Free n-dinilpotent doppelsemigroups
A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic K-theory. In this paper we consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra...
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| Cite this: | Free n-dinilpotent doppelsemigroups / A.V. Zhuchok, M. Demko // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 304-316. — Бібліогр.: 23 назв. — англ. |
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irk-123456789-1557352019-06-18T01:26:27Z Free n-dinilpotent doppelsemigroups Zhuchok, A.V. Demko, M. A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic K-theory. In this paper we consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra. We construct a freen-dinilpotent doppelsemigroup and study separately freen-dinilpotentdoppelsemigroups of rank 1. Moreover,we characterize the least n-dinilpotent congruence on a free doppelsemigroup, establish that the semigroups of the freen-dinilpotentdoppelsemigroup are isomorphic and the automorphism group of the freen-dinilpotent doppelsemigroup is isomorphic to the symmetric group. We also give different examples of doppelsemigroups andprove that a system of axioms of a doppelsemigroup is independent. 2016 Article Free n-dinilpotent doppelsemigroups / A.V. Zhuchok, M. Demko // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 304-316. — Бібліогр.: 23 назв. — англ. 1726-3255 2010 MSC:08B20, 20M10, 20M50, 17A30. http://dspace.nbuv.gov.ua/handle/123456789/155735 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic K-theory. In this paper we consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra. We construct a freen-dinilpotent doppelsemigroup and study separately freen-dinilpotentdoppelsemigroups of rank 1. Moreover,we characterize the least n-dinilpotent congruence on a free doppelsemigroup, establish that the semigroups of the freen-dinilpotentdoppelsemigroup are isomorphic and the automorphism group of the freen-dinilpotent doppelsemigroup is isomorphic to the symmetric group. We also give different examples of doppelsemigroups andprove that a system of axioms of a doppelsemigroup is independent. |
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Zhuchok, A.V. Demko, M. Free n-dinilpotent doppelsemigroups Algebra and Discrete Mathematics |
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Zhuchok, A.V. Demko, M. |
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Free n-dinilpotent doppelsemigroups |
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Free n-dinilpotent doppelsemigroups |
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Free n-dinilpotent doppelsemigroups |
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Free n-dinilpotent doppelsemigroups |
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Free n-dinilpotent doppelsemigroups |
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free n-dinilpotent doppelsemigroups |
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Free n-dinilpotent doppelsemigroups / A.V. Zhuchok, M. Demko // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 304-316. — Бібліогр.: 23 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT zhuchokav freendinilpotentdoppelsemigroups AT demkom freendinilpotentdoppelsemigroups |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 22 (2016). Number 2, pp. 304–316
© Journal “Algebra and Discrete Mathematics”
Free n-dinilpotent doppelsemigroups
Anatolii V. Zhuchok∗ and Milan Demko∗∗
Abstract. A doppelalgebra is an algebra defined on a vector
space with two binary linear associative operations. Doppelalgeb-
ras play a prominent role in algebraic K-theory. In this paper
we consider doppelsemigroups, that is, sets with two binary asso-
ciative operations satisfying the axioms of a doppelalgebra. We
construct a free n-dinilpotent doppelsemigroup and study sepa-
rately free n-dinilpotent doppelsemigroups of rank 1. Moreover,
we characterize the least n-dinilpotent congruence on a free dop-
pelsemigroup, establish that the semigroups of the free n-dinilpotent
doppelsemigroup are isomorphic and the automorphism group of the
free n-dinilpotent doppelsemigroup is isomorphic to the symmetric
group. We also give different examples of doppelsemigroups and
prove that a system of axioms of a doppelsemigroup is independent.
1. Introduction
The notion of a doppelalgebra was considered by Richter [1] in the
context of algebraic K-theory. She defined this notion as a vector space
over a field equipped with two binary linear associative operations ⊣
and ⊢ satisfying the axioms (x ⊣ y) ⊢ z = x ⊣ (y ⊢ z), (x ⊢ y) ⊣
z = x ⊢ (y ⊣ z). Observe that any doppelalgebra gives rise to a Lie
algebra by [x, y] = x ⊢ y + x ⊣ y − y ⊢ x − y ⊣ x and conversely, any
∗The paper was written during the research stay of the first author at the University
of Presov under the National Scholarship Programme of the Slovak Republic.
