Free n-nilpotent dimonoids
We construct a free n-nilpotent dimonoid and describe its structure. We also characterize the least n-nilpotent congruence on a free dimonoid, construct a new class of dimonoids with zero and give examples of nilpotent dimonoids of nilpotency index 2.
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Free n-nilpotent dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 299–309. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-1523562019-06-12T01:25:24Z Free n-nilpotent dimonoids Zhuchok, A.V. We construct a free n-nilpotent dimonoid and describe its structure. We also characterize the least n-nilpotent congruence on a free dimonoid, construct a new class of dimonoids with zero and give examples of nilpotent dimonoids of nilpotency index 2. 2013 Article Free n-nilpotent dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 299–309. — Бібліогр.: 20 назв. — англ. 1726-3255 2010 MSC:08B20, 20M10, 20M50, 17A30, 17A32. http://dspace.nbuv.gov.ua/handle/123456789/152356 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We construct a free n-nilpotent dimonoid and describe its structure. We also characterize the least n-nilpotent congruence on a free dimonoid, construct a new class of dimonoids with zero and give examples of nilpotent dimonoids of nilpotency index 2. |
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Zhuchok, A.V. |
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Zhuchok, A.V. Free n-nilpotent dimonoids Algebra and Discrete Mathematics |
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Zhuchok, A.V. |
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Zhuchok, A.V. |
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Free n-nilpotent dimonoids |
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Free n-nilpotent dimonoids |
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Free n-nilpotent dimonoids |
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Free n-nilpotent dimonoids |
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Free n-nilpotent dimonoids |
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free n-nilpotent dimonoids |
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Інститут прикладної математики і механіки НАН України |
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2013 |
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Free n-nilpotent dimonoids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 299–309. — Бібліогр.: 20 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT zhuchokav freennilpotentdimonoids |
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2025-07-13T02:54:05Z |
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2025-07-13T02:54:05Z |
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1837498629879758848 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 2. pp. 299 – 310
c© Journal “Algebra and Discrete Mathematics”
Free n-nilpotent dimonoids
Anatolii V. Zhuchok
Communicated by V. I. Sushchansky
Abstract. We construct a free n-nilpotent dimonoid and
describe its structure. We also characterize the least n-nilpotent
congruence on a free dimonoid, construct a new class of dimonoids
with zero and give examples of nilpotent dimonoids of nilpotency
index 2.
1. Introduction
The notion of a dialgebra is based on the notion of a dimonoid [1].
Therefore all results obtained for dimonoids can be applied to dialgebras.
This connection between dimonoids and dialgebras shows that dimonoids
are very natural objects to study. Another reason for our interest in
dimonoids is their connection with the notions of interassociativity [2],
strong interassociativity [3], related semigroups [4] and doppelalgebras
[5]. Note also that the notion of an n-tuple semigroup, which was used in
[6] to study properties of n-tuple algebras of associative type, is related
to commutative dimonoids [7] in the case n = 2.
The notion of a nilpotent semigroup was introduced by Malcev [8]
and independently by Neuman and Taylor [9]. They showed that nilpo-
tent groups can be defined by using semigroup identities. Further, the
nilpotency in semigroups has been extensively studied by many authors.
In particular, properties of nilpotent semigroups have been investigated
by Lallement [10]. The relationships between nilpotent semigroups and
2010 MSC: 08B20, 20M10, 20M50, 17A30, 17A32.
Key words and phrases: n-nilpotent dimonoid, free n-nilpotent dimonoid, 0-
diband of subdimonoids, dimonoid, semigroup.
300 Free n-nilpotent dimonoids
semigroup algebras were studied by Jespers and Okninski [11]. The nilpo-
tency in algebras with two binary associative operations was considered
too (see, e.g., [12]).
In this paper we continue researches from [13 – 16] developing the
variety theory of dimonoids. The main focus of our paper is to study
nilpotent dimonoids.
In Section 2 we present a new class of dimonoids with zero and give
examples of nilpotent dimonoids of nilpotency index 2.
