Abelian doppelsemigroups
A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain identities. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as doppelalgebras, duplexes, interassociative semigroups,...
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irk-123456789-1884152023-02-28T01:27:08Z Abelian doppelsemigroups Zhuchok, A.V. Knauer, K. A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain identities. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as doppelalgebras, duplexes, interassociative semigroups, restrictive bisemigroups, dimonoids and trioids. This paper is devoted to the study of abelian doppelsemigroups. We show that every abelian doppelsemigroup can be constructed from a left and right commutative semigroup and describe the free abelian doppelsemigroup. We also characterize the least abelian congruence on the free doppel-semigroup, give examples of abelian doppelsemigroups and find conditions under which the operations of an abelian doppelsemi-group coincide. 2018 Article Abelian doppelsemigroups / A.V. Zhuchok, K. Knauer // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 290–304. — Бібліогр.: 31 назв. — англ. 1726-3255 2010 MSC: 08B20, 20M10, 20M50, 17A30. http://dspace.nbuv.gov.ua/handle/123456789/188415 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain identities. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as doppelalgebras, duplexes, interassociative semigroups, restrictive bisemigroups, dimonoids and trioids. This paper is devoted to the study of abelian doppelsemigroups. We show that every abelian doppelsemigroup can be constructed from a left and right commutative semigroup and describe the free abelian doppelsemigroup. We also characterize the least abelian congruence on the free doppel-semigroup, give examples of abelian doppelsemigroups and find conditions under which the operations of an abelian doppelsemi-group coincide. |
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Zhuchok, A.V. Knauer, K. Abelian doppelsemigroups Algebra and Discrete Mathematics |
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Zhuchok, A.V. Knauer, K. |
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Abelian doppelsemigroups |
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Abelian doppelsemigroups |
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Abelian doppelsemigroups |
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Abelian doppelsemigroups |
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Abelian doppelsemigroups |
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abelian doppelsemigroups |
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Інститут прикладної математики і механіки НАН України |
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Abelian doppelsemigroups / A.V. Zhuchok, K. Knauer // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 290–304. — Бібліогр.: 31 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT zhuchokav abeliandoppelsemigroups AT knauerk abeliandoppelsemigroups |
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2025-07-16T10:26:56Z |
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1837798908867117056 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 26 (2018). Number 2, pp. 290–304
c© Journal “Algebra and Discrete Mathematics”
Abelian doppelsemigroups∗
Anatolii V. Zhuchok and Kolja Knauer
Abstract. A doppelsemigroup is an algebraic system con-
sisting of a set with two binary associative operations satisfying
certain identities. Doppelsemigroups are a generalization of semi-
groups and they have relationships with such algebraic structures
as doppelalgebras, duplexes, interassociative semigroups, restrictive
bisemigroups, dimonoids and trioids. This paper is devoted to the
study of abelian doppelsemigroups. We show that every abelian
doppelsemigroup can be constructed from a left and right commuta-
tive semigroup and describe the free abelian doppelsemigroup. We
also characterize the least abelian congruence on the free doppel-
semigroup, give examples of abelian doppelsemigroups and find
conditions under which the operations of an abelian doppelsemi-
group coincide.
1. Introduction
For semigroups, Drouzy introduced the notion of interassociativity
[3]: two semigroups are interassociative if certain axioms relating opera-
tions of these semigroups are satisfied. Interassociativity of semigroups
was studied in [1–9, 31]. In this paper, we consider doppelsemigroups
which are sets with two binary associative operations satisfying axioms
of interassociativity. Doppelalgebras introduced by Richter [15] in the
context of algebraic K-theory are linear analogs of doppelsemigroups and
∗The paper was written during the research stay of the first author at the University
of Aix-Marseille as a part of the French Government fellowship.
2010 MSC: 08B20, 20M10, 20M50, 17A30.
Key words and phrases: doppelsemigroup, abelian doppelsemigroup, free abelian
doppelsemigroup, free doppelsemigroup, interassociativity, semigroup, congruence, dop-
pelalgebra.
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A. V. Zhuchok, K. Knauer 291
commutative dimonoids in the sense of Loday [10, 18] are examples of
doppelsemigroups. Therefore, doppelsemigroup theory has connections
to doppelalgebra theory and dimonoid theory. Trioids were introduced in
the paper of Loday and Ronco [11] and were studied in different papers
(see, e.g., [28, 29]). These algebras were used in [11] to describe classes
of trialgebras. The system of axioms of a trioid includes some axioms of
a doppelsemigroup. Bisemigroups were considered in the work of Schein
[16] and, in particularly, restrictive bisemigroups were investigated in [17].
