Interassociativity and three-element doppelsemigroups
In the paper we characterize all interassociates of some non-inverse semigroups and describe up to isomorphism all three-element (strong) doppelsemigroups and their automorphism groups. We prove that there exist 75 pairwise non-isomorphic three-element doppelsemigroups among which 41 doppelsemigroup...
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| Cite this: | Interassociativity and three-element doppelsemigroups / V. Gavrylkiv, D. Rendziak // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 224–247. — Бібліогр.: 20 назв. — англ. |
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Gavrylkiv, V. Rendziak, D. 2023-03-02T19:24:26Z 2023-03-02T19:24:26Z 2019 Interassociativity and three-element doppelsemigroups / V. Gavrylkiv, D. Rendziak // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 224–247. — Бібліогр.: 20 назв. — англ. 1726-3255 2010 MSC: 08B20, 20M10, 20M50, 17A30. https://nasplib.isofts.kiev.ua/handle/123456789/188491 In the paper we characterize all interassociates of some non-inverse semigroups and describe up to isomorphism all three-element (strong) doppelsemigroups and their automorphism groups. We prove that there exist 75 pairwise non-isomorphic three-element doppelsemigroups among which 41 doppelsemigroups are commutative. Non-commutative doppelsemigroups are divided into 17 pairs of dual doppelsemigroups. Also up to isomorphism there are 65 strong doppelsemigroups of order 3, and all non-strong doppelsemigroups are not commutative. The authors would like to express their sincere thanks to the anonymous referee for a very careful reading of the paper and for all its insightful comments and valuable suggestions, which improve considerably the presentation of this paper. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Interassociativity and three-element doppelsemigroups Article published earlier |
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Interassociativity and three-element doppelsemigroups |
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Interassociativity and three-element doppelsemigroups Gavrylkiv, V. Rendziak, D. |
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Interassociativity and three-element doppelsemigroups |
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Interassociativity and three-element doppelsemigroups |
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Interassociativity and three-element doppelsemigroups |
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Interassociativity and three-element doppelsemigroups |
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interassociativity and three-element doppelsemigroups |
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Gavrylkiv, V. Rendziak, D. |
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In the paper we characterize all interassociates of some non-inverse semigroups and describe up to isomorphism all three-element (strong) doppelsemigroups and their automorphism groups. We prove that there exist 75 pairwise non-isomorphic three-element doppelsemigroups among which 41 doppelsemigroups are commutative. Non-commutative doppelsemigroups are divided into 17 pairs of dual doppelsemigroups. Also up to isomorphism there are 65 strong doppelsemigroups of order 3, and all non-strong doppelsemigroups are not commutative.
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Interassociativity and three-element doppelsemigroups / V. Gavrylkiv, D. Rendziak // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 224–247. — Бібліогр.: 20 назв. — англ. |
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AT gavrylkivv interassociativityandthreeelementdoppelsemigroups AT rendziakd interassociativityandthreeelementdoppelsemigroups |
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2025-11-24T05:05:37Z |
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2025-11-24T05:05:37Z |
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1850842320406052864 |
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“adm-n4” — 2020/1/24 — 13:02 — page 224 — #74
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 28 (2019). Number 2, pp. 224–247
c© Journal “Algebra and Discrete Mathematics”
Interassociativity and three-element
doppelsemigroups
Volodymyr Gavrylkiv and Diana Rendziak
Communicated by A. V. Zhuchok
Abstract. In the paper we characterize all interassociates
of some non-inverse semigroups and describe up to isomorphism
all three-element (strong) doppelsemigroups and their automor-
phism groups. We prove that there exist 75 pairwise non-isomorphic
three-element doppelsemigroups among which 41 doppelsemigroups
are commutative. Non-commutative doppelsemigroups are divided
into 17 pairs of dual doppelsemigroups. Also up to isomorphism
there are 65 strong doppelsemigroups of order 3, and all non-strong
doppelsemigroups are not commutative.
Introduction
Given a semigroup (S,⊣), consider a semigroup (S,⊢) defined on
the same set. We say that (S,⊢) is an interassociate of (S,⊣) provided
(x ⊣ y) ⊢ z = x ⊣ (y ⊢ z) and (x ⊢ y) ⊣ z = x ⊢ (y ⊣ z) for all x, y, z ∈ S. In
1971, Zupnik [20] coined the term interassociativity in a general groupoid
setting. However, he required only one of the two defining equations to
hold. The present concept of interassociativity for semigroups originated in
1986 in Drouzy [4], where it is noted that every group is isomorphic to each
of its interassociates. In 1983, Gould and Richardson [8] introduced strong
interassociativity, defined by the above equations along with x ⊣ (y ⊢ z) =
x ⊢ (y ⊣ z). J. B. Hickey in 1983 [9] and 1986 [10] dealt with the special
case of interassociativity in which the operation ⊢ is defined by specifying
2010 MSC: 08B20, 20M10, 20M50, 17A30.
Key words and phrases: semigroup, interassociativity, doppelsemigroup, strong
doppelsemigroup.
“adm-n4” — 2020/1/24 — 13:02 — page 225 — #75
V. Gavrylkiv, D. Rendziak 225
a ∈ S and stipulating that x ⊢ y = x ⊣ a ⊣ y for all x, y ∈ S. Clearly
(S,⊢), which Hickey calls a variant of (S,⊣), is a semigroup that is an
interassociate of (S,⊣). It is easy to show that if (S,⊣) is a monoid, every
interassociate (S,⊢) must satisfy the condition x ⊢ y = x ⊣ a ⊣ y for some
fixed element a ∈ S and for all x, y ∈ S, that is (S,⊢) is a variant of (S,⊣).
Methods of constructing interassociates were developed, for semigroups
in general and for specific classes of semigroups, in 1997 by Boyd, Gould
and Nelson [1]. The description of all interassociates of finite monogenic
semigroups was presented by Gould, Linton and Nelson in 2004, see [7].
A doppelsemigroup is an algebraic structure (D,⊣,⊢) consisting of
a non-empty set D equipped with two associative binary operations ⊣ and
⊢ satisfying the following axioms:
(x ⊣ y) ⊢ z = x ⊣ (y ⊢ z), (D1)
(x ⊢ y) ⊣ z = x ⊢ (y ⊣ z). (D2)
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.
A doppelsemigroup (D,⊣,⊢) is called commutative [13] if both semigroups
(D,⊣) and (D,⊢) are commutative. A doppelsemigroup (D,⊣,⊢) is said
to be strong [15] if it satisfies the axiom x ⊣ (y ⊢ z) = x ⊢ (y ⊣ z).