In Section 3 we construct a free n-nilpotent dimonoid of an arbitrary
rank and consider separately free n-nilpotent dimonoids of rank 1.
In Section 4 we introduce the notion of a 0-diband of subdimonoids
and in terms of 0-dibands of subdimonoids describe the structure of free
n-nilpotent dimonoids.
In the final section we characterize the least n-nilpotent congruence
on a free dimonoid.
2. Dimonoids with zero
In this section we construct a new class of dimonoids with zero and
give examples of nilpotent dimonoids of nilpotency index 2.
An element 0 of a dimonoid (D,⊣,⊢) (see, e.g., [17]) will be called
zero, if x ∗ 0 = 0 = 0 ∗ x for all x ∈ D and ∗ ∈ {⊣,⊢}.
Let D = (D,⊣,⊢) be an arbitrary dimonoid and I be an arbitrary
nonempty set. Define operations ⊣
′
and ⊢
′
on D′ = (I ×D × I) ∪ {0} by
(i, a, j) ∗
′
(k, b, t) =
{
(i, a ∗ b, t), j = k,
0, j 6= k,
(i, a, j) ∗
′
0 = 0 ∗
′
(i, a, j) = 0 ∗
′
0 = 0
for all (i, a, j), (k, b, t) ∈ D′\{0} and ∗ ∈ {⊣,⊢}. The algebra (D′,⊣
′
,⊢
′
)
will be denoted by B[D, I].
Proposition 1. B[D, I] is a dimonoid with zero.
Proof. By [18] operations ⊣′ and ⊢′ are associative. Let (i, a, j) , (k, b, t) ,
(m, c, n) ∈ D′\{0}. If j 6= k or t 6= m, then, obviously, all axioms of a
dimonoid hold. If j = k and t = m, then
(
(i, a, j) ⊣′ (k, b, t)
)
⊣′ (m, c, n) = (i, a⊣b, t) ⊣′ (m, c, n) =
= (i, (a⊣b) ⊣c, n) = (i, a⊣(b⊢c), n) =
A. V. Zhuchok 301
= (i, a, j) ⊣′ (k, b⊢c, n) = (i, a, j) ⊣′
(
(k, b, t) ⊢′ (m, c, n)
)
,
(
(i, a, j) ⊢′ (k, b, t)
)
⊣′ (m, c, n) = (i, a⊢b, t) ⊣′ (m, c, n) =
= (i, (a⊢b) ⊣ c, n) = (i, a⊢(b ⊣ c), n) = (i, a, j) ⊢
′
(k, b ⊣ c, n) =
= (i, a, j) ⊢
′
((k, b, t) ⊣
′
(m, c, n)),
(
(i, a, j) ⊣′ (k, b, t)
)
⊢′ (m, c, n) = (i, a⊣b, t) ⊢′ (m, c, n) =
= (i, (a⊣b) ⊢c, n) = (i, a⊢(b⊢c), n) = (i, a, j) ⊢
′
(k, b ⊢ c, n) =
= (i, a, j) ⊢
′
((k, b, t) ⊢
′
(m, c, n))
according to the axioms of a dimonoid D.
The proofs of the remaining cases are obvious. Thus, B[D, I] is a
dimonoid with zero 0.
Observe that if operations of a dimonoid D coincides and it is a
group G, then any Brandt semigroup [18] is isomorphic to some semi-
group B[G, I]. So, B[D, I] generalizes the semigroup B[G, I]. We call the
dimonoid B[D, I] a Brandt dimonoid.
As usual, N denotes the set of all positive integers.
A dimonoid (D,⊣,⊢) with zero will be called nilpotent, if for some
n ∈ N and any xi ∈ D, 1 ≤ i ≤ n + 1, and ∗j ∈ {⊣,⊢}, 1 ≤ j ≤ n, any
parenthesizing of
x1 ∗1 x2 ∗2 . . . ∗n xn+1 (∗)
gives 0 ∈ D. The least such n we shall call the nilpotency index of (D,⊣,⊢).