∗∗The second author acknowledges the support of the Slovak VEGA Grant
No. 1/0063/14.
2010 MSC: 08B20, 20M10, 20M50, 17A30.
Key words and phrases: doppelalgebra, interassociativity, doppelsemigroup,
free n-dinilpotent doppelsemigroup, free doppelsemigroup, semigroup, congruence.
A. V. Zhuchok, M. Demko 305
Lie algebra has a universal enveloping doppelalgebra (see [1]). Moreover,
for any doppelalgebra a new operation · defined by x · y = x ⊢ y +
x ⊣ y is associative and so, there exists a functor from the category of
doppelalgebras to the category of associative algebras. Later Pirashvili [2]
considered duplexes which are sets equipped with two binary associative
operations and constructed a free duplex of an arbitrary rank. He also
considered duplexes with operations satisfying the axioms of a doppelalgeb-
ra denoting obtained category by Duplexes2. Such algebraic structures
are called doppelsemigroups [3]. A free doppelsemigroup of rank 1 is given
in [1] (see also [2]). Operations of the free doppelsemigroup of rank 1
are used in [4]. Doppelalgebras appeared in [5] as algebras over some
operad. Doppelalgebras and doppelsemigroups have relationships with
interassociativity for semigroups originated by Drouzy [6] and investigated
in [7–11], strong interassociativity for semigroups introduced by Gould and
Richardson [12] and dimonoids introduced by Loday [13] (see also [14–18],
[20–22]). Doppelsemigroups are a generalization of semigroups and all
results obtained for doppelsemigroups can be applied to doppelalgebras.
For further details and background see [1].
The free product of doppelsemigroups, the free doppelsemigroup, the
free commutative doppelsemigroup and the free n-nilpotent doppelsemi-
group were constructed in [3]. The paper [17] gives a classification of
relatively free dimonoids, in particular, therein the free n-dinilpotent
dimonoid [18] is presented. In this paper we continue researches from [3,
18] developing the variety theory of doppelsemigroups. The main focus of
our paper is to study dinilpotent doppelsemigroups.
In Section 3 we present different examples of doppelsemigroups.
In Section 4 we prove that a system of axioms of a doppelsemigroup
is independent.
In Section 5 we construct a free n-dinilpotent doppelsemigroup of an
arbitrary rank and consider separately free n-dinilpotent doppelsemigroups
of rank 1. We also establish that the semigroups of the free n-dinilpotent
doppelsemigroup are isomorphic and the automorphism group of the free
n-dinilpotent doppelsemigroup is isomorphic to the symmetric group.
In the final section we characterize the least n-dinilpotent congruence
on a free doppelsemigroup.
2. Preliminaries
Recall that a doppelalgebra [1, 2] is a vector space V over a field
equipped with two binary linear operations ⊣ and ⊢: V ⊗V → V , satisfying
306 Free n-dinilpotent doppelsemigroups
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)
A nonempty set with two binary operations ⊣ and ⊢ satisfying the
axioms (D1)–(D4) is called a doppelsemigroup [3].
Given a semigroup (D, ⊣), consider a semigroup (D, ⊢) defined on
the same set. Recall that (D, ⊢) is an interassociate of (D, ⊣) [6], if the
axioms (D1) and (D2) hold. Strong interassociativity [12] is defined by
the axioms (D1) and (D2) along with
x ⊢ (y ⊣ z) = x ⊣ (y ⊢ z). (D5)
Thus, we can see that in any doppelsemigroup (D, ⊣, ⊢), (D, ⊢) is
an interassociate of (D, ⊣), and conversely, if a semigroup (D, ⊢) is an
interassociate of a semigroup (D, ⊣), then (D, ⊣, ⊢) is a doppelsemigroup
[3]. Moreover, a semigroup (D, ⊢) is a strong interassociate of a semi-
group (D, ⊣) if and only if (D, ⊣, ⊢) is a doppelsemigroup satisfying the
axiom (D5).
Descriptions of all interassociates of a monogenic semigroup and of the
free commutative semigroup are presented in [7] and [8, 10], respectively.
More recently, the paper [11] was devoted to studying interassociates
of the bicyclic semigroup. Methods of constructing interassociates for
semigroups were developed in [19].
Recall the definition of a k-nilpotent semigroup (see also [14, 17, 18]).
