idempotent dimonoid or diband,
if x ≺ x = x = x ≻ x for all x ∈ D.
If ' : S → T is a homomorphism of dimonoids, then corresponding
congruence on S will be denoted by Δ'.
Let S be an arbitrary dimonoid, J be some idempotent dimonoid. If
there exists homomorphism
� : S → J : x 7→ x�,
then every class of congruence Δ� is a subdimonoid of the dimonoid S,
and dimonoid S itself is a union of such dimonoids S�, � ∈ J that
x� = � ⇔ x ∈ S� = Δx
� = {t ∈ S ∣(x; t) ∈ Δ�},
S� ≺ S" ⊆ S�≺ ", S� ≻ S" ⊆ S�≻",
� ∕= "⇒ S�
∩
S" = ∅.
In this case we say that S is decomposable into a diband of subdimonoids
(or S is a diband J of subdimonoids S� (� ∈ J)). If J is an idempo-
tent semigroup (band), then we say that S is a band J of subdimonoids
S� (� ∈ J).
As usual N denotes the set of positive integers.
Let (D,≺, ≻) be a dimonoid, a ∈ D, n ∈ N . Denote by an (re-
spectively, n a) the degree n of an element a concerning the operation ≺
(respectively, ≻).
118 Commutative dimonoids
Lemma 1. Let (D,≺, ≻) be a dimonoid with a commutative operation
≺. For all b, c ∈ D, m ∈ N, m > 1,
(b ≺ c)m = bm ≻ cm = (b ≻ c)m.
Proof. For any b, c ∈ D we have
(b ≺ c)m = bm ≺ cm = bm ≺ cm−1 ≺ c =
= (c ≺ bm) ≺ cm−1 = c ≺ (bm ≻ cm−1) =
= (bm ≻ cm−1) ≺ c = bm ≻ (cm−1 ≺ c) = bm ≻ cm
according to the commutativity of ≺ and axioms of dimonoid.
We prove that bm ≻ cm = (b ≻ c)m using an induction on m. For
m = 2 we have
b2 ≻ c2 = (b ≺ b) ≻ (c ≺ c) = b ≻ (b ≻ (c ≺ c)) =
= b ≻ ((b ≻ c) ≺ c) = b ≻ (c ≺ (b ≻ c)) =
= (b ≻ c) ≺ (b ≻ c) = (b ≻ c)2
according to the commutativity of ≺ and axioms of dimonoid.
Let bk ≻ ck = (b ≻ c)k for m = k. Then for m = k + 1 we obtain
(b ≻ c)k+1 = (b ≻ c)k ≺ (b ≻ c) =
= (bk ≻ ck) ≺ (b ≻ c) = bk ≻ (ck ≺ (b ≻ c)) =
= bk ≻ ((b ≻ c) ≺ ck) = bk ≻ (b ≻ (c ≺ ck)) =
= bk ≻ (b ≻ ck+1) = (bk ≺ b) ≻ ck+1 = bk+1 ≻ ck+1
according to the supposition, the commutativity of ≺ and axioms of di-
monoid.
Thus, bm ≻ cm = (b ≻ c)m for m > 1.
A dimonoid (D,≺, ≻) will be called commutative, if its both opera-
tions are commutative.
Lemma 2. In 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.
A. V. Zhuchok 119
Proof. We have for all x, y, z ∈ D
(x ≺ y) ≻ z = z ≻ (x ≺ y) =
= (z ≻ x) ≺ y = (x ≻ z) ≺ y =
= x ≻ (z ≺ y) = x ≻ (y ≺ z) = (x ≻ y) ≺ z,
(x ≻ y) ≺ z = z ≺ (x ≻ y) = (z ≺ x) ≺ y =
= (x ≺ y) ≺ z = x ≺ (y ≻ z)
according to the commutativity and axioms of dimonoid. Hence,
(x ≺ y) ≻ z = x ≺ (y ≻ z).
From Lemma 2 it follows that the operations ≺ and ≻ of a commu-
tative dimonoid (D,≺,≻) is indistinguishable for three and more multi-
pliers and the product of these elements doesn’t depend on the parenthe-
sizing.
Lemma 3. The operations of any commutative dimonoid (D,≺, ≻) with
an idempotent operation ≺ coincide.
Proof. We have for all x, y, z ∈ D
(x ≺ y) ≻ z = (x ≺ y) ≺ z
according to Lemma 2. Hence, setting x = y, we obtain x ≺ z = x ≻ z
for all x, z ∈ D.