For k ∈ N a nilpotent dimonoid of nilpotency index ≤ k is said to be
k-nilpotent.
Note that from (∗) it follows that operations of any 1-nilpotent di-
monoid coincide and it is a zero semigroup.
We finish this section with the consideration of examples of nilpotent
dimonoids of nilpotency index 2.
Let X and Y be arbitrary disjoint sets, 0 ∈ X, and let
ϕ : Y × Y → X, ψ : Y × Y → X
be arbitrary different maps. Define operations ⊣ and ⊢ on X ∪ Y by
x ⊣ y =
{
(x, y)ϕ, x, y ∈ Y,
0 otherwise,
x ⊢ y =
{
(x, y)ψ, x, y ∈ Y,
0 otherwise
for all x, y ∈ X ∪ Y .
302 Free n-nilpotent dimonoids
Proposition 2. (X∪Y,⊣,⊢) is a nilpotent dimonoid of nilpotency index 2.
Proof. By [7] (X∪Y,⊣,⊢) is a dimonoid. From the definition of operations
⊣ and ⊢ it follows that (X ∪ Y,⊣,⊢) is not a 1-nilpotent dimonoid. As
(x ∗1 y) ∗2 z = 0 = x ∗1 (y ∗2 z)
for all x, y, z ∈ X ∪ Y , ∗1, ∗2 ∈ {⊣,⊢}, then the dimonoid (X ∪ Y,⊣,⊢) is
nilpotent of nilpotency index 2.
Recall that a dimonoid is called commutative, if its both operations
are commutative.
Let X be an arbitrary set such that 0, a, b, c, d ∈ X and a 6= b,
b 6= c, c 6= d, d 6= a. Define operations ⊣ and ⊢ on X, assuming
x ⊣ y =
{
b, x = y = a,
0 otherwise,
x ⊢ y =
{
d, x = y = c,
0 otherwise
for all x, y ∈ X.
Proposition 3. If b 6= 0 or d 6= 0, then (X,⊣,⊢) is a nilpotent commu-
tative dimonoid of nilpotency index 2.
Proof. By [7] (X,⊣,⊢) is a commutative dimonoid. If b 6= 0 or d 6= 0, then,
similar to Proposition 2, the fact that (X,⊣,⊢) is nilpotent of nilpotency
index 2 can be proved.
3. Constructions
In this section we construct a free n-nilpotent dimonoid of an arbitrary
rank and consider separately free n-nilpotent dimonoids of rank 1.
Note that the class of all n-nilpotent dimonoids is a subvariety of the
variety of all dimonoids. Indeed, this class is a subclass of the variety
of all dimonoids which is closed under taking of homomorphic images,
subdimonoids and Cartesian products. A dimonoid which is free in the
variety of n-nilpotent dimonoids will be called a free n-nilpotent dimonoid.
The necessary information about varieties of dimonoids can be found
in [13].
A. V. Zhuchok 303
Let X be an alphabet, F [X] be the free semigroup over X. Denote
the length of a word w ∈ F [X] by lw. Fix n ∈ N and assume
FNn = {(w,m) ∈ F [X] × N |m ≤ lw ≤ n} ∪ {0}.
Define operations ⊣ and ⊢ on FNn by
(w1,m1) ⊣ (w2,m2) =
{
(w1w2,m1) , lw1w2
≤n,
0, lw1w2
> n,
(w1,m1) ⊢ (w2,m2) =
{
(w1w2, lw1
+m2) , lw1w2
≤n,
0, lw1w2
> n,
(w1,m1) ∗ 0 = 0 ∗ (w1,m1) = 0 ∗ 0 = 0
for all (w1,m1) , (w2,m2) ∈ FNn\{0} and ∗ ∈ {⊣,⊢}. The algebra
(FNn,⊣,⊢) will be denoted by FNn(X).
Theorem 1. FNn(X) is the free n-nilpotent dimonoid.
Proof. Let (w1,m1), (w2,m2) , (w3,m3) ∈ FNn\{0}. In order to prove
that FNn(X) is a dimonoid we consider the following cases.