The latter algebras have applications in the theory of binary relations
[17]. It turns out that the varieties of commutative doppelsemigroups
and of commutative bisemigroups coincide. A doppelsemigroup can also
be determined by using the notion of a duplex [14]. Free duplexes were
constructed in [14]. It should be noted that doppelsemigroups satisfy the
hyperidentity of associativity [12, 13]. If operations of a doppelsemigroup
coincide, we obtain the notion of a semigroup.
The class of all doppelsemigroups forms a variety. Relatively free
doppelsemigroups [25] play a crucial role in studying the variety of dop-
pelsemigroups. This motivates us to investigate explicit constructions of
free algebras in the variety of doppelsemigroups. It should be noted that
some relatively free doppelsemigroups were studied recently: the construc-
tions of the free (strong) doppelsemigroup, of the free commutative (strong)
doppelsemigroup and of the free n-nilpotent (strong) doppelsemigroup
were presented in [23, 26]. The free n-dinilpotent (strong) doppelsemi-
group was constructed in [22,26]. In [21], the first author described the
free left n-dinilpotent doppelsemigroup. The problem is to study abelian
doppelsemigroups and construct the free object in the variety of abelian
doppelsemigroups. This is the main focus of the present paper.
The paper is organized as follows. In Section 2, we present various
notions and results used in the paper, and show that every abelian dop-
pelsemigroup can be constructed from a left and right commutative semi-
group. In Section 3, examples of abelian doppelsemigroups are given. In
Section 4, we construct the free abelian doppelsemigroup of arbitrary
rank and, as consequences, obtain the free abelian doppelsemigroup of
rank 1 and the free left and right commutative semigroup. We also estab-
lish that the automorphism group of the free abelian doppelsemigroup
is isomorphic to the symmetric group. In Section 5, we characterize the
least abelian congruence on the free doppelsemigroup. In the final section,
some properties of (abelian) doppelsemigroups are established.
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292 Abelian doppelsemigroups
2. Preliminaries
Recall that a nonempty set D equipped with two binary associative
operations ⊣ and ⊢ satisfying the axioms
(x ⊣ y) ⊢ z = x ⊣ (y ⊢ z), (D1)
(x ⊢ y) ⊣ z = x ⊢ (y ⊣ z) (D2)
for all x, y, z ∈ D is called a doppelsemigroup [23]. A doppelsemigroup
(D,⊣,⊢) is called strong [26] if it satisfies the axiom
x ⊣ (y ⊢ z) = x ⊢ (y ⊣ z).
A doppelsemigroup is called commutative [23] if both its operations are
commutative. A nonempty set equipped with two binary associative opera-
tions ⊣ and ⊢ satisfying the axioms (D2) and
(x ⊣ y) ⊣ z = x ⊣ (y ⊢ z), (D3)
(x ⊣ y) ⊢ z = x ⊢ (y ⊢ z) (D4)
is called a dimonoid [10]. For more information on dimonoids see [19,20,27].
A dimonoid (D,⊣,⊢) is called abelian [30] if
x ⊣ y = y ⊢ x (1)
for all x, y ∈ D. Examples of abelian dimonoids and, in particulary, free
abelian dimonoids were presented in [30].
A doppelsemigroup will be called abelian if it satisfies the identity
(1). The class of all abelian doppelsemigroups forms a subvariety of the
variety of doppelsemigroups which does not coincide with the variety
of commutative doppelsemigroups. A doppelsemigroup which is free in
the variety of abelian doppelsemigroups will be called the free abelian
doppelsemigroup. If ρ is a congruence on a doppelsemigroup (D,⊣,⊢) such
that (D,⊣,⊢)/ρ is an abelian doppelsemigroup, we say that ρ is an abelian
congruence.
Recall that a semigroup S is called left (respectively, right) commutative
if it satisfies the identity xya = yxa (respectively, axy = ayx). It is clear
that a left commutative semigroup satisfies any identity of the form
x1x2 . . . xna = x1πx2π . . . xnπa, (2)
where π is a permutation of {1, 2, . . . , n}. Dually, a right commutative
semigroup satisfies any identity of the form
ax1x2 . . . xn = ax1πx2π . . . xnπ, (3)
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A. V. Zhuchok, K. Knauer 293
where π is a permutation of {1, 2, . . . , n}. The class of all left and right
commutative semigroups forms a variety. A semigroup which is free in the
variety of left and right commutative semigroups will be called the free
left and right commutative semigroup.