Let (D,≺, ≻) be a dimonoid, n ∈ N . Recall that we denote by n a
the degree n of an element a ∈ D concerning the operation ≻.
Lemma 4. Let (D,≺, ≻) be a dimonoid with a commutative operation
≻. For all b ∈ D, m ∈ N ,
2 bm = (2m) b.
Proof. We use an induction on m. For m = 1, obviously, the equality is
correct. For m = 2 we have
2 b2 = (b ≺ b) ≻ (b ≺ b) = ((b ≺ b) ≻ b) ≺ b =
= (b ≻ (b ≻ b)) ≺ b = b ≻ ((b ≻ b) ≺ b) =
= ((b ≻ b) ≺ b) ≻ b = (b ≻ b) ≻ (b ≻ b) = 2b ≻ 2b = 4 b
120 Commutative dimonoids
according to the axioms of dimonoid and the commutativity of ≻.
Let 2 bk = (2k) b for m = k. Then for m = k + 1 we obtain
(2(k + 1)) b = (2k + 2)b = (2k) b ≻ 2b =
= 2 bk ≻ 2b = ( bk ≻ b) ≻ ( bk ≻ b) =
= bk ≻ (b ≻ ( bk ≻ b)) = ( bk ≺ b) ≻ ( bk ≻ b) =
= bk+1 ≻ ( bk ≻ b) = bk ≻ (b ≻ bk+1) =
= (bk ≺ b) ≻ bk+1 = bk+1 ≻ bk+1 = 2bk+1
according to the supposition, axioms of dimonoid and the commutativity
of ≻. Thus, 2 bm = (2m) b.
Now we give an example of dimonoid.
Let X and Y be arbitrary disjoint sets, 0 ∈ X, and let
' : Y × Y → X, : Y × Y → X
be arbitrary different maps. Define the operations ≺, ≻ on X
∪
Y by
x ≺ y =
{
(x, y)', x, y ∈ Y,
0 otherwise,
x ≻ y =
{
(x, y) , x, y ∈ Y,
0 otherwise
for all x, y ∈ X
∪
Y . It is immediate to check that (X
∪
Y,≺, ≻) is a
dimonoid.
3. Main results
In this section we describe the least idempotent congruence on the di-
monoid with a commutative operation ≺ and show that every commuta-
tive dimonoid (D,≺, ≻) is a semilattice Y of archimedean subdimonoids
Di, i ∈ Y .
We will call a commutative idempotent semigroup as a semilattice.
If � is a congruence on the dimonoid D such that D/� is an idempotent
dimonoid, then we say that � is an idempotent congruence.
Let (D,≺, ≻) be a dimonoid with a commutative operation ≺ (re-
spectively, ≻), a, b ∈ D. We say that a ≺-divide b (respectively, a ≻-
divide b) and write a≺∣b (respectively, a≻∣b), if there exists such element
x ∈ (D,≺) (respectively, x ∈ (D,≻)) with an identity that a ≺ x = b
(respectively, a ≻ x = b).
Define a relation � on the dimonoid (D,≺, ≻) with a commutative
operation ≺ by
A. V. Zhuchok 121
a�b if and only if there exist positive integers
m, n, m ∕= 1, n ∕= 1 such that a≺∣b
m, b≺∣a
n.
Theorem 1. The relation � on the dimonoid (D,≺, ≻) with a commu-
tative operation ≺ is the least idempotent congruence, and (D,≺, ≻)/�
is a commutative idempotent dimonoid which is a semilattice.
Proof. The fact that the relation � is a congruence on the semigroup
(D,≺) has been proved by Tamura and Kimura [2]. Show that � is
compatible concerning the operation ≻.
Let a�b, a, b, c ∈ D. Then a ≺ c�b ≺ c. It means that there exist
x, y ∈ (D,≺), m,n ∈ N∖{1}, for which
(a ≺ c) ≺ x = (b ≺ c)m, (1)
(b ≺ c) ≺ y = (a ≺ c)n. (2)
Considering both parts of the equality (1), we obtain
(a ≺ c) ≺ x = (x ≺ a) ≺ c =
= x ≺ (a ≻ c) = (a ≻ c) ≺ x
according to the commutativity of ≺ and an axiom of dimonoid, and
(b ≺ c)m = (b ≻ c)m by Lemma 1. Hence, (a ≻ c) ≺ x = (b ≻ c)m. Thus,
a ≻ c≺∣(b ≻ c)m. Analogously, from the equality (2) we obtain that
b ≻ c≺∣(a ≻ c)n. Together with preceding it means that a ≻ c�b ≻ c.