If lw1w2
> n or lw2w3
> n, then the proof is straightforward. Similar
to [19], the case lw1w2w3
≤ n can be considered. In the case lw1w2
≤ n,
lw2w3
≤ n and lw1w2w3
> n we have
((w1,m1)∗1 (w2,m2))∗2 (w3,m3) = 0 = (w1,m1)∗1 ((w2,m2)∗2 (w3,m3))
for ∗1, ∗2 ∈ {⊣,⊢}. The proofs of the remaining cases are obvious. Thus,
FNn(X) is a dimonoid.
As for any (wi,mi) ∈ FNn\{0}, 1 ≤ i ≤ n + 1, and ∗j ∈ {⊣,⊢},
1 ≤ j ≤ n, any parenthesizing of
(w1,m1) ∗1 (w2,m2) ∗2 . . . ∗n (wn+1,mn+1)
gives 0, then FNn(X) is nilpotent. At the same time for any (xi, 1) ∈
FNn\{0}, where xi ∈ X, 1 ≤ i ≤ n,
(x1, 1) ⊣ (x2, 1) ⊣ . . . ⊣ (xn, 1) = (x1x2 . . . xn, 1) 6= 0.
It means that FNn(X) has nilpotency index n.
Let us show that FNn(X) is free.
304 Free n-nilpotent dimonoids
Let
(
T,⊣′,⊢′
)
be an arbitrary n-nilpotent dimonoid and β : X → T
be an arbitrary map. Define a map
π : FNn(X) → (T,⊣′,⊢′) : u 7→ uπ,
assuming
uπ =
x1β⊢′ . . .⊢′xlβ⊣′ . . .⊣′xsβ, if
u = (x1 . . . xi . . . xs, l),
xi ∈ X, 1 ≤ i ≤ s,
0, if u = 0.
Show that π is a homomorphism. For arbitrary elements
(x1 . . . xi . . . xs, l) , (y1 . . . yj . . . yr, t) ∈ FNn\{0},
where xi, yj ∈ X, 1 ≤ i ≤ s, 1 ≤ j ≤ r, we obtain
((x1 . . . xi . . . xs, l) ⊣ (y1 . . . yj . . . yr, t))π =
=
{
(x1 . . . xsy1 . . . yr, l)π, s+ r≤n,
0π, s+ r > n,
((x1 . . . xi . . . xs, l) ⊢ (y1 . . . yj . . . yr, t))π =
=
{
(x1 . . . xsy1 . . . yr, s+ t)π, s+ r≤n,
0π, s+ r > n.
If s+ r≤n, then, using the axioms of a dimonoid, we have
(x1 . . . xsy1 . . . yr, l)π =
= x1β⊢′ . . .⊢′xlβ⊣′ . . .⊣′xsβ⊣′y1β⊣′ . . .⊣′yrβ =
= (x1β⊢′ . . .⊢′xlβ⊣′ . . .⊣′xsβ)⊣′(y1β⊢′ . . .⊢′ytβ⊣′ . . .⊣′yrβ) =
= (x1 . . . xi . . . xs, l)π⊣′ (y1 . . . yj . . . yr, t)π,
(x1 . . . xsy1 . . . yr, s+ t)π =
= x1β⊢′ . . .⊢′xsβ⊢′y1β⊢′ . . .⊢′ytβ⊣′ . . .⊣′yrβ =
=
(
x1β⊢′ . . .⊢′xlβ⊣′ . . .⊣′xsβ
)
⊢′
(
y1β⊢′ . . .⊢′ytβ⊣′ . . .⊣′yrβ
)
=
= (x1 . . . xi . . . xs, l)π⊢′ (y1 . . . yj . . . yr, t)π.
In the case s+ r > n,
0π = 0 = (x1 . . . xi . . . xs, l)π ∗′ (y1 . . . yj . . . yr, t)π
for ∗ ∈ {⊣,⊢}.
The proofs of the remaining cases are obvious. Thus, π is a homomor-
phism. This completes the proof of Theorem 1.