Many classes of doppelsemigroups were studied by A. Zhuchok and
Y. Zhuchok. The free product of doppelsemigroups, the free (strong) dop-
pelsemigroup, the free commutative (strong) doppelsemigroup, the free
n-nilpotent (strong) doppelsemigroup and the free rectangular doppelsemi-
group were constructed in [13,15,19]. Relatively free doppelsemigroups
were studied in [17]. The free n-dinilpotent (strong) doppelsemigroup
was constructed in [12, 15]. In [14] A. Zhuchok described the free left
n-dinilpotent doppelsemigroup. Representations of ordered doppelsemi-
groups by binary relations were studied by Y. Zhuchok and J. Koppitz [18].
Until now, the task of describing all pairwise non-isomorphic (strong)
doppelsemigroups of order 3 has not been solved. The goal of the present
work is to characterize all interassociates of some non-inverse semigroups,
and use these characterizations in describing up to isomorphism all three-
element (strong) doppelsemigroups and their automorphism groups.
1. Preliminaries
A semigroup S is called an inflation of its subsemigroup T (see [3],
Section 3.2) provided that there is an surjective map r : S → T such that
“adm-n4” — 2020/1/24 — 13:02 — page 226 — #76
226 Interassociativity and 3-element doppelsemigroups
r2 = r and r(a)r(b) = ab for all a, b ∈ S. In the described situation S is
often referred to as an inflation of T with an associated map r (or just
with a map r). It is immediate that if S is an inflation of T then T is
a retract of S (that is the image under a retraction r in the sense that
r(a) = a for all a ∈ T ) and S2 ⊂ T .
A semigroup S is called monogenic if it is generated by some element
a ∈ S in the sense that S = {an}n∈N. If a monogenic semigroup is infinite
then it is isomorphic to the additive semigroup N of positive integer
numbers. A finite monogenic semigroup S = 〈a〉 also has simple structure,
see [11]. There are positive integer numbers r and m called the index and
the period of S such that
• S = {a, a2, . . . , ar+m−1} and r +m− 1 = |S|;
• ar+m = ar;
• Cm := {ar, ar+1, . . . , ar+m−1} is a cyclic and maximal subgroup of
S with the neutral element e = an ∈ Cm and generator an+1, where
n ∈ (m · N) ∩ {r, . . . , r +m− 1}.
We denote by Mr,m a finite monogenic semigroup of index r and
period m.
Recall that an isomorphism between (S, ∗) and (S′, ◦) is a bijective
function ψ : S → S′ such that ψ(x ∗ y) = ψ(x) ◦ ψ(y) for all x, y ∈ S. If
there exists an isomorphism between (S, ∗) and (S′, ◦) then (S, ∗) and
(S′, ◦) are said to be isomorphic, denoted (S, ∗) ∼= (S′, ◦). An isomorphism
between (S, ∗) and (S, ∗) is called an automorphism of a semigroup (S, ∗).
By Aut(S, ∗) we denote the automorphism group of a semigroup (S, ∗).
An element e of a semigroup (S, ∗) is called an idempotent if e ∗ e = e.
The semigroup is a band, if all its elements are idempotents. Commutative
bands are called semilattices. By Ln we denote the linear semilattice
{0, 1, . . . , n− 1} of order n, endowed with the operation of minimum.
If (S, ∗) is a semigroup then the semigroup (S, ∗d) with operation
x∗dy = y ∗ x is called dual to (S, ∗).
A non-empty subset I of a semigroup (S, ∗) is called an ideal if I ∗S ∪
S ∗ I ⊂ I. An element z of a semigroup S is called a zero (resp. a left zero,
a right zero) in S if a ∗ z = z ∗ a = z (resp. z ∗ a = z, a ∗ z = z) for any
a ∈ S. If (D,⊣,⊢) is a doppelsemigroup and z ∈ D is a zero (resp. a left
zero, a right zero) of a semigroup (D,⊣) then (D1) and (D2) imply that
z is a zero (resp. a left zero, a right zero) of a semigroup (D,⊢), and vice
versa. Thus, any interassociate of a semigroup with zero is a semigroup
with zero as well.
A semigroup (S, ∗) is called a null semigroup if there exists an element
z ∈ S such that x ∗ y = z for any x, y ∈ S. In this case z is a zero of S.
“adm-n4” — 2020/1/24 — 13:02 — page 227 — #77
V. Gavrylkiv, D. Rendziak 227
All null semigroups on the same set are isomorphic. By OX we denote
a null semigroup on a set X. If X is finite of cardinality |X| = n then
instead of OX we use On. It is easy to see that a null semigroup is a strong
interassociate of each semigroup with the same zero.
Let X be a set, z ∈ X and A ⊂ X \ {z}. Define the binary operation
∗ on X in the following way:
x ∗ y =
{
x if y = x ∈ A
z otherwise.
It is easy to check that a set X endowed with the operation ∗ is
a semigroup with zero z, and we denote this semigroup by OAX . If A =
X \ {z} then OAX is a semilattice. In the case A = ∅, OAX coincides with
a null semigroup with zero z. The semigroups OAX and OBY are isomorphic
if and only if |X| = |Y | and |A| = |B|. If X is a finite set of cardinality
|X| = n and |A| = m then we use Omn instead of OAX .
Let (S, ∗) be a semigroup and e /∈ S. The binary operation ∗ defined
on S can be extended to S∪{e} putting e∗s = s∗e = s for all s ∈ S∪{e}.
The notation (S, ∗)+1 denotes a monoid (S ∪ {e}, ∗) obtained from (S, ∗)
by adjoining the extra identity e (regardless of whether (S, ∗) is or is not
a monoid).
Let (S, ∗) be a semigroup and 0 /∈ S. The binary operation ∗ defined
on S can be extended to S∪{0} putting 0∗s = s∗0 = 0 for all s ∈ S∪{0}.
The notation (S, ∗)+0 denotes a semigroup (S ∪ {0}, ∗) obtained from
(S, ∗) by adjoining the extra zero 0 (regardless of whether (S, ∗) has or
has not the zero).
Let (M, ∗) be a monoid with identity e, and 1̃ /∈ M . The binary
operation ∗ defined on M can be extended to M ∪ {1̃} putting 1̃ ∗ 1̃ = e
and 1̃ ∗m = m ∗ 1̃ = m for all m ∈M . The notation (M, ∗)1̃ denotes the
semigroup obtained from (M, ∗) by adjoining an extra element 1̃. Note
that (M, ∗)1̃ is not a monoid and (M, ∗)1̃ is an inflation of a monoid (M, ∗).