If (D,⊣) is a semigroup, then D with an operation ⊢, defined by (1), is
a semigroup. The semigroup (D,⊢) is called dual to the semigroup (D,⊣).
If (D,⊢) is dual to (D,⊣), then it is clear that (D,⊣) is anti-isomorphic to
(D,⊢). Obviously, a binary relation ̺ on a semigroup (D,⊣) is a congruence
if and only if ̺ is a congruence on the semigroup (D,⊢) dual to (D,⊣).
From here, for an arbitrary abelian doppelsemigroup (D,⊣,⊢), the set
of all congruences on (D,⊣,⊢) coincides with the set of all congruences
on (D,⊣).
The following lemma establishes necessary and sufficient conditions
under which two dual semigroups give rise to an abelian doppelsemigroup.
Lemma 2.1. Let (D,⊣) be an arbitrary semigroup and (D,⊢) the dual
semigroup to (D,⊣). Then (D,⊣,⊢) is an abelian doppelsemigroup if and
only if (D,⊣) or (D,⊢) is a left and right commutative semigroup.
Proof. Let (D,⊣,⊢) be a doppelsemigroup and (D,⊢) dual to (D,⊣). Then
for all x, y, z ∈ D,
x ⊣ y ⊣ z = x ⊣ (z ⊢ y) = (x ⊣ z) ⊢ y = y ⊣ x ⊣ z,
x ⊣ y ⊣ z = (y ⊢ x) ⊣ z = y ⊢ (x ⊣ z) = x ⊣ z ⊣ y
according to (1) and the axioms of a doppelsemigroup. Hence (D,⊣) is a
left and right commutative semigroup.
Conversely, let (D,⊣) be a left and right commutative semigroup,
(D,⊢) dual to (D,⊣) and x, y, z ∈ D. Then
x ⊣ z ⊣ y = x ⊣ (y ⊢ z), z ⊣ x ⊣ y = (x ⊣ y) ⊢ z,
y ⊣ x ⊣ z = (x ⊢ y) ⊣ z, y ⊣ z ⊣ x = x ⊢ (y ⊣ z)
according to (1). Hence, using left commutativity and right commutativity
of (D,⊣), we obtain (D1) and (D2). So, (D,⊣,⊢) is a doppelsemigroup.
It is abelian because (1) holds.
The remaining case is considered in a similar way.
Corollary 2.2. Let (D,⊣) be a left and right commutative semigroup,
and (D,⊢) the dual semigroup to (D,⊣). Then (D,⊣,⊢) is an abelian
doppelsemigroup. Conversely, any abelian doppelsemigroup (D,⊣,⊢) can
be constructed in this way.
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294 Abelian doppelsemigroups
Using Corollary 2.2 and the remark above, we conclude that the set
of all congruences on an abelian doppelsemigroup (D,⊣,⊢) coincides with
the set of all congruences on a left and right commutative semigroup
(D,⊣). So, the problem of the description of congruences on an abelian
doppelsemigroup is reduced to the description of congruences on a left
and right commutative semigroup.
Corollary 2.3. Let (D,⊢) be an arbitrary semigroup and (D,⊣) the dual
semigroup to (D,⊢). Then (D,⊣,⊢) is an abelian doppelsemigroup if and
only if (D,⊢) or (D,⊣) is a left and right commutative semigroup.
Corollary 2.4. Let (D,⊢) be a left and right commutative semigroup,
and (D,⊣) the dual semigroup to (D,⊢). Then (D,⊣,⊢) is an abelian
doppelsemigroup. Conversely, any abelian doppelsemigroup (D,⊣,⊢) can
be constructed in this way.
As usual, N denotes the set of all positive integers.
Lemma 2.5. In an abelian doppelsemigroup (D,⊣,⊢), for any n > 1 with
n ∈ N, and any xi ∈ D, 1 6 i 6 n+ 1, and ∗j ∈ {⊣,⊢}, 1 6 j 6 n,
x1 ∗1 x2 ∗2 · · · ∗n xn+1 = x1π ∗1 x2π ∗2 · · · ∗n x(n+1)π
= x1 ⊣ x2 ⊣ · · · ⊣ xn+1,
where π is a permutation of {1, 2, . . . , n+ 1}.