A. V. Zhuchok 305
Corollary 1. The free n-nilpotent dimonoid FNn(X) generated by a
finite set X is finite. Specifically, |FNn(X)| =
∑n
i=1 i|X|i + 1.
Now we construct a dimonoid which is isomorphic to the free n-
nilpotent dimonoid of rank 1.
Fix n ∈ N and let Ñn = {(m, t) ∈ N × N | t ≤ m ≤ n} ∪ {0}. Define
operations ⊣ and ⊢ on Ñn by
(m1, t1) ⊣ (m2, t2) =
{
(m1 +m2, t1) , m1 +m2≤n,
0, m1 +m2 > n,
(m1, t1) ⊢ (m2, t2) =
{
(m1 +m2,m1 + t2) , m1 +m2≤n,
0, m1 +m2 > n,
(m1, t1) ∗ 0 = 0 ∗ (m1, t1) = 0 ∗ 0 = 0
for all (m1, t1) , (m2, t2) ∈ Ñn\{0} and ∗ ∈ {⊣,⊢}. An immediate verifica-
tion shows that axioms of a dimonoid hold concerning operations ⊣ and
⊢. So, (Ñn,⊣,⊢) is a dimonoid. Denote it by Nn.
Lemma 1. If |X| = 1, then FNn(X) ∼= Nn.
Proof. Let X = {a}. Define a map
δ : FNn(X) → Nn,
assuming
uδ =
{
(k, l), u = (ak, l),
0, u = 0.
An easy verification shows that δ is an isomorphism.
4. 0-diband decompositions of FNn(X)
In this section we introduce the notion of a 0-diband of subdimonoids
and in terms of 0-dibands of subdimonoids describe the structure of free
n-nilpotent dimonoids.
For dimonoids with zero there exists a natural analog of the notion of
a diband of subdimonoids [17].
A dimonoid S with zero 0 (see Sect. 2) will be called a 0-diband of
subdimonoids Sβ, β ∈ B, where B is an idempotent dimonoid [17], if
S =
⋃
β∈B Sβ , Sβ∩Sγ = {0} for β 6= γ and Sβ⊣Sγ ⊆ Sβ⊣γ , Sβ⊢Sγ ⊆ Sβ⊢γ
306 Free n-nilpotent dimonoids
for any β, γ ∈ B. If B is an idempotent semigroup (band), then we say
that S is a 0-band of subdimonoids Sβ, β ∈ B.
Observe that the notion of a 0-diband of subdimonoids generalizes
the notion of a 0-band of semigroups [20] which plays an important role
in the structural theory of semigroups.
Let w = x1 . . . xi . . . xs ∈ F [X], where xi ∈ X, 1 ≤ i ≤ s (see Sect. 3).
Denote the set of all letters occurring in w by c (w).
Consider the semigroups Xrb,Xℓz,Xrz,B(X),Bℓz(X),Brz(X) and the
dimonoids FRct(X), Xℓz,rb, Xrb,rz, Xℓz,rz, Bℓz,rz(X) which were defined
in [14] and [15]. It is well-known that the first six semigroups are relatively
free bands. In [14] and [15] it was shown that the second five dimonoids
are relatively free dibands.
Let
P(a,b,c) = {(x1...xi...xs,m) ∈ FNn(X) | (x1, xm, xs) = (a, b, c)} ∪ {0}
for (a, b, c) ∈ FRct(X), n > 2 and |X| > 1;
P(a,b] = {(x1...xi...xs,m) ∈ FNn(X) | (x1, xm) = (a, b)} ∪ {0}
for (a, b) ∈ Xℓz,rb, n > 1 and |X| > 1;
P[b,c) = {(x1...xi...xs,m) ∈ FNn(X) | (xm, xs) = (b, c)} ∪ {0}
for (b, c) ∈ Xrb,rz, n > 1 and |X| > 1;
P(b] = {(x1...xi...xs,m) ∈ FNn(X) |xm = b} ∪ {0}
for b ∈ Xℓz,rz, n > 1 and |X| > 1;
P Y(b] = {(x1...xi...xs,m) ∈ FNn(X) |(xm, c(x1...xi...xs)) = (b, Y )} ∪ {0}
for (b, Y ) ∈ Bℓz,rz(X), n > 1 and 1 < |X| ≤ n.