Let (D,⊣,⊢) be a doppelsemigroup and 0 /∈ D. The binary operations
defined on D can be extended to D ∪ {0} putting 0 ⊣ d = d ⊣ 0 =
0 = 0 ⊢ d = d ⊢ 0 for all d ∈ D ∪ {0}. The notation (D,⊣,⊢)+0 denotes
a doppelsemigroup (D∪{0},⊣,⊢) obtained from (D,⊣,⊢) by adjoining the
extra zero 0. If (D,⊣,⊢) is a strong doppelsemigroup then (D,⊣,⊢)+0 is
a strong doppelsemigroup as well. It is easy to see that Aut((D,⊣,⊢)+0) ∼=
Aut(D,⊣,⊢).
A semigroup (S, ∗) is said to be a left (right) zero semigroup if a∗b = a
(a ∗ b = b) for any a, b ∈ S. By LOX and ROX we denote a left zero
“adm-n4” — 2020/1/24 — 13:02 — page 228 — #78
228 Interassociativity and 3-element doppelsemigroups
semigroup and a right zero semigroup on a set X, respectively. It is easy
to see that the semigroups LOX and ROX are dual. If X is finite of
cardinality |X| = n then instead of LOX and ROX we use LOn and ROn,
respectively.
Let X be a set, A ⊂ X and 0 /∈ X. Define the binary operation ∗ on
X0 = X ∪ {0} in the following way:
x ∗ y =
{
x if y ∈ A
0 if y ∈ X0 \A.
It is easy to check that a set X0 endowed with the operation ∗ is
a semigroup with zero 0, and we denote this semigroup by LO∼0A←X . If
A = X then LO∼0A←X coincides with LO+0
X . In the case A = ∅, LO∼0A←X
coincides with a null semigroup OX0 with zero 0. The semigroups LO∼0A←X
and LO∼0B←Y are isomorphic if and only if |X| = |Y | and |A| = |B|. If X
is a finite set of cardinality |X| = n and |A| = m then we use LO∼0m←n
instead of LO∼0A←X .
Let X be a set, A ⊂ X and 0 /∈ X. Define the binary operation ∗ on
X0 = X ∪ {0} in the following way:
x ∗ y =
{
y if x ∈ A
0 if x ∈ X0 \A.
It is easy to check that a set X0 endowed with the operation ∗ is
a semigroup with zero 0, and we denote this semigroup by RO∼0A←X . If
A = X then RO∼0A←X coincides with RO+0
X . In the case A = ∅, RO∼0A←X
coincides with a null semigroup on X0 with zero 0. Semigroups RO∼0A←X
and RO∼0B←Y are isomorphic if and only if |X| = |Y | and |A| = |B|. If X
is a finite set of cardinality |X| = n and |A| = m then we use RO∼0m←n
instead of RO∼0A←X .
It is easy to see that the semigroups LO∼0A←X and RO∼0A←X are dual.
Let a and c be different elements of a set X. Define the associative
binary operation ⊣ac on X in the following way:
x ⊣ac y =
x if x 6= c
a if x = c and y 6= c
c if x = y = c.
It follows that (X,⊣ac ) is a non-commutative band in which all elements
z 6= c are left zeros.
“adm-n4” — 2020/1/24 — 13:02 — page 229 — #79
V. Gavrylkiv, D. Rendziak 229
It is not difficult to check that for any different b, d ∈ X, the semi-
groups (X,⊣ac ) and (X,⊣bd) are isomorphic. We denote by LOBX a model
semigroup of the class of semigroups isomorphic to (X,⊣ac ). If X is a finite
set of cardinality |X| = n then we use LOBn instead of LOBX .
The semigroup ROBX is defined dually.
Let a and c be different elements of a set X. Define the associative
binary operation ⊢ac on X in the following way:
x ⊢ac y =
{
x if x 6= c
a if x = c.
It follows that (X,⊢ac ) is a non-commutative non-regular semigroup in
which all elements z 6= c are left zeros.
It is not difficult to check that for any b 6= c, the semigroups (X,⊢ac )
and (X,⊢bc) are isomorphic. We denote by LOX\{c}←X a model semigroup
of the class of semigroups isomorphic to (X,⊢ac ). If X is a finite set of
cardinality |X| = n then we use LO(n−1)←n instead of LOX\{c}←X .
Dually we define the semigroups ROX\{c}←X and RO(n−1)←n.
A transformation l : S → S of a semigroup (S, ∗) is called a left
translation if l(x ∗ y) = l(x) ∗ y for all x, y ∈ S. By Corollary 2.2. from [1]
for any semigroup (S, ∗) and for any its left translation l, the semigroup
(S, ∗l), where x ∗l y = x ∗ l(y), is an interassociate of (S, ∗). Thus, (S, ∗, ∗l)
is a doppelsemigroup for any left translation l : S → S.
The following lemma was proved in [1].
Lemma 1.1. Let (S, ∗) be an inflation of an inverse Clifford semigroup
(A, ∗) and let r : S → A denote the associated retraction. If (S, ◦) is
a semigroup that is an interassociate of (S, ∗) then A is an ideal of (S, ◦)
and (A, ◦) = (A, ∗l) for some left translation l of (A, ∗). Moreover, r is
a homomorphism of (S, ◦) onto (A, ◦).
2. Isomorphisms of doppelsemigroups
A bijective map ψ : D1 → D2 is called an isomorphism of doppelsemi-
groups (D1,⊣1,⊢1) and (D2,⊣2,⊢2) if
ψ(a ⊣1 b) = ψ(a) ⊣2 ψ(b) and ψ(a ⊢1 b) = ψ(a) ⊢2 ψ(b)
for all a, b ∈ D1.
If there exists an isomorphism between the doppelsemigroups
(D1,⊣1,⊢1) and (D2,⊣2,⊢2) then (D1,⊣1,⊢1) and (D2,⊣2,⊢2) are said
“adm-n4” — 2020/1/24 — 13:02 — page 230 — #80
230 Interassociativity and 3-element doppelsemigroups
to be isomorphic, denoted (D1,⊣1,⊢1) ∼= (D2,⊣2,⊢2). An isomorphism
ψ : D → D is called an automorphism of a doppelsemigroup (D,⊣,⊢). By
Aut(D,⊣,⊢) we denote the automorphism group of a doppelsemigroup
(D,⊣,⊢).
Proposition 2.1. Let (D1,⊣1,⊢1) and (D2,⊣2,⊢2) be doppelsemigroups
such that (D1,⊣1) and (D2,⊣2) are null semigroups. If the semigroups
(D1,⊢1) and (D2,⊢2) are isomorphic then the doppelsemigroups (D1,⊣1,⊢1)
and (D2,⊣2,⊢2) are isomorphic as well.