Proof. Lemma 3.1 from [23] states that in any doppelsemigroup the
product of three and more multipliers does not depend on parenthesizing.
The proof follows from Lemmas 3.1 [23], 2.1 and the identities (1)–(3).
Corollary 2.6. Every abelian doppelsemigroup is strong.
3. Examples of abelian doppelsemigroups
In this section, we give examples of abelian doppelsemigroups.
Let X be an arbitrary set such that |X| > 4 and a, b, x, y, 0 pairwise
distinct elements from X. Define binary operations ⊣ and ⊢ on X by
f ⊣ g =
a, f = x, g = y,
b, f = y, g = x,
0 otherwise,
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A. V. Zhuchok, K. Knauer 295
f ⊢ g =
a, f = y, g = x,
b, f = x, g = y,
0 otherwise
for all f, g ∈ X.
Proposition 3.1. (X,⊣,⊢) is an abelian doppelsemigroup.
Proof. It is immediate to check that (X,⊣,⊢) is an abelian doppelsemi-
group.
Let X be an arbitrary set, |X| > 1 and F ∗ [X] the free commutative
semigroup on X. Fix x, y ∈ X such that x 6= y. Define binary operations
⊣ and ⊢ on
K = (F ∗ [X] \X) ∪ {x, y} ∪ {(x, y)}
by
w ⊣ u =
{
(x, y), w = x, u = y,
wu otherwise,
w ⊢ u =
{
(x, y), w = y, u = x,
wu otherwise,
w ⊣ (x, y) = w ⊢ (x, y) = wxy,
(x, y) ⊣ w = (x, y) ⊢ w = xyw,
(x, y) ⊣ (x, y) = (x, y) ⊢ (x, y) = xyxy
for all w, u ∈ K\{(x, y)}.
Proposition 3.2. (K,⊣,⊢) is an abelian doppelsemigroup.
Proof. An immediate verification shows that four identities of a dop-
pelsemigroup hold concerning the operations ⊣ and ⊢ and thus, (K,⊣,⊢)
is a doppelsemigroup. It is clear that the operations ⊣ and ⊢ satisfy (1).
Let X1 and X2 be arbitrary nonempty subsets of N such that
X1 ∩X2 = ∅. Define binary operations ⊣ and ⊢ on N ∪ (X1 ×X2) by
m ⊣ k =
{
(m, k), m ∈ X1, k ∈ X2,
m+ k otherwise,
m ⊢ k =
{
(m, k), m ∈ X2, k ∈ X1,
m+ k otherwise,
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296 Abelian doppelsemigroups
m ⊣ (x, y) = m ⊢ (x, y) = m+ x+ y,
(x, y) ⊣m = (x, y) ⊢m = x+ y +m,
(x, y) ⊣ (a, b) = (x, y) ⊢ (a, b) = x+ y + a+ b
for all m, k ∈ N and (x, y), (a, b) ∈ X1 ×X2.
Proposition 3.3. (N ∪ (X1 ×X2),⊣,⊢) is an abelian doppelsemigroup.
Proof. The proof follows by a direct verification.
4. Free objects
In this section, we construct the free abelian doppelsemigroup of
arbitrary rank and consider separately free abelian doppelsemigroups of
rank 1. We also show that the automorphism group of the free abelian
doppelsemigroup is isomorphic to the symmetric group and present the
free left and right commutative semigroup.
Let X be an arbitrary nonempty set, F [X] the free semigroup on
X and F ∗ [X] the free commutative semigroup on X. The length of an
arbitrary word ω in the alphabet X will be denoted by lω. Consider the
sets
A1 = {w ∈ F [X] | lw ∈ {1, 2}} and A2 = {w ∈ F ∗[X] | lw > 2}.
Define binary operations ⊣ and ⊢ on A1 ∪A2 by
w ⊣ u = wu and w ⊢ u = uw
for all w, u ∈ A1∪A2. The algebra obtained in this way will be denoted by
FAD(X). Note that the semigroup (A1 ∪A2,⊢) is dual to the semigroup
(A1 ∪A2,⊣).
Theorem 4.1. FAD(X) is the free abelian doppelsemigroup.
Proof. For any w, u, ω ∈ FAD(X) and ∗, ◦ ∈ {⊣,⊢}, we have
(w ∗ u) ◦ ω = w ∗ (u ◦ ω) = wuω,
hence FAD(X) is a doppelsemigroup. Moreover, w ⊣ u = wu = u ⊢w and
so, FAD(X) is abelian.