As usual, N denotes the set of all positive integers. A semigroup S is
called nilpotent, if Sn+1 = 0 for some n ∈ N. The least such n is called
the nilpotency index of S. For k ∈ N a nilpotent semigroup of nilpotency
index 6 k is called k-nilpotent.
An element 0 of a doppelsemigroup (D, ⊣, ⊢) is called zero [3], if
x ∗ 0 = 0 = 0 ∗ x for all x ∈ D and ∗ ∈ {⊣, ⊢}. A doppelsemigroup
(D, ⊣, ⊢) with zero will be called dinilpotent, if (D, ⊣) and (D, ⊢) are
nilpotent semigroups. A dinilpotent doppelsemigroup (D, ⊣, ⊢) will be
called n-dinilpotent, if (D, ⊣) and (D, ⊢) are n-nilpotent semigroups. If
ρ is a congruence on a doppelsemigroup (D, ⊣, ⊢) such that (D, ⊣, ⊢)/ρ
is an n-dinilpotent doppelsemigroup, we say that ρ is an n-dinilpotent
congruence.
A. V. Zhuchok, M. Demko 307
Note that operations of any 1-dinilpotent doppelsemigroup coincide
and it is a zero semigroup. The class of all n-dinilpotent doppelsemigroups
forms a subvariety of the variety of doppelsemigroups. It is not difficult to
check that the variety of n-nilpotent doppelsemigroups [3] is a subvariety
of the variety of n-dinilpotent doppelsemigroups. A doppelsemigroup
which is free in the variety of n-dinilpotent doppelsemigroups will be
called a free n-dinilpotent doppelsemigroup.
Lemma 1 ([3], Lemma 3.1). In a doppelsemigroup (D, ⊣, ⊢) for any
n > 1, 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
gives the same element from D.
The free doppelsemigroup is given in [3]. Recall this construction.
Let X be an arbitrary nonempty set and let ω be an arbitrary word in
the alphabet X. The length of ω will be denoted by lω. Let further F [X] be
the free semigroup on X, T the free monoid on the two-element set {a, b}
and θ ∈ T the empty word. By definition, the length lθ of θ is equal to 0.
Define operations ⊣ and ⊢ on F = {(w, u) ∈ F [X] × T | lw − lu = 1} by
(w1, u1) ⊣ (w2, u2) = (w1w2, u1au2),
(w1, u1) ⊢ (w2, u2) = (w1w2, u1bu2)
for all (w1, u1), (w2, u2) ∈ F . The algebra (F, ⊣, ⊢) is denoted by FDS(X).
Theorem 1 ([3], Theorem 3.5). FDS(X) is the free doppelsemigroup.
If f : D1 → D2 is a homomorphism of doppelsemigroups, the cor-
responding congruence on D1 will be denoted by ∆f . Denote the symmet-
ric group on X by ℑ[X] and the automorphism group of a doppelsemigroup
M by Aut M .
3. Some examples
In this section we give different examples of doppelsemigroups.
a) Every semigroup can be considered as a doppelsemigroup (see [3]).
b) Recall that a dimonoid [13–18, 20–22] is a nonempty set equipped
with two binary operations ⊣ and ⊢ satisfying the axioms (D2)–(D4) and
(x ⊣ y) ⊣ z = x ⊣ (y ⊢ z),
(x ⊣ y) ⊢ z = x ⊢ (y ⊢ z).
308 Free n-dinilpotent doppelsemigroups
A dimonoid is called commutative [20], if both its operations are commu-
tative. The following assertion gives relationships between commutative
dimonoids and doppelsemigroups (this assertion was formulated without
the proof in [3] and [16]).
Proposition 1. Every commutative dimonoid is a doppelsemigroup.
Proof. Let (D, ⊣, ⊢) be a commutative dimonoid. Then, by definition,
(D, ⊣, ⊢) satisfies the axioms (D2)–(D4). From Lemma 2 of [20] it follows
that (D, ⊣, ⊢) satisfies the axiom (D1). So, it is a doppelsemigroup.
Examples of commutative dimonoids can be found in [14, 20].
c) Let (D, ⊣, ⊢) be a doppelsemigroup and a, b ∈ D. Define operations
⊣a and ⊢b on D by
x ⊣a y = x ⊣ a ⊣ y, x ⊢b y = x ⊢ b ⊢ y
for all x, y ∈ D. By a direct verification (D, ⊣a, ⊢b) is a doppelsemigroup.