Further we will deal with 0-diband decompositions and 0-band decom-
positions of free n-nilpotent dimonoids.
The following structure theorem gives decompositions of free n-nilpotent
dimonoids into 0-dibands of subdimonoids.
Theorem 2. The free n-nilpotent dimonoid FNn(X) is a 0-diband of
subdimonoids
(i) P(a,b,c), (a, b, c) ∈ FRct(X), if n > 2 and |X| > 1;
(ii) P(a,b], (a, b) ∈ Xℓz,rb, if n > 1 and |X| > 1;
(iii) P[b,c), (b, c) ∈ Xrb,rz, if n > 1 and |X| > 1;
(iv) P(b], b ∈ Xℓz,rz, if n > 1 and |X| > 1;
(v) P Y(b], (b, Y ) ∈ Bℓz,rz(X), if n > 1 and 1 < |X| ≤ n.
A. V. Zhuchok 307
Proof. We prove (v). It is clear that in the case n > 1 and 1 < |X| ≤ n,
P Y(b]\{0} 6= ∅ for all (b, Y ) ∈ Bℓz,rz(X). Moreover, P Y(b], (b, Y ) ∈ Bℓz,rz(X),
is a subdimonoid of FNn(X). Obviously,
FNn(X) =
⋃
(b,Y )∈Bℓz,rz(X) P
Y
(b] and P Y(b] ∩ PΛ
(y] = {0}
for (b, Y ) 6= (y,Λ). It is immediate to check that
P Y(b]⊣P
Λ
(y] ⊆ P Y ∪Λ
(b] and P Y(b]⊢P
Λ
(y] ⊆ P Y ∪Λ
(y]
for any (b, Y ), (y,Λ) ∈ Bℓz,rz(X). Thus, FNn(X) is a 0-diband of subdi-
monoids P Y(b], (b, Y ) ∈ Bℓz,rz(X).
The proofs of the remaining cases are similar.
Assume
P(a,c) = {(x1...xi...xs,m) ∈ FNn(X) | (x1, xs) = (a, c)} ∪ {0}
for (a, c) ∈ Xrb, n > 1 and |X| > 1;
P(a) = {(x1...xi...xs,m) ∈ FNn(X) |x1 = a} ∪ {0}
for a ∈ Xℓz, n > 1 and |X| > 1;
P[c] = {(x1...xi...xs,m) ∈ FNn(X) |xs = c} ∪ {0}
for c ∈ Xrz, n > 1 and |X| > 1;
PY = {(x1...xi...xs,m) ∈ FNn(X) | c(x1...xi...xs) = Y } ∪ {0}
for Y ∈ B(X), n > 1 and 1 < |X| ≤ n;
P Y(a) = {(x1...xi...xs,m) ∈ FNn(X) | (x1, c(x1...xi...xs)) = (a, Y )} ∪ {0}
for (a, Y ) ∈ Bℓz(X), n > 1 and 1 < |X| ≤ n;
P Y[c] = {(x1...xi...xs,m) ∈ FNn(X) | (xs, c(x1...xi...xs)) = (c, Y )} ∪ {0}
for (c, Y ) ∈ Brz(X), n > 1 and 1 < |X| ≤ n.
The following structure theorem gives decompositions of free n-nilpotent
dimonoids into 0-bands of subdimonoids.
308 Free n-nilpotent dimonoids
Theorem 3. The free n-nilpotent dimonoid FNn(X) is a 0-band of
subdimonoids
(i) P(a,c), (a, c) ∈ Xrb, if n > 1 and |X| > 1;
(ii) P(a), a ∈ Xℓz, if n > 1 and |X| > 1;
(iii) P[c], c ∈ Xrz, if n > 1 and |X| > 1;
(iv) PY , Y ∈ B(X), if n > 1 and 1 < |X| ≤ n;
(v) P Y(a), (a, Y ) ∈ Bℓz(X), if n > 1 and 1 < |X| ≤ n;
(vi) P Y[c], (c, Y ) ∈ Brz(X), if n > 1 and 1 < |X| ≤ n.