Proof. Let z1 and z2 be zeros of null semigroups (D1,⊣1) and (D2,⊣2),
respectively. Then z1 and z2 are zeros of the semigroups (D1,⊢1) and
(D2,⊢2), respectively. Let ψ : D1 → D2 is an isomorphism of the semi-
groups (D1,⊢1) and (D2,⊢2). Since zeros are preserved by isomorphisms
of semigroups, ψ(z1) = z2. Taking into account that |D1| = |D2| and any
map between two null semigroups of the same order that preserves zeros is
a isomorphism of these semigroups, we conclude that ψ : D1 → D2 is an
isomorphism of the doppelsemigroups (D1,⊣1,⊢1) and (D2,⊣2,⊢2).
Proposition 2.2. Let (D1,⊣1,⊢1) and (D2,⊣2,⊢2) be doppelsemigroups,
and (D1,⊢) ∼= (D1,⊢1) implies ⊢ = ⊢1 for any interassociate
(D1,⊢) of (D1,⊣1). If (D2,⊣2) ∼= (D1,⊣1) and (D2,⊢2) ∼= (D1,⊢1) then
(D2,⊣2,⊢2) ∼= (D1,⊣1,⊢1).
Proof. Let ψ : D2 → D1 be an isomorphism of semigroups (D2,⊣2) and
(D1,⊣1). For any a, b ∈ D1 define the operation ⊢ψ on D1 in the following
way:
a ⊢ψ b = ψ(ψ−1(a) ⊢2 ψ
−1(b)).
It follows that ψ : D2 → D1 is an isomorphism from (D2,⊣2,⊢2)
to (D1,⊣1,⊢ψ), and thus (D1,⊣1,⊢ψ) is a doppelsemigroup as
an isomorphic image of the doppelsemigroup (D2,⊣2,⊢2). Taking
into account that (D1,⊢ψ) is an interassociate of (D1,⊣1) and
(D1,⊢ψ) ∼= (D2,⊢2) ∼= (D1,⊢1), we conclude that ⊢ψ = ⊢1. Therefore,
(D2,⊣2,⊢2) ∼= (D1,⊣1,⊢1).
Proposition 2.3. If (D,⊣,⊢) is a doppelsemigroup such that (D,⊣) is
a null semigroup then Aut(D,⊣,⊢) = Aut(D,⊢).
Proof. Let z be a zero of a null semigroup (D,⊣). Then z is a zero of
(D,⊢). If ψ : D → D is an automorphism of (D,⊢) then ψ(z) = z. Using
the similar arguments as in the proof of Proposition 2.1, we conclude that
ψ : D → D is an automorphism of (D,⊣). It follows that ψ ∈ Aut(D,⊣,⊢).
Therefore, Aut(D,⊣,⊢) = Aut(D,⊢).
“adm-n4” — 2020/1/24 — 13:02 — page 231 — #81
V. Gavrylkiv, D. Rendziak 231
Using the fact that all bijections of a left (right) zero semigroup are its
automorphisms and the similar arguments as in the proof of Proposition 2.3,
one can prove the following proposition.
Proposition 2.4. If (D,⊣,⊢) is a doppelsemigroup such that the semigroup
(D,⊣) is isomorphic to LO+0
X or RO+0
X then Aut(D,⊣,⊢) = Aut(D,⊢).
3. Interassociates of some non-inverse semigroups
In this section we characterize all interassociates of some non-inverse
semigroups which we shall use in section 4 for describing all three-element
(strong) (commutative) doppelsemigroups up to isomorphism.
In the following Propositions 3.1 and 3.2 we use Lemma 1.1 to recognize
all (strong) interassociates of the semigroups O+0
X and OAX .
Given a semigroup (S, ·), let Int(S, ·) denote the set of all semigroups
that are interassociates of (S, ·).
Proposition 3.1. Let O+0
X be a semigroup obtained from a null semigroup
OX = (X,⊣) with zero z by adjoining an extra zero 0 /∈ X. The set
Int(O+0
X ) consists of a null semigroup OX∪{0} with zero 0 and semigroups
(X,⊢)+0 for all semigroups (X,⊢) with zero z. All interassociates of O+0
X
are strong.
Proof. The semigroup O+0
X is an inflation of its subsemilattice A = {0, z}
with the associated retraction r : O+0
X → A,
r(x) =
{
0, x = 0,
z, x ∈ X.
Let l : A → A be a left translation of the semilattice (A,⊣). Then
l(0) = l(0 ⊣ 0) = l(0) ⊣ 0 = 0. So, there are two left translations of A:
l1(x) = 0 and l2(x) = x for all x ∈ A. Let (X0,⊢) be any interassociate
of O+0
X , where X0 = X ∪ {0}. By Lemma 1.1, A is an ideal of (X0,⊢), r
is a homomorphism from (X0,⊢) onto (A,⊢), and the semigroup (A,⊢)
is equal to (A,⊣l1), where x ⊢ y = x ⊣l1 y = x ⊣ l1(y) = 0 for all x, y ∈ A,
or (A,⊢) is equal to (A,⊣l2), where x ⊢ y = x ⊣l2 y = x ⊣ l2(y) = x ⊣ y
for all x, y ∈ A. It follows that (A,⊢) is a null semigroup with zero 0 or
(A,⊢) = (A,⊣).
If (A,⊢) is a null semigroup then r(x⊢ y) = r(x)⊢ r(y) = 0. Therefore,
the definition of r implies x ⊢ y = 0 for all x, y ∈ X0. Consequently, in
this case (X0,⊢) is a null semigroup with zero 0.
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232 Interassociativity and 3-element doppelsemigroups
Let (A,⊢) = (A,⊣). Taking into account that r is a homomorphic
retraction and A ∋ 0 is an ideal, we conclude that 0 ⊢ x = r(0 ⊢ x) =
r(0)⊢ r(x) = 0⊣ r(x) = 0 and x⊢0 = r(x⊢0) = r(x)⊢ r(0) = r(x)⊣0 = 0
for all x ∈ X0. If x, y ∈ X then r(x ⊢ y) = r(x) ⊢ r(y) = z ⊢ z = z ⊣ z = z.
Thus, the definition of r implies x ⊢ y ∈ X for all x, y ∈ X. Consequently,
(X,⊢) is an interassociate of a null semigroup (X,⊣) with zero z. It follows
that (X,⊢) is an arbitrary semigroup with zero z, and (X0,⊢) = (X,⊢)+0.