We call the doppelsemigroup (D, ⊣a, ⊢b) a variant of (D, ⊣, ⊢), or, al-
ternatively, the sandwich doppelsemigroup of (D, ⊣, ⊢) with respect to
the sandwich elements a and b, or the doppelsemigroup with deformed
multiplications.
d) The direct product
∏
i∈I Di of doppelsemigroups Di, i ∈ I, is,
obviously, a doppelsemigroup.
e) Now we give a new class of doppelsemigroups with zero.
Let D = (D, ⊣, ⊢) be an arbitrary doppelsemigroup and I 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 2. B(D, I) is a doppelsemigroup with zero.
Proof. The proof is similar to the proof of Proposition 1 from [21].
Observe that if operations of a doppelsemigroup D coincide and it
is a group G, then any Brandt semigroup [23] is isomorphic to some
semigroup B(G, I). So, B(D, I) generalizes the semigroup B(G, I). We
call the doppelsemigroup B(D, I) a Brandt doppelsemigroup.
A. V. Zhuchok, M. Demko 309
4. Independence of axioms of a doppelsemigroup
In this section for a doppelsemigroup we prove the following theorem.
Theorem 2. A system of axioms (D1)–(D4) as defined above is indepen-
dent.
Proof. Let X be an arbitrary nonempty set, |X| > 1. Define operations
⊣ and ⊢ on X by
x ⊣ y = x, x ⊢ y = y
for all x, y ∈ X. The model (X, ⊣, ⊢) satisfies the axioms (D2)–(D4) but
does not satisfy (D1). Indeed, for all x, y, z ∈ X,
(x ⊢ y) ⊣ z = y = x ⊢ (y ⊣ z),
(x ⊣ y) ⊣ z = x = x ⊣ (y ⊣ z),
(x ⊢ y) ⊢ z = z = x ⊢ (y ⊢ z).
Since |X| > 1, there is x, z ∈ X such that x 6= z. Consequently, for all
y ∈ X,
(x ⊣ y) ⊢ z = z 6= x = x ⊣ (y ⊢ z).
Put
x ⊣ y = y, x ⊢ y = x
for all x, y ∈ X. As in the previous case, we can show that (X, ⊣, ⊢)
satisfies the axioms (D1), (D3), (D4) but does not satisfy (D2).
Let N
0 be the set of all positive integers with zero and let
x ⊣ y = 2x, z ⊣ 0 = 0 = 0 ⊣ z, z ⊢ c = 0
for all x, y ∈ N and z, c ∈ N
0. In this case the model (N0, ⊣, ⊢) satisfies
the axioms (D1), (D2), (D4) but does not satisfy (D3). Indeed, for all
z, c, a ∈ N
0,
(z ⊣ c) ⊢ a = 0 = z ⊣ (c ⊢ a),
(z ⊢ c) ⊣ a = 0 = z ⊢ (c ⊣ a),
(z ⊢ c) ⊢ a = 0 = z ⊢ (c ⊢ a).
In addition, for all x, y, b ∈ N we get
(x ⊣ y) ⊣ b = 2x ⊣ b = 4x 6= 2x = x ⊣ 2y = x ⊣ (y ⊣ b).
Put
z ⊣ c = 0, x ⊢ y = 2y, z ⊢ 0 = 0 = 0 ⊢ z
for all z, c ∈ N
0 and x, y ∈ N. As in the previous case, we can show that
(N0, ⊣, ⊢) satisfies the axioms (D1)–(D3) but does not satisfy (D4).
310 Free n-dinilpotent doppelsemigroups
5. Constructions
In this section we construct a free n-dinilpotent doppelsemigroup of an
arbitrary rank and consider separately free n-dinilpotent doppelsemigroups
of rank 1. We also establish that the semigroups of the free n-dinilpotent
doppelsemigroup are isomorphic and the automorphism group of the free
n-dinilpotent doppelsemigroup is isomorphic to the symmetric group.
As in Section 2, let F [X] be the free semigroup on X, T the free
monoid on the two-element set {a, b} and θ ∈ T the empty word. For
x ∈ {a, b} and all u ∈ T , the number of occurrences of an element x in u
is denoted by dx(u). Obviously, dx(θ) = 0. Fix n ∈ N and assume
Mn = {(w, u) ∈ F [X] × T | lw − lu = 1, dx(u) + 1 6 n, x ∈ {a, b}} ∪ {0}.