Let us show that FAD(X) is free in the variety of abelian doppelsemi-
groups.
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A. V. Zhuchok, K. Knauer 297
Obviously, FAD(X) is generated by X. Let (V,⊣′,⊢′) be an arbitrary
abelian doppelsemigroup. Let β : X → V be an arbitrary map. Define a
map
φ : FAD(X) → (V,⊣′,⊢′)
by
(x1x2 . . . xn)φ = x1β ⊣′ x2β ⊣′ · · · ⊣′ xnβ, x1, x2, . . . , xn ∈ X.
According to Lemma 2.5 and (1) φ is well-defined. We will now show that
φ is a homomorphism by using Lemma 2.5 and (1) again.
Let y1y2 . . . ys ∈ FAD(X), where y1, y2, . . . , ys ∈ X. Consider the
following two cases. If n = s = 1, then
(x1 ⊢ y1)φ = (y1x1)φ = y1β ⊣′ x1β = x1β ⊢′ y1β = x1φ ⊢′ y1φ.
In the case n 6= 1 or s 6= 1 we get
((x1x2 . . . xn) ⊢ (y1y2 . . . ys))φ = (x1x2 . . . xny1y2 . . . ys)φ
= x1β ⊣′ x2β ⊣′ · · · ⊣′ xnβ ⊣′ y1β ⊣′ y2β ⊣′ · · · ⊣′ ysβ
= x1β ⊣′ x2β ⊣′ · · · ⊣′ xnβ ⊢′ y1β ⊣′ y2β ⊣′ · · · ⊣′ ysβ
= (x1x2 . . . xn)φ ⊢′ (y1y2 . . . ys)φ.
A direct verification shows that
((x1x2 . . . xn) ⊣ (y1y2 . . . ys))φ = (x1x2 . . . xn)φ ⊣′ (y1y2 . . . ys)φ
for all x1x2 . . . xn, y1y2 . . . ys ∈ FAD(X). So, φ is a homomorphism.
Clearly, xφ = xβ for all x ∈ X. Since X generates FAD(X), the
uniqueness of such homomorphism φ is obvious. Thus, FAD(X) is free in
the variety of abelian doppelsemigroups.
Corollary 4.2. The operations of a singly generated free abelian dop-
pelsemigroup coincide and it is the additive semigroup of positive integers.
Corollary 4.3. (A1 ∪A2,⊣) is the free left and right commutative semi-
group.
The free abelian doppelsemigroup FAD(X) is determined uniquely
up to isomorphism by cardinality of the set X. Hence the automorphism
group of FAD(X) is isomorphic to the symmetric group on X.
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298 Abelian doppelsemigroups
5. The least abelian congruence on the free
doppelsemigroup
In this section, we characterize the least abelian congruence on the
free doppelsemigroup.
The free doppelsemigroup is given in [23]. Recall this construction.
We will use notations from Section 4.
Let T be 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)
and
(w1, u1) ⊢ (w2, u2) = (w1w2, u1bu2)
for all (w1, u1), (w2, u2) ∈ F . The algebra (F,⊣,⊢) is denoted by FDS(X).
Theorem 5.1. ([23], 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 .
Theorem 5.2. LetFDS(X) be the free doppelsemigroup, (x1x2 . . . xn, u)∈
FDS(X), x1, x2, . . . , xn ∈ X and FAD(X) the free abelian doppelsemi-
group. Then the map
µ : FDS(X) → FAD(X),
(x1x2 . . . xn, u) 7→ (x1x2 . . . xn, u)µ =
{
x2x1, n = 2, u = b,
x1x2 . . . xn otherwise
is an epimorphism inducing the least abelian congruence on FDS(X).
Proof. For arbitrary elements (x1x2 . . . xn, u), (y1y2 . . . ys, t) ∈ FDS(X),
where x1, x2, . . . , xn, y1, y2, . . . , ys ∈ X, we consider the following two
cases.
Case 1: n = s = 1. Then
((x1, θ) ⊣ (y1, θ))µ = (x1y1, a)µ = x1y1 = x1 ⊣ y1 = (x1, θ)µ ⊣ (y1, θ)µ,
((x1, θ) ⊢ (y1, θ))µ = (x1y1, b)µ = y1x1 = x1 ⊢ y1 = (x1, θ)µ ⊢ (y1, θ)µ.