To show that all interassociates of O+0
X are strong, it is sufficient to
use the following two facts:
• if (X,⊢) is a strong interassociate of (X,⊣) then (X,⊢)+0 is a strong
interassociate of (X,⊣)+0;
• all interassociates of a null semigroup are strong.
Proposition 3.2. A semigroup (X,⊢) with zero z is an interassociate of
OAX together with the operation ⊣ if and only if the following conditions
hold:
1) (A0,⊢) coincides with OBA0 for some B ⊂ A, where A0 = A ∪ {z};
2) A ⊢ (X \A) = (X \A) ⊢A = {z};
3) X \A is a subsemigroup with zero z of (X,⊢).
All interassociates of OAX are strong.
Proof. Note that OAX is an inflation of its subsemilattice A0 with the
associated retraction r : OAX → A0,
r(x) =
{
x, x ∈ A,
z, x /∈ A.
Let l : A0 → A0 be a left translation of (A0,⊣). Then l(z) = l(z ⊣ z) =
l(z) ⊣ z = z. If a ∈ A, l(a) = b ∈ A0 and b 6= a then a ⊣ b = z. Therefore,
b = b ⊣ b = l(a) ⊣ b = l(a ⊣ b) = l(z) = z. It follows that l(a) ∈ {z, a} for
any a ∈ A. On the other hand, it is clear that for any B ⊂ A the map
lB : A0 → A0,
lB(x) =
{
x, x ∈ B,
z, x ∈ A0 \B
is a left translation of A0.
Let (X,⊢) be any interassociate of OAX . By Lemma 1.1, A0 is an ideal
of (X,⊢), r is a homomorphism from (X,⊢) onto (A0,⊢), and (A0,⊢) is
equal to (A0,⊣lB ), where
x ⊢ y = x ⊣lB y = x ⊣ lB(y) =
{
x if x = y ∈ B,
z otherwise
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V. Gavrylkiv, D. Rendziak 233
for all x, y ∈ A0. This implies (A0,⊢) coincides with OBA0 for B ⊂ A.
Since A0 is an ideal of (X,⊢), a⊢x, x⊢a ∈ A0 for all a ∈ A0, x ∈ X \A.
Taking into account that r(a⊢x) = r(a)⊢r(x) = r(a)⊢z = z and r(x⊢a) =
r(x) ⊢ r(a) = z ⊢ r(a) = z for all a ∈ A0, x ∈ X \ A, we conclude that
x⊢a, a⊢x ∈ (X\A)∩A0 = {z}. Therefore,A⊢(X\A) = (X\A)⊢A = {z}.
Let us show that X \ A is a subsemigroup of (X,⊢). Indeed, since
r is a homomorphism, r((X \ A) ⊢ (X \ A)) = r(X \ A) ⊢ r(X \ A) =
{z}⊢{z} = {z}, and the definition of r implies (X \A)⊢ (X \A) ⊂ X \A.
Since (X \ A,⊣) is a null semigroup with zero z, (X \ A,⊢) is any
semigroup with the same zero z.
To show that a semigroup (X,⊢) for which the conditions 1)-3) hold
is a strong interassociate of OAX , it is sufficient to note the following two
facts:
• an element s ∈ {x ⊣ (y ⊢ z), (x ⊣ y) ⊢ z, x ⊢ (y ⊣ z), (x ⊢ y) ⊣ z} is
non-zero if and only if x = y = z ∈ B for some B ⊂ A;
• b ⊣ (b ⊢ b) = (b ⊣ b) ⊢ b = b ⊢ (b ⊣ b) = (b ⊢ b) ⊣ b = b for any b ∈ B
for some B ⊂ A.
In the following Proposition 3.3 we recognize all interassociates of the
semigroup (M,⊣)1̃ for any monoid (M,⊣).
Proposition 3.3. Let (M,⊣) be a monoid with identity e, and M 1̃ =
M ∪ {1̃}, where 1̃ /∈ M . If (M 1̃,⊢) is an interassociate of (M,⊣)1̃ then
(M 1̃,⊢) = (M,⊣)+1 or (M 1̃,⊢) is a variant of (M,⊣)1̃ with the sandwich
operation x⊢y = x⊣a⊣y, where a = 1̃⊢ 1̃ ∈M . If (M,⊣) is a commutative
monoid then all interassociates of (M,⊣)1̃ are strong interassociate with
each other.
Proof. Let (M 1̃,⊢) be an interassociate of the semigroup (M,⊣)1̃. Then
for any x, y ∈M we have the following equalities:
x ⊢ y = (x ⊣ 1̃) ⊢ (1̃ ⊣ y) = x ⊣ (1̃ ⊢ 1̃) ⊣ y = x ⊣ a ⊣ y,
where a = 1̃ ⊢ 1̃ ∈M 1̃.
Consider two cases.
(1) Let a = 1̃. Then x ⊢ y = x ⊣ 1̃ ⊣ y = x ⊣ y for all x, y ∈M . Taking
into account that 1̃ ⊢ 1̃ = 1̃ and for any x ∈ M the following equalities
hold:
x ⊢ 1̃ = (x ⊣ 1̃) ⊢ 1̃ = x ⊣ (1̃ ⊢ 1̃) = x ⊣ 1̃ = x,
1̃ ⊢ x = 1̃ ⊢ (1̃ ⊣ x) = (1̃ ⊢ 1̃) ⊣ x = 1̃ ⊣ x = x,
we conclude that in this case (M 1̃,⊢) = (M,⊣)+1.
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234 Interassociativity and 3-element doppelsemigroups
(2) Let a 6= 1̃, and thus a ∈ M . We claim that 1̃ ⊢ x, x ⊢ 1̃ ∈ M for
any x ∈ M 1̃. Suppose that 1̃ ⊢ c = 1̃ for some c ∈ M . Then e = 1̃ ⊣ 1̃ =
(1̃ ⊢ c) ⊣ 1̃ = 1̃ ⊢ (c ⊣ 1̃) = 1̃ ⊢ c = 1̃, and we have a contradiction. By
analogy, x ⊢ 1̃ ∈M for any x ∈M 1̃.
For any x ∈M 1̃ we have that
x ⊢ 1̃ = (x ⊢ 1̃) ⊣ 1̃ = x ⊢ (1̃ ⊣ 1̃) = x ⊢ e,
1̃ ⊢ x = 1̃ ⊣ (1̃ ⊢ x) = (1̃ ⊣ 1̃) ⊢ x = e ⊢ x.