Define operations ⊣ and ⊢ on Mn by
(w1, u1) ⊣ (w2, u2) =
{
(w1w2, u1au2), dx(u1au2) + 1 6 n, x ∈ {a, b},
0, in all other cases,
(w1, u1) ⊢ (w2, u2) =
{
(w1w2, u1bu2), dx(u1bu2) + 1 6 n, x ∈ {a, b},
0, in all other cases,
(w1, u1) ∗ 0 = 0 ∗ (w1, u1) = 0 ∗ 0 = 0
for all (w1, u1), (w2, u2) ∈ Mn\{0} and ∗ ∈ {⊣, ⊢}. The obtained algebra
will be denoted by FDDSn(X).
Theorem 3. FDDSn(X) is the free n-dinilpotent doppelsemigroup.
Proof. First prove that FDDSn(X) is a doppelsemigroup. Let (w1, u1),
(w2, u2), (w3, u3) ∈ Mn\{0}. For x, y, z ∈ {a, b} it is clear that
dx(u1yu2zu3) + 1 6 n
implies
dx(u1yu2) + 1 6 n, (1)
dx(u2zu3) + 1 6 n. (2)
Let dx(u1au2au3) + 1 6 n for all x ∈ {a, b}. Then, using (1), (2),
we get
((w1, u1) ⊣ (w2, u2)) ⊣ (w3, u3) = (w1w2, u1au2) ⊣ (w3, u3)
= (w1w2w3, u1au2au3)
= (w1, u1) ⊣ (w2w3, u2au3)
= (w1, u1) ⊣ ((w2, u2) ⊣ (w3, u3)).
A. V. Zhuchok, M. Demko 311
If dx(u1au2au3) + 1 > n for some x ∈ {a, b}, then, obviously,
((w1, u1) ⊣ (w2, u2)) ⊣ (w3, u3) = 0 = (w1, u1) ⊣ ((w2, u2) ⊣ (w3, u3)).
So, the axiom (D3) of a doppelsemigroup holds.
If dx(u1au2bu3) + 1 6 n for all x ∈ {a, b}, then, using (1), (2), obtain
((w1, u1) ⊣ (w2, u2)) ⊢ (w3, u3) = (w1w2, u1au2) ⊢ (w3, u3)
= (w1w2w3, u1au2bu3)
= (w1, u1) ⊣ (w2w3, u2bu3)
= (w1, u1) ⊣ ((w2, u2) ⊢ (w3, u3)).
Let dx(u1au2bu3) + 1 > n for some x ∈ {a, b}. Then, clearly,
((w1, u1) ⊣ (w2, u2)) ⊢ (w3, u3) = 0 = (w1, u1) ⊣ ((w2, u2) ⊢ (w3, u3)).
Thus, the axiom (D1) of a doppelsemigroup holds. Similarly, one can
check the axioms (D2) and (D4). Thus, FDDSn(X) is a doppelsemigroup.
Take arbitrary elements (wi, ui) ∈ Mn\{0}, 1 6 i 6 n + 1. It is clear
that
da(u1au2a . . . aun+1) + 1 > n.
From here
(w1, u1) ⊣ (w2, u2) ⊣ . . . ⊣ (wn+1, un+1) = 0.
At the same time, assuming y0 = θ for y ∈ {a, b}, for any (xi, θ) ∈ Mn\{0},
where xi ∈ X, 1 6 i 6 n, get
(x1, θ) ⊣ (x2, θ) ⊣ . . . ⊣ (xn, θ) = (x1x2 . . . xn, an−1) 6= 0.
From the last arguments we conclude that (Mn, ⊣) is a nilpotent semigroup
of nilpotency index n. Analogously, we can prove that (Mn, ⊢) is a nilpotent
semigroup of nilpotency index n. So, by definition, FDDSn(X) is an
n-dinilpotent doppelsemigroup.
Let us show that FDDSn(X) is free in the variety of n-dinilpotent
doppelsemigroups.