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A. V. Zhuchok, K. Knauer 299
Case 2: n 6= 1 or s 6= 1. Then
((x1x2 . . . xn, u) ⊣ (y1y2 . . . ys, t))µ = (x1x2 . . . xny1y2 . . . ys, uat)µ
= x1x2 . . . xny1y2 . . . ys = (x1x2 . . . xn, u)µ ⊣ (y1y2 . . . ys, t)µ,
((x1x2 . . . xn, u) ⊢ (y1y2 . . . ys, t))µ = (x1x2 . . . xny1y2 . . . ys, ubt)µ
= x1x2 . . . xny1y2 . . . ys = (x1x2 . . . xn, u)µ ⊢ (y1y2 . . . ys, t)µ.
Thus, µ is a homomorphism.
For arbitrary a1a2 . . . an ∈ FAD(X) there exists (a1a2 . . . an, a
n−1) ∈
FDS(X) such that
(a1a2 . . . an, a
n−1)µ = a1a2 . . . an,
where a0 = θ. So, µ is surjective. By Theorem 4.1, FAD(X) is the free
abelian doppelsemigroup. Then ∆µ is the least abelian congruence on
FDS(X).
6. Some properties
In this section, we show that the variety of abelian doppelsemigroups
is a subclass of the variety of abelian dimonoids, give conditions under
which the operations of an abelian doppelsemigroup coincide and establish
that the varieties of commutative doppelsemigroups and of commutative
bisemigroups coincide.
A dimonoid (D,⊣,⊢) is a left zero and right zero dimonoid [24] provided
that (D,⊣) is a left zero semigroup, and (D,⊢) is a right zero semigroup.
Proposition 6.1. Every abelian doppelsemigroup is an abelian dimonoid.
The converse statement is not true in general.
Proof. We need to prove that in an abelian doppelsemigrooup (D,⊣,⊢)
the identities (D3) and (D4) hold. For all x, y, z ∈ D, we have
(x ⊣ y) ⊣ z = x ⊣ (y ⊢ z)
and
x ⊢ (y ⊢ z) = (x ⊣ y) ⊢ z
by Lemma 2.5.
Consider a left zero and right zero dimonoid (D,⊣,⊢). Obviously, it is
abelian. Moreover, if |D| > 1, x, y, z ∈ D and x 6= z, then
(x ⊣ y) ⊢ z = z 6= x = x ⊣ (y ⊢ z).
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300 Abelian doppelsemigroups
Corollary 6.2. The variety of abelian doppelsemigroups is a subclass of
the variety of abelian dimonoids.
A semigroup S is called separative if for any s, t ∈ S, s2 = st = t2
implies s = t. A semigroup S is called a left cancellative semigroup if for
any a, b, c ∈ S, ca = cb implies a = b. Right cancellative semigroups are
defined dually. A semigroup S is called a two-sided cancellative semigroup
if it is a left cancellative semigroup and a right cancellative semigroup. A
semigroup S is called globally idempotent if S2 = S. A semigroup, which
satisfies two quasiidentities
x2 = xy, y2 = yx =⇒ x = y,
x2 = yx, y2 = xy =⇒ x = y,
is called weakly cancellative.
The following statement establishes necessary and sufficient conditions
under which the operations of two dual semigroups coincide.
Proposition 6.3. Let (D,⊣) be an arbitrary semigroup and (D,⊢) the
dual semigroup to (D,⊣). The operations of (D,⊣,⊢) coincide if and only
if ⊣ is commutative.
Proof. Let ⊣ = ⊢. Then, using (1), we get x ⊣ y = y ⊢ x = y ⊣ x for all
x, y ∈ D and so, ⊣ is commutative.
Conversely, let ⊣ be commutative. Since (D,⊢) is the dual semigroup
to (D,⊣), we have x⊣ y = y ⊣ x = x⊢ y for all x, y ∈ D. Thus, ⊣ = ⊢.
Corollary 6.4. The operations of an abelian doppelsemigroup (D,⊣,⊢)
coincide if (D,⊣) or (D,⊢) is
(i) a commutative semigroup,
(ii) an idempotent semigroup,
(iii) a separative semigroup,
(iv) a globally idempotent semigroup,
(v) a left (right, two-sided) cancellative semigroup,
(vi) a weakly cancellative semigroup, or
(vii) a monoid.