Dually, a left compatibility of the relation � concerning the operation
≻ can be proved. So, � is a congruence on (D,≺, ≻).
Obviously, a�a ≺ a. Then a ≺ x1 = (a ≺ a)m1 , (a ≺ a) ≺ x2 = am2
for some x1, x2 ∈ (D,≺) and m1, m2 ∈ N∖{1}. From two last equalities
it follows that
a ≺ x1 = (a ≺ a)m1 = (a ≻ a)m1 ,
(a ≺ a) ≺ x2 = (x2 ≺ a) ≺ a =
= x2 ≺ (a ≻ a) = (a ≻ a) ≺ x2 = am2 ,
whence a�a ≻ a and so � is an idempotent. From the commutativity of
the operation ≺ it follows that a ≺ b� b ≺ a. Hence, (a ≺ b) ≺ t1 = ( b ≺
a)n1 , (b ≺ a) ≺ t2 = (a ≺ b)n2 for some t1, t2 ∈ (D,≺), n1, n2 ∈ N∖{1}
and
(a ≺ b) ≺ t1 = (t1 ≺ a) ≺ b = t1 ≺ (a ≻ b) =
= (a ≻ b) ≺ t1 = ( b ≺ a)n1 = ( b ≻ a)n1 ,
122 Commutative dimonoids
(b ≺ a) ≺ t2 = (t2 ≺ b) ≺ a = t2 ≺ (b ≻ a) =
= (b ≻ a) ≺ t2 = (a ≺ b)n2 = (a ≻ b)n2
according to the commutativity of the operation ≺, an axiom of dimonoid
and Lemma 1. That is, a ≻ b� b ≻ a. So, (D,≺, ≻)/� is a commutative
idempotent dimonoid. From Lemma 3 it follows that (D,≺, ≻)/� is a
semilattice.
The proof will be completed, if we show that � is contained in every
idempotent congruence � on (D,≺, ≻).
Let a�b, a, b ∈ D. Then a ≺ z = bk, b ≺ d = al for some z, d ∈ (D,≺)
and k, l ∈ N∖{1}. Since a� a ≺ a, b� b ≺ b by the idempotentity of �,
then we have a � b ≺ d, b� a ≺ z. So,
a� b ≺ d� b ≺ b ≺ d� b ≺ al� b ≺ a� bk ≺ a� a ≺ z ≺ a� a ≺ z � b.
Thus, a�b and � ⊆ �.
We say that a commutative dimonoid is archimedean, if its both semi-
groups are archimedean.
Theorem 2. A semigroup (D,≺) of a commutative dimonoid D = (D,≺
, ≻) is archimedean (respectively, regular) if and only if a semigroup
(D,≻) of the commutative dimonoid D is archimedean (respectively, reg-
ular). Every commutative dimonoid D is a semilattice Y of archimedean
subdimonoids Di, i ∈ Y .
Proof. Let (D,≺) be an archimedean semigroup, a, b ∈ D. Then a ≺
x = bm, b ≺ y = an for some x, y ∈ D, m,n ∈ N . Multiply both parts of
the equality a ≺ x = bm by bm and the equality b ≺ y = an by an:
(a ≺ x) ≻ bm = a ≻ (x ≺ bm) =
= a ≻ (bm ≺ x) = bm ≻ bm = (2m)b,
(b ≺ y) ≻ an = b ≻ (y ≺ an) =
= b ≻ (an ≺ y) = an ≻ an = (2n)a
according to Lemmas 2 and 4 and the commutativity of ≺. So, a≻∣(2m)b,
b≻∣(2n)a. That is, (D,≻) is an archimedean semigroup.
Conversely, let a ≻ x = mb, b ≻ y = na for some x, y ∈ D, m, n ∈ N .
Take k, p ∈ N∖{1} such that k + m, p + n is even and multiply both
parts of the equality a ≻ x = mb by k b and the equality b ≻ y = na by
p a:
A. V. Zhuchok 123
kb ≻ (a ≻ x) = (kb ≺ a) ≻ x = x ≻ (kb ≺ a) =
= (x ≻ kb) ≺ a = a ≺ (x ≻ kb)
according to the axioms and the commutativity of dimonoid,
kb ≻ mb = (k +m)b = 2b
k+m
2 =
= b
k+m
2 ≻ b
k+m
2 = (b ≺ b)
k+m
2 =
= (b2)
k+m
2 = bk+m
by Lemmas 1 and 4. Thus, a≺∣b
k+m. Analogously, b ≺ (y ≻ pb) = ap+n,
that is, b≺∣a
p+n.