Obviously, FDDSn(X) is generated by X × {θ}. Let (K, ⊣
′
, ⊢
′
) be an
arbitrary n-dinilpotent doppelsemigroup. Let β : X × {θ} → K be an
arbitrary map. Consider a map α : X → K such that xα = (x, θ)β for all
x ∈ X and define a map
π : FDDSn(X) → (K, ⊣
′
, ⊢
′
)
312 Free n-dinilpotent doppelsemigroups
by
ωπ =
x1αỹ1x2αỹ2 . . . ỹs−1xsα, if ω = (x1x2 . . . xs, y1y2 . . . ys−1),
xd ∈ X, 1 6 d 6 s, yp ∈ {a, b},
1 6 p 6 s − 1, s > 1,
x1α, if ω = (x1, θ), x1 ∈ X,
0, if ω = 0,
where
ỹp =
{
⊣
′
, yp = a,
⊢
′
, yp = b
for all 1 6 p 6 s − 1, s > 1. According to Lemma 1 π is well-defined.
To show that π is a homomorphism we will use the axioms of a
doppelsemigroup and the identities of an n-dinilpotent doppelsemigroup.
If s = 1, we will regard the sequence y1y2 . . . ys−1 ∈ T as θ. For
arbitrary elements
(w1, u1) = (x1x2 . . . xs, y1y2 . . . ys−1),
(w2, u2) = (z1z2 . . . zk, c1c2 . . . ck−1) ∈ FDDSn(X),
where xd, zi ∈ X, 1 6 d 6 s, 1 6 i 6 k, yp, cj ∈ {a, b}, 1 6 p 6 s − 1,
1 6 j 6 k − 1, in the case dx(u1au2) + 1 6 n for all x ∈ {a, b}, we get
((x1x2 . . . xs, y1y2 . . . ys−1) ⊣ (z1z2 . . . zk, c1c2 . . . ck−1))π
= (x1 . . . xsz1 . . . zk, y1 . . . ys−1ac1 . . . ck−1)π
= x1αỹ1 . . . ỹs−1xsαãz1αc̃1 . . . c̃k−1zkα
= (x1αỹ1 . . . ỹs−1xsα) ⊣
′
(z1αc̃1 . . . c̃k−1zkα)
= (x1x2 . . . xs, y1y2 . . . ys−1)π ⊣
′
(z1z2 . . . zk, c1c2 . . . ck−1)π.
If dx(u1au2) + 1 > n for some x ∈ {a, b}, then
((x1x2 . . . xs, y1y2 . . . ys−1) ⊣ (z1z2 . . . zk, c1c2 . . . ck−1))π = 0π = 0.
Since (K, ⊣
′
, ⊢
′
) is n-dinilpotent, we have
0 = x1αỹ1 . . . ỹs−1xsαãz1αc̃1 . . . c̃k−1zkα
= (x1αỹ1 . . . ỹs−1xsα) ⊣
′
(z1αc̃1 . . . c̃k−1zkα)
= (x1x2 . . . xs, y1y2 . . . ys−1)π ⊣
′
(z1z2 . . . zk, c1c2 . . . ck−1)π.
A. V. Zhuchok, M. Demko 313
So,
((w1, u1) ⊣ (w2, u2))π = (w1, u1)π ⊣
′
(w2, u2)π
for all (w1, u1), (w2, u2) ∈ FDDSn(X).
Similarly for ⊢. So, π is a homomorphism. Clearly, (x, θ)π = (x, θ)β for
all (x, θ) ∈ X × {θ}. Since X × {θ} generates FDDSn(X), the uniqueness
of such homomorphism π is obvious. Thus, FDDSn(X) is free in the
variety of n-dinilpotent doppelsemigroups.
Now we construct a doppelsemigroup which is isomorphic to the free
n-dinilpotent doppelsemigroup of rank 1.
Fix n ∈ N and assume
Φn = {u ∈ T | dx(u) + 1 6 n, x ∈ {a, b}} ∪ {0}.
Define operations ⊣ and ⊢ on Φn by
u1 ⊣ u2 =
{
u1au2, dx(u1au2) + 1 6 n, x ∈ {a, b},
0, in all other cases,
u1 ⊢ u2 =
{
u1bu2, dx(u1bu2) + 1 6 n, x ∈ {a, b},
0, in all other cases,
u1 ∗ 0 = 0 ∗ u1 = 0 ∗ 0 = 0
for all u1, u2 ∈ Φn\{0} and ∗ ∈ {⊣, ⊢}. The obtained algebra will be
denoted by Φn. Obviously, Φn is a doppelsemigroup.
Lemma 2. If |X| = 1, then Φn
∼= FDDSn(X).