Proof. Let (D,⊣,⊢) be an abelian doppelsemigroup. We prove all cases
for (D,⊣). The remaining cases are considered in a similar way.
(i) Obviously, in (D,⊣,⊢), the operation ⊣ is commutative if and only
if ⊢ is commutative. Hence, by Proposition 6.3, ⊣ = ⊢.
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A. V. Zhuchok, K. Knauer 301
(ii) By Lemma 2.5,
x ⊣ y ⊣ z = y ⊣ x ⊣ z (4)
for all x, y, z ∈ D. If y = z, then, using the idempotent property of ⊣ and
Lemma 2.5, we obtain x ⊣ y = y ⊣ x for all x, y ∈ D. Hence, according to
Proposition 6.3, ⊣ = ⊢.
(iii) Let x, y ∈ D. Assume that a = x⊣ y, b = y ⊣ x. Using Lemma 2.5,
we have
a2 = (x ⊣ y) ⊣ (x ⊣ y) = (x ⊣ y)2, (5)
a ⊣ b = (x ⊣ y) ⊣ (y ⊣ x) = (x ⊣ y)2 (6)
and
b2 = (y ⊣ x) ⊣ (y ⊣ x) = (x ⊣ y)2. (7)
Since the semigroup (D,⊣) is separative, a2 = a ⊣ b = b2 implies a = b.
Thus, ⊣ is commutative and by Proposition 6.3, ⊣ = ⊢.
(iv) Let x, y ∈ D. By global idempotentity of (D,⊣), y = y1 ⊣ y2 for
some y1, y2 ∈ D. Then
x ⊣ y = x ⊣ (y1 ⊣ y2) = (y1 ⊣ y2) ⊣ x = y ⊣ x
according to Lemma 2.5. By virtue of Proposition 6.3, ⊣ = ⊢.
(v) For all x, y, z ∈ D, using Lemma 2.5, we have
z ⊣ x ⊣ y = z ⊣ y ⊣ x.
Hence, by left cancellativity of (D,⊣), we obtain x ⊣ y = y ⊣ x for all
x, y ∈ D and so, ⊣ = ⊢ by Proposition 6.3.
The case of right cancellativity is proved in a similar way. The proof
for the case of two-sided cancellativity follows from the above.
(vi) For arbitrary elements x, y ∈ D assume that a = x⊣y and b = y⊣x.
By virtue of Lemma 2.5, we obtain (5), (6), (7) and
b ⊣ a = (y ⊣ x) ⊣ (x ⊣ y) = (x ⊣ y)2.
Due to weak cancellativity of (D,⊣), a2 = a ⊣ b, b2 = b ⊣ a implies a = b.
Thus, ⊣ is commutative. Then, by Proposition 6.3, ⊣ = ⊢.
(vii) Let (D,⊣) be a monoid with the identity element e. For all
x, y, z ∈ D, we have (4) according to Lemma 2.5. If we substitute z = e
into (4), we obtain x ⊣ y = y ⊣ x for all x, y ∈ D. From here ⊣ = ⊢ by
Proposition 6.3.
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302 Abelian doppelsemigroups
At the end of the paper we establish a relation between doppelsemi-
groups and bisemigroups [16].
Recall that a nonempty set B with the operations ⊣ and ⊢ is called a
bisemigroup if (B,⊣) and (B,⊢) are semigroups and (D2) holds. Restric-
tive bisemigroups were studied by Schein (see, e.g., [17]) and they have
applications in the theory of binary relations. A bisemigroup (B,⊣,⊢) will
be called commutative if the operations ⊣ and ⊢ are commutative.
It is not hard to prove the following statement.
Proposition 6.5. The varieties of commutative doppelsemigroups and
of commutative bisemigroups coincide.
Observe that doppelalgebras are linear analogs of doppelsemigroups,
therefore all results obtained for doppelsemigroups hold for doppelalgebras.
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Contact information
Anatolii V.
Zhuchok
Department of Algebra and System Analysis,
Luhansk Taras Shevchenko National University,
Gogol square, 1, Starobilsk, 92703, Ukraine
E-Mail(s): zhuchok.av@gmail.com
Kolja Knauer Laboratory of Computer Science and Systems,
Aix-Marseille University, LIS UMR 7020, Case
Courrier 901, 163, avenue de Luminy 13288,
Marseille Cedex 9, France
E-Mail(s): kolja.knauer@lis-lab.fr
Received by the editors: 07.09.2018.
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