The corresponding statement about regularity of the semigroup of a
dimonoid follows immediately from Lemma 2.
Now we shall prove the second part of the theorem. By Theorem 1
� is the least idempotent congruence on D, D/� is a semilattice and
D → D/� : x 7→ [x] is a homomorphism ([x] is a class of the congruence �,
which contains x). By Tamura and Kimura [2] it follows that every class
A of the congruence � is a archimedean subsemigroup of the semigroup
(D,≺). According to the preceding computation A is an archimedean
subsemigroup of the semigroup (D,≻).
If the operations of a commutative dimonoid coincide, then from The-
orem 2 we obtain the theorem by Tamura and Kimura [2] about the de-
composition of a commutative semigroup into a semilattice of archimedean
subsemigroups.
4. Some examples
In this section we construct the examples of different dimonoids. First
we consider the examples of dimonoids with one and two commutative
operations.
a) Let (X,≺) be a zero semigroup, (X, ≻) be a right zero semigroup.
Then (X,≺, ≻) is dimonoid with commutative operation ≺. It is easy
to see that the least idempotent congruence � = X ×X on (X,≺, ≻).
b) Let (S,≺) be a zero semigroup (0 is a zero), (S, ≻) be a commuta-
tive semigroup with a zero 0 such that S ≻ S ≻ S = 0. Then (S,≺, ≻)
is commutative dimonoid. It is easy to see that the least idempotent
congruence � = S × S on (S,≺, ≻).
We give an example of such dimonoid.
124 Commutative dimonoids
Let (X,≺) be a zero semigroup, that is, x ≺ y = 0 for all x, y ∈ X.
Fix elements a, b of the set X, a ∕= b and define on X the operation ≻,
assuming
x ≻ y =
{
a, x = y = b,
0 otherwise
for all x, y ∈ X. It is easy to see that (X,≺,≻) is a commutative di-
monoid.
c) Let X be an arbitrary set such that 0, a, b, c, d ∈ X and a ∕=
b, b ∕= c, c ∕= d, d ∕= a. Define on the set X the operations ≺ and ≻,
assuming
x ≺ y =
{
b, x = y = a,
0 otherwise,
x ≻ y =
{
d, x = y = c,
0 otherwise
for all x, y ∈ X. It is immediate to check that (X,≺,≻) is commutative
dimonoid. In this case the least idempotent congruence � = X ×X.
Now we construct the examples of dimonoids which are not necessarily
commutative.
d) Let S be a semigroup and let f be its idempotent endomorphism.
On S define the multiplications by
x ≺ y = x(yf), x ≻ y = (xf)y
for all x, y ∈ S.
Proposition 1. (S,≺,≻) is dimonoid.
Proof. For any x, y, z ∈ S we obtain
(x ≺ y) ≺ z = x(yf) ≺ z = x(yf)(zf) = x((yz)f),
x ≺ (y ≺ z) = x ≺ y(zf) = x(y(zf))f =
= x(yf)(zf2) = x(yf)(zf) = x((yz)f),
x ≺ (y ≻ z) = x ≺ ((yf)z) = x((yf)z)f =
= x(yf2)(zf) = x(yf)(zf) = x((yz)f),
(x ≻ y) ≻ z = (xf)y ≻ z = (xf2)(yf)z =
= (xf)(yf)z = (xy)fz,
x ≻ (y ≻ z) = x ≻ ((yf)z) = (xf)(yf)z = (xy)fz,
A. V. Zhuchok 125
(x ≺ y) ≻ z = x(yf) ≻ z = (x(yf))fz =
= (xf)(yf2)z = (xf)(yf)z = (xy)fz,
(x ≻ y) ≺ z = (xf)y ≺ z = (xf)y(zf),
x ≻ (y ≺ z) = x ≻ (y(zf)) = (xf)y(zf).
Comparing these expressions, we conclude that (S,≺,≻) is dimonoid.
e) Let S and T be semigroups, � : T → S is a homomorphism. On
S × T define the multiplications by
(s, t) ≺ (p, g) = (s, tg), (s, t) ≻ (p, g) = ((t�)p, tg)
for all (s, t), (p, g) ∈ S × T .