Taking into account that for a = 1̃ ⊢ 1̃ ∈M
1̃ ⊢ 1̃ = 1̃ ⊣ (1̃ ⊢ 1̃) ⊣ 1̃ = 1̃ ⊣ a ⊣ 1̃
and for any x ∈M
1̃ ⊢ x = e ⊢ x = e ⊣ a ⊣ x = 1̃ ⊣ a ⊣ x,
x ⊢ 1̃ = x ⊢ e = x ⊣ a ⊣ e = x ⊣ a ⊣ 1̃,
we conclude that (M 1̃,⊢) is a variant of (M,⊣)1̃ with the sandwich opera-
tion x ⊢ y = x ⊣ a ⊣ y, where a = 1̃ ⊢ 1̃ ∈M .
Let (M,⊣) be a commutative monoid. Taking into account that for
each a ∈ M the variants with respect to a of (M,⊣)1̃ and (M,⊣)+1
coincide, and the set of interassociates of (M,⊣)1̃ consists of (M,⊣)+1
and variants of (M,⊣)1̃ with respect to all a ∈ M , we conclude that
each interassociate of (M,⊣)1̃ is an interassociate of (M,⊣)+1. Since
(M,⊣)+1 is a monoid, all of its interassociates are variants. Consequently,
Int((M,⊣)1̃) = Int((M,⊣)+1). Let (M 1̃,⊢1) and (M 1̃,⊢2) be any two
interassociate of (M,⊣)+1. Then x⊢1 y = x⊣a1 ⊣y and x⊢2 y = x⊣a2 ⊣y
for some a1, a2 ∈ M 1̃ and any x, y ∈ M 1̃. Taking into account that
(M,⊣)+1 is commutative and hence x ⊢1 (y ⊢2 z) = x ⊢1 (y ⊣ a2 ⊣ z) =
x ⊣ a1 ⊣ y ⊣ a2 ⊣ z = x ⊣ a2 ⊣ y ⊣ a1 ⊣ z = x ⊢2 (y ⊣ a1 ⊣ z) = x ⊢2 (y ⊢1 z),
we conclude that ⊢1 and ⊢2 are strong interassociate.
In the following Propositions 3.4 and 3.5 we recognize all (strong)
interassociates of the semigroups LO+0
X and RO+0
X .
Proposition 3.4. The set Int(LO+0
X ) consists of all semigroups LO∼0A←X ,
where A ⊂ X. Any two interassociates of LO+0
X are interassociate with each
other. The semigroup LO∼0A←X is a strong interassociate of the semigroup
LO∼0B←X if and only if A = B or A = ∅ or B = ∅.
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V. Gavrylkiv, D. Rendziak 235
Proof. Let (X0,⊢) be an interassociate of the semigroup LO+0
X with
operation ⊣.
If a ⊢ b = 0 for some a, b ∈ X then
x ⊢ b = (x ⊣ a) ⊢ b = x ⊣ (a ⊢ b) = x ⊣ 0 = 0
for any x ∈ X0.
If c ⊢ d 6= 0 for some c, d ∈ X then
x ⊢ d = (x ⊣ c) ⊢ d = x ⊣ (c ⊢ d) = x
for any x ∈ X0.
Let A = {a ∈ X | x ⊢ a 6= 0 for any x ∈ X}. It follows that (X0,⊢)
coincides with LO∼0A←X .
Let us show that for any A,B ⊂ X the semigroups LO∼0A←X with
operation ⊣A and LO∼0B←X with operation ⊢B are interassociate with each
other.
To prove x ⊢B (y ⊣A z) = (x ⊢B y) ⊣A z consider the following two
cases:
• if z ∈ A then x ⊢B (y ⊣A z) = x ⊢B y = (x ⊢B y) ⊣A z for any
x, y ∈ X0;
• if z ∈ X0 \ A then x ⊢B (y ⊣A z) = x ⊢B 0 = 0 = (x ⊢B y) ⊣A z for
any x, y ∈ X0.
To prove x ⊣A (y ⊢B z) = (x ⊣A y) ⊢B z consider the following two
cases:
• if z ∈ B then x ⊣A (y ⊢B z) = x ⊣A y = (x ⊣A y) ⊢B z for any
x, y ∈ X0;
• if z ∈ X0 \B then x ⊣A (y ⊢B z) = x ⊣A 0 = 0 = (x ⊣A y) ⊢B z for
any x, y ∈ X0.
Let us prove that a semigroup LO∼0A←X is a strong interassociate of
a semigroup LO∼0B←X if and only if A = B or A = ∅ or B = ∅.
If A = B then LO∼0A←X = LO∼0B←X . So, LO∼0A←X is a strong interasso-
ciate of a semigroup LO∼0B←X .
If A = ∅ or B = ∅ then LO∼0A←X or LO∼0B←X is a null semigroup. Since
a null semigroup is a strong interassociate of any semigroup with zero, in
this case, LO∼0A←X and LO∼0B←X are strong interassociate with each other.
Let A and B are different non-empty subsets of X. Show that LO∼0A←X
and LO∼0B←X are not strong interassociate with each other. For this, it is
sufficient to consider the following two cases.
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236 Interassociativity and 3-element doppelsemigroups
• There are exist a ∈ A and b ∈ B \A. Then a⊢B (a⊣A b) = a⊢B 0 = 0
while a ⊣A (a ⊢B b) = a ⊣A a = a 6= 0.
• There are exist b ∈ B and a ∈ A\B. Then b⊣A (b⊢B a) = b⊣A 0 = 0
while b ⊢B (b ⊣A a) = b ⊢B b = b 6= 0.
Taking into account that (X,⊣) is an interassociate of (X,⊢) if and only
if (X,⊣d) is an interassociate of (X,⊢d), and for each A ⊂ X the semigroup
LO∼0A←X is dual to RO∼0A←X , we conclude the following proposition.
Proposition 3.5. The set Int(RO+0
X ) consists of all semigroups RO∼0A←X ,
where A ⊂ X. Any two interassociates of RO+0
X are interassociate with
each other. The semigroup RO∼0A←X is a strong interassociate of the semi-
group RO∼0B←X if and only if A = B or A = ∅ or B = ∅.
Let a and c be different elements of a set X. Consider the semigroup
LOBX = (X,⊣ac ), where the binary operation ⊣ac on X is defined in the
following way:
x ⊣ac y =
x if x 6= c
a if x = c and y 6= c
c if x = y = c.
Proposition 3.6. If (X,⊢) is an interassociate of (X,⊣ac ) then (X,⊢) =
(X,⊣ac ) or (X,⊢) = LOX\{c}←X = (X,⊢ac ), where
x ⊢ac y =
{
x, x 6= c,
a, x = c.
All interassociates of (X,⊣ac ) are strong.
Proof. Since each element z ∈ X \ {c} is a left zero of the semigroup
(X,⊣ac ), z is a left zero of (X,⊢).