Proof. We prove (v). It is easy to see that in the case n > 1 and 1 < |X| ≤
n, P Y(a)\{0} 6= ∅ for all (a, Y ) ∈ Bℓz(X). Besides, P Y(a), (a, Y ) ∈ Bℓz(X),
is a subdimonoid of FNn(X). Clearly,
FNn (X) =
⋃
(a,Y )∈Bℓz(X) P
Y
(a) and P Y(a) ∩ PΛ
(x) = {0}
for (a, Y ) 6= (x,Λ). One can check that
P Y(a)⊣P
Λ
(x) ⊆ P Y ∪Λ
(a) and P Y(a)⊢P
Λ
(x) ⊆ P Y ∪Λ
(a)
for any (a, Y ), (x,Λ) ∈ Bℓz(X). Consequently, FNn(X) is a 0-band of
subdimonoids P Y(a), (a, Y ) ∈ Bℓz(X).
The proofs of the remaining cases are similar.
5. The least n-nilpotent congruence on a free dimonoid
In this section we present the least n-nilpotent congruence on a free
dimonoid.
If f : D1 → D2 is a homomorphism of dimonoids, then the correspond-
ing congruence on D1 will be denoted by ∆f . If ρ is a congruence on a
dimonoid (D,⊣,⊢) such that (D,⊣,⊢) /ρ is an n-nilpotent dimonoid (see
Sect. 2), then we say that ρ is an n-nilpotent congruence.
Let F̆ [X] be the free dimonoid over X (see [14], [19]). Fix n ∈ N and
define a relation ξn on F̆ [X] by
(w1,m1)ξn(w2,m2) if and only if
(w1,m1) = (w2,m2) or lw1
> n, lw2
> n.
Theorem 4. The relation ξn on the free dimonoid F̆ [X] is the least
n-nilpotent congruence.
A. V. Zhuchok 309
Proof. Define a map ψ : F̆ [X] → FNn(X) by
(w,m)ψ =
{
(w,m) , lw≤n,
0, lw > n,
(w,m) ∈ F̆ [X] .
Let (w1,m1) , (w2,m2) ∈ F̆ [X] and lw1w2
≤n. From lw1w2
≤n it follows
that lw1
<n and lw2
<n. Then
((w1,m1) ⊣ (w2,m2))ψ = (w1w2,m1)ψ = (w1w2,m1) =
= (w1,m1) ⊣ (w2,m2) = (w1,m1)ψ⊣ (w2,m2)ψ,
((w1,m1) ⊢ (w2,m2))ψ =
= (w1w2, lw1
+m2)ψ = (w1w2, lw1
+m2) =
= (w1,m1) ⊢ (w2,m2) = (w1,m1)ψ⊢ (w2,m2)ψ.
If lw1w2
> n, then
((w1,m1) ⊣ (w2,m2))ψ = (w1w2,m1)ψ = 0 =
= (w1,m1)ψ⊣ (w2,m2)ψ,
((w1,m1) ⊢ (w2,m2))ψ = (w1w2, lw1
+m2)ψ = 0 =
= (w1,m1)ψ⊢ (w2,m2)ψ.
Thus, ψ is a surjective homomorphism. By Theorem 1 FNn(X) is the
free n-nilpotent dimonoid. Then ∆ψ is the least n-nilpotent congruence
on F̆ [X]. From the definition of ψ it follows that ∆ψ = ξn.
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Contact information
A. V. Zhuchok Department of Mathematical Analysis and Algebra,
Luhansk Taras Shevchenko National University,
Oboronna str., 2, Luhansk, 91011, Ukraine
E-Mail: zhuchok_a@mail.ru
Received by the editors: 02.07.2013
and in final form 27.08.2013.
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