Proposition 2. (S × T,≺,≻) is dimonoid.
Proof. Obviously, the operation ≺ is associative. For any (s, t), (p, g), (a, b) ∈
S × T we have
((s, t) ≻ (p, g)) ≻ (a, b) = ((t�)p, tg) ≻ (a, b) = ((tg)�a, tgb),
(s, t) ≻ ((p, g) ≻ (a, b)) = (s, t) ≻ ((g�)a, gb) =
= ((t�)(g�)a, tgb) = ((tg)�a, tgb),
((s, t) ≺ (p, g)) ≻ (a, b) = (s, tg) ≻ (a, b) = ((tg)�a, tgb),
(s, t) ≺ ((p, g) ≻ (a, b)) = (s, t) ≺ ((g�)a, gb) =
= (s, tgb) = ((s, t) ≺ (p, g)) ≺ (a, b),
((s, t) ≻ (p, g)) ≺ (a, b) = ((t�)p, tg) ≺ (a, b) = ((t�)p, tgb),
(s, t) ≻ ((p, g) ≺ (a, b)) = (s, t) ≻ (p, gb) = ((t�)p, tgb).
Comparing these expressions, we conclude that (S × T,≺,≻) is di-
monoid.
f) Let X∗ be a set of words in the alphabet X. If w ∈ X∗, then
the first (respectively, the last) letter of a word w we denote by w(0)
(respectively, w(1)).
Assume the operations ≺, ≻ on the set X∗ by
w ≺ u = w(0)w(1), w ≻ u = u(0)u(1)
for all w, u ∈ X∗.
126 Commutative dimonoids
Proposition 3. (X∗,≺,≻) is dimonoid.
Proof. Obviously, the operations ≺ and ≻ are associative. For any w, u, ! ∈
X∗ we obtain
(w ≺ u) ≺ ! = w(0)w(1) ≺ ! = w(0)w(1) = w ≺ (u ≻ !),
w ≻ (u ≻ !) = w ≻ !(0)!(1) = !(0)!(1) = (w ≺ u) ≻ !,
(w ≻ u) ≺ ! = u(0)u(1) ≺ ! = u(0)u(1) =
= w ≻ u(0)u(1) = w ≻ (u ≺ !).
Let (X ×X,≺
′
,≻
′
) be an idempotent dimonoid with operations
(x, y) ≺
′
(a, b) = (x, y), (x, y) ≻
′
(a, b) = (a, b)
for all (x, y), (a, b) ∈ X × X. Denote this dimonoid by X̃ and for all
i, j ∈ X assume
A(i,j) = {w ∈ X∗∣(w(0), w(1)) = (i, j)}.
The next assertion describes the structure of the dimonoid (X∗,≺,≻).
Proposition 4. The dimonoid (X∗,≺,≻) is a diband X̃ of zero semi-
groups A(i,j), (i, j) ∈ X̃.
Proof. Define a map � by
� : (X∗,≺,≻) → X̃ : w 7→ (w(0), w(1)).
The map � is a homomorphism. Indeed, if w, u ∈ X∗, then
(w ≺ u)� = (w(0)w(1))� = (w(0), w(1)) =
= (w(0), w(1)) ≺
′
(u(0), u(1)) = w� ≺
′
u�,
(w ≻ u)� = (u(0)u(1))� = (u(0), u(1)) =
= (w(0), w(1)) ≻
′
(u(0), u(1)) = w� ≻
′
u�.
It is clear that A(i,j), (i, j) ∈ X̃ is an arbitrary class of the congruence
△� . Moreover, if w, u ∈ A(i,j), then w ≺ u = w ≻ u = ij, hence A(i,j) is
zero semigroup with the zero ij.
A. V. Zhuchok 127
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. Tamura, N. Kimura, On decomposition of a commutative semigroup, Kodai
Math. Sem. Rep., N.4, 1954, pp.109-112.
[3] A.H. Clifford, Bands of semigroups, Proc. Amer. Math. Soc., N.5, 1954, pp.499-
504.
[4] A.V. Zhuchok, Decompositions of free products, Visnyk Kyiv. Univ., Ser. Phis.-
Math. Nauk, N.2, 2002, pp.41-49 (In Ukrainian).
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: 18.11.2008
and in final form 26.07.2009.
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