Proof. Let X = {r}. One can show that a map γ : Φn → FDDSn(X),
defined by the rule
uγ =
{
(rlu+1, u), u ∈ Φn\{0},
0, u = 0,
is an isomorphism.
The following lemma establishes a relationship between semigroups of
the free n-dinilpotent doppelsemigroup FDDSn(X).
Lemma 3. The semigroups (Mn, ⊣) and (Mn, ⊢) are isomorphic.
314 Free n-dinilpotent doppelsemigroups
Proof. Let â = b, b̂ = a and define a map σ : (Mn, ⊣) → (Mn, ⊢) by
putting
tσ =
(w, ŷ1ŷ2 . . . ŷm), t = (w, y1y2 . . . ym) ∈ Mn\{0},
yp ∈ {a, b}, 1 6 p 6 m,
t, in all other cases.
An immediate verification shows that σ is an isomorphism.
Since the set X × {θ} is generating for FDDSn(X), we obtain the
following description of the automorphism group of the free n-dinilpotent
doppelsemigroup.
Lemma 4. Aut FDDSn(X) ∼= ℑ[X].
6. The least n-dinilpotent congruence on a free doppel-
semigroup
In this section we present the least n-dinilpotent congruence on a free
doppelsemigroup.
Let FDS(X) be the free doppelsemigroup (see Section 2) and n ∈ N.
Define a relation µ(n) on FDS(X) by
(w1, u1)µ(n)(w2, u2) if and only if (w1, u1) = (w2, u2) or
{
dx(u1) + 1 > n for some x ∈ {a, b},
dy(u2) + 1 > n for some y ∈ {a, b}.
Theorem 4. The relation µ(n) on the free doppelsemigroup FDS(X) is
the least n-dinilpotent congruence.
Proof. Define a map ϕ : FDS(X) → FDDSn(X) by
(w, u)ϕ =
{
(w, u), if dx(u) + 1 6 n for all x ∈ {a, b},
0, in all other cases
((w, u) ∈ FDS(X)). Show that ϕ is a homomorphism.
Let (w1, u1), (w2, u2) ∈ FDS(X) and dx(u1au2)+16n for all x∈{a, b}.
From the last inequality it follows that dx(u1) + 1 6 n and dx(u2) + 1 6 n
for all x ∈ {a, b}. Then
((w1, u1) ⊣ (w2, u2))ϕ = (w1w2, u1au2)ϕ = (w1w2, u1au2)
= (w1, u1) ⊣ (w2, u2) = (w1, u1)ϕ ⊣ (w2, u2)ϕ.
A. V. Zhuchok, M. Demko 315
If dx(u1au2) + 1 > n for some x ∈ {a, b}, then
((w1, u1) ⊣ (w2, u2))ϕ = (w1w2, u1au2)ϕ = 0 = (w1, u1)ϕ ⊣ (w2, u2)ϕ.
Let further dx(u1bu2) + 1 6 n for all x ∈ {a, b}. Then dx(u1) + 1 6 n,
dx(u2) + 1 6 n for all x ∈ {a, b} and
((w1, u1) ⊢ (w2, u2))ϕ = (w1w2, u1bu2)ϕ = (w1w2, u1bu2)
= (w1, u1) ⊢ (w2, u2) = (w1, u1)ϕ ⊢ (w2, u2)ϕ.
If dx(u1bu2) + 1 > n for some x ∈ {a, b}, then
((w1, u1) ⊢ (w2, u2))ϕ = (w1w2, u1bu2)ϕ = 0 = (w1, u1)ϕ ⊢ (w2, u2)ϕ.
Thus, ϕ is a surjective homomorphism. By Theorem 3 FDDSn(X) is
the free n-dinilpotent doppelsemigroup. Then ∆ϕ is the least n-dinilpotent
congruence on FDS(X). From the definition of ϕ it follows that ∆ϕ = µ(n).
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Contact information
A. V. Zhuchok Department of Algebra and System Analysis,
Luhansk Taras Shevchenko National University,
Gogol square, 1, Starobilsk, 92703, Ukraine
E-Mail(s): zhuchok_a@mail.ru
M. Demko Department of Physics, Mathematics and Tech-
niques, University of Presov, Slovakia, 17. novem-
bra 1, Presov, 08116, Slovakia
E-Mail(s): milan.demko@unipo.sk
Received by the editors: 03.10.2016
and in final form 30.11.2016.
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