For each x ∈ X we have
c ⊢ x = (c ⊣ac c) ⊢ x = c ⊣ac (c ⊢ x) ∈ {a, c}.
If x 6= c then for each y ∈ X the following equalities hold:
c ⊢ x = c ⊢ (x ⊣ac y) = (c ⊢ x) ⊣ac y.
It follows that c ⊢ x is a left zero, and therefore, c ⊢ x ∈ X \ {c} for all
x 6= c. Consequently, c ⊢ x = a for all x 6= c.
If c ⊢ c = c then (X,⊢) = (X,⊣ac ). If c ⊢ c = a then (X,⊢) = (X,⊢ac ).
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V. Gavrylkiv, D. Rendziak 237
Let us show that (X,⊢ac ) is a strong interassociate of (X,⊣ac ). Since
each element x ∈ X\{c} is a left zero of (X,⊢ac ) and (X,⊣ac ), x⊣
a
c (y⊢
a
c z) =
x = x ⊢ac (y ⊣
a
c z) for any x ∈ X \ {c} and y, z ∈ X. Taking into account
that c⊣ac (y ⊢
a
c z) ∈ c⊣ac (X \ {c}) = {a} and c⊢ac (y ⊣
a
c z) = a, we conclude
that c ⊣ac (y ⊢
a
c z) = c ⊢ac (y ⊣
a
c z) for any y, z ∈ X.
Dually one can characterize all interassociates of the semigroup ROBX .
4. Three-element doppelsemigroups and their
automorphism groups
In this section we describe up to isomorphism all (strong) doppelsemi-
groups with at most three elements and their automorphism groups.
Firstly, recall some useful facts which we shall often use in this section.
In fact, each semigroup (S,⊣) can be consider as a (strong) doppelsemi-
group (S,⊣,⊣) with the automorphism group Aut(S,⊣,⊣) = Aut(S,⊣),
and we denote this trivial doppelsemigroup by S. As always, we denote
by (S,⊣a) a variant of a semigroup (S,⊣), where x ⊣a y = x ⊣ a ⊣ y. If
the semigroups (S,⊣a) and (S,⊣b) are variants of a commutative semi-
group (S,⊣) then the doppelsemigroup (S,⊣a,⊣b) is strong. If semigroup is
a monoid then all of its interassociates are variants. A semigroup coincides
with each of its interassociates if and only if it is a rectangular band, see
[1, Lemma 5.5]. Every group is isomorphic to each of its interassociates,
see [4]. Following the algebraic tradition, we take for a model of the class
of cyclic groups of order n the multiplicative group Cn = {z ∈ C : zn = 1}
of n-th roots of 1.
Let (D1,⊣1,⊢1) be such a doppelsemigroup that for each doppelsemi-
group (D2,⊣2,⊢2) the isomorphisms (D2,⊣2) ∼= (D1,⊣1) and (D2,⊢2) ∼=
(D1,⊢1) imply (D2,⊣2,⊢2) ∼= (D1,⊣1,⊢1). If S and T are model semi-
groups of classes of semigroups isomorphic to (D1,⊣1) and (D1,⊢1), re-
spectively, then by S ≬ T we denote a model doppelsemigroup of the class
of doppelsemigroups isomorphic to (D1,⊣1,⊢1).
Note that if (D,⊣,⊢) is a (strong) doppelsemigroup then (D,⊢,⊣) is
a (strong) doppelsemigroup as well. In general case, the doppelsemigroups
(D,⊣,⊢) and (D,⊢,⊣) are not isomorphic. It is clear that Aut(D,⊣,⊢) =
Aut(D,⊢,⊣).
It is well-known that there are exactly five pairwise non-isomorphic
semigroups having two elements: C2, L2, O2, LO2, RO2.
Consider the cyclic group C2 = {−1, 1} and find up to isomorphism
all doppelsemigroups (D,⊣,⊢) with (D,⊣) ∼= C2. Because C2 is a monoid,
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238 Interassociativity and 3-element doppelsemigroups
all of its interassociates are variants. Since every group is isomorphic to
each of its interassociates, in this case there are two (strong) doppelsemi-
groups up to isomorphism: C2 and C2 ≬ C
−1
2 = ({−1, 1}, ·, ·−1). These
doppelsemigroups are not isomorphic. Indeed, let ψ is an isomorphism
from ({−1, 1}, ·, ·−1) to ({−1, 1}, ·, ·). Taking into account that −1 is a neu-
tral element of the group ({−1, 1}, ·−1) and ψ must preserve the neutral
elements of both groups ({−1, 1}, ·) and ({−1, 1}, ·−1) of the doppelsemi-
group ({−1, 1}, ·, ·−1), we conclude that ψ(1) = 1 and ψ(−1) = 1, which
contradicts the assertion that ψ is an isomorphism. Since Aut(C2) ∼= C1,
Aut(C2 ≬ C
−1
2 ) ∼= C1.
Since LO2 and RO2 are rectangular bands, all their interassociates
coincide with them, and therefore, in this case there are only two dop-
pelsemigroups: LO2 and RO2.
It is well-known that a null semigroup OX is an interassociate of
each semigroup on X with the same zero. Consequently, O2 has two
non-isomorphic interassociates: O2 and L2. Taking into account that the
semilattice L2 is the monoid ({0, 1},min), we conclude that L2 has two non-
isomorphic interassociates: L2 and ({0, 1},min0) = O2. By Propositions 2.1
and 2.2, it follows that the last four non-isomorphic doppelsemigroups are
O2, O2 ≬ L2, L2 and L2 ≬O2. Note that commutativity of L2 implies that
all these doppelsemigroups are strong. By Proposition 2.3, Aut(L2 ≬O2) =
Aut(O2 ≬ L2) = Aut(L2) ∼= C1.
Consequently, there exist 6 pairwise non-isomorphic commutative
two-element doppelsemigroups and 2 non-isomorphic non-commutative
doppelsemigroups of order 2. All two-element doppelsemigroups are strong.
In the following table we present up to isomorphism all two-element
doppelsemigroups and their automorphism groups.
D C2 O2 L2 C2 ≬ C
−1
2 O2 ≬ L2 L2 ≬O2 LO2 RO2
Aut(D) C1 C1 C1 C1 C1 C1 C2 C2
Table 1. Two-element doppelsemigroups and their automorphism groups
In the remaining part of the paper we concentrate on describing up to
isomorphism all three-element (strong) doppelsemigroups.
Among 19683 different binary operations on a three-element set S
there are exactly 113 operations which are associative. In other words,
there exist exactly 113 three-element semigroups, and many of these are
isomorphic so that there are essentially only 24 pairwise non-isomorphic
semigroups of order 3, see [2, 5, 6].
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V. Gavrylkiv, D. Rendziak 239
Among 24 pairwise non-isomorphic semigroups of order 3 there are
12 commutative semigroups. The rest 12 pairwise non-isomorphic non-
commutative three-element semigroups are divided into the pairs of dual
semigroups that are antiisomorphic. The automorphism groups of dual
semigroups coincide.
List of all pairwise non-isomorphic semigroups of order 3 and their
automorphism groups are presented in Table 2 and Table 3 taken from [6].
S C3 O3 M2,2 C+1
2 C 1̃
2 M3,1 O+1
2 O+0
2 L3 C+0
2 O2
3 O1
3
Aut(S) C2 C2 C1 C1 C1 C1 C1 C1 C1 C1 C2 C1
Table 2. Commutative semigroups S of order 3 and their automorphism
groups
S LO3, RO3 LO+0
2 , RO+0
2 LO∼01←2, RO
∼0
1←2 LO+1
2 , RO+1
2 LOB3, ROB3 LO2←3, RO2←3
Aut(S) S3 C2 C1 C2 C1 C2
Table 3. Non-commutative three-element semigroups and their automorphism
groups
In the sequel, we divide our investigation into cases. In the case of
a semigroup S we shall find all doppelsemigroups (D,⊣,⊢) such that (D,⊣)
is isomorphic to S.
Case C3. Up to isomorphism, the multiplicative group C3 = {1, a, a−1},
where a = e2πi/3, is a unique group of order 3. Since C3 is a monoid,
all of its interassociates are variants. Because C3 is commutative, all of
its interassociates are strong. Since every group is isomorphic to each
of its interassociates, in this case there are exactly three (strong) dop-
pelsemigroups: C3, (C3, ·, ·a) and (C3, ·, ·a−1). It is easy to check the map
ψ : C3 → C3, ψ(g) = g−1 (where g−1 is the inverse of g in the group
(C3, ·)), is an isomorphism from (C3, ·, ·a) to (C3, ·, ·a−1). We denote by
C3 ≬C
−1
3 the doppelsemigroup (C3, ·, ·a−1). By the same arguments as for
the group C2, we conclude that the doppelsemigroups C3 and C3 ≬ C
−1
3
are non-isomorphic. Let ψ : C3 → C3 is an automorphism of the dop-
pelsemigroup (C3, ·, ·a−1). Taking into account that 1 is the identity of
the group (C3, ·) and a is the identity of the group (C3, ·a−1), we conclude
that ψ(1) = 1 and ψ(a) = a. Consequently, ψ(a−1) = a−1, and ψ is the
identity automorphism. It follows that Aut(C3 ≬ C
−1
3 ) ∼= C1.
Case O3. A null semigroup O3 is a (strong) interassociate of each three-
element semigroup with the same zero. Thus, up to isomorphism there
are the following 12 (strong) doppelsemigroups: O3, O3 ≬M3,1, O3 ≬O
+1
2 ,
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240 Interassociativity and 3-element doppelsemigroups
O3 ≬ O
+0
2 , O3 ≬ L3, O3 ≬ C
+0
2 , O3 ≬ O
2
3, O3 ≬ O
1
3, O3 ≬ LO
+0
2 , O3 ≬ RO
+0
2 ,
O3≬LO
∼0
1←2, O3≬RO
∼0
1←2. According to Proposition 2.1, up to isomorphism
there are no other doppelsemigroups (D,⊣,⊢) such that (D,⊣) ∼= O3. By
Proposition 2.3, Aut(O3 ≬ S) ∼= Aut(S) for any three-element semigroup
S with zero.
Case M2,2. Consider the monogenic semigroup M2,2 = {a, a2, a3 | a4 =
a2}. There are three interassociates of this semigroup: (M2,2, ∗k), where
ax ∗k a
y = ax+y+k−2 for every ax, ay ∈ M2,2 and k ∈ {1, 2, 3}, see [7,
Theorem 1.1]. It easy to check that (M2,2, ∗1) = {a2, a3}+1 ∼= C+1
2 ,
(M2,2, ∗2) = (M2,2, ∗) and (M2,2, ∗3) = {a2, a3}1̃ ∼= C 1̃
2 . So, in this case
there are three doppelsemigroups: M2,2, M2,2 ≬ C
+1
2 and M2,2 ≬ C
1̃
2 . Since
all three interassociates of M2,2 are pairwise non-isomorphic, according to
Proposition 2.2 we conclude that up to isomorphism there are no other dop-
pelsemigroups (D,⊣,⊢) such that (D,⊣) ∼= M2,2. Since Aut(M2,2) ∼= C1,
Aut(M2,2 ≬ C
+1
2 ) ∼= C1 and Aut(M2,2 ≬ C
1̃
2 )
∼= C1.
Case C+1
2 . Since C+1
2 is a monoid, all of its interassociates are variants.
Let e be an extra identity adjoined to C2 = {−1, 1}. Then ({−1, 1, e}, ·e) =
C+1
2 , ({−1, 1, e}, ·1) ∼= C 1̃
2 and ({−1, 1, e}, ·−1) ∼= M2,2. Therefore, there
are three doppelsemigroups: C+1
2 , C+1
2 ≬C 1̃
2 and C+1
2 ≬M2,2. Since C+1
2 is
a commutative monoid, all these doppelsemigroups are strong. Taking into
account that all three interassociates of C+1
2 are pairwise non-isomorphic,
by Proposition 2.2 we conclude that up to isomorphism there are no other
doppelsemigroups (D,⊣,⊢) such that (D,⊣) ∼= C+1
2 . Since Aut(C+1
2 ) ∼=
C1, Aut(C
+1
2 ≬ C 1̃
2 )
∼= C1 and Aut(C+1
2 ≬M2,2) ∼= C1.
Case C 1̃
2 . According to Proposition 3.3 the semigroup C 1̃
2 has three
interassociates. As we have seen in previous cases, these interassociates
must be isomorphic to C 1̃
2 , C
+1
2 and M2,2. Taking into account that, by
Proposition 3.3, all interassociates of C 1̃
2 are strong, we conclude that in
this case there are three pairwise non-isomorphic strong doppelsemigroups:
C 1̃
2 , C
1̃
2 ≬ C+1
2 and C 1̃
2 ≬M2,2. Since Aut(C 1̃
2)
∼= C1, Aut(C
1̃
2 ≬ C+1
2 ) ∼= C1
and Aut(C 1̃
2 ≬M2,2) ∼= C1.
Case M3,1. Consider the monogenic semigroup M3,1 = {a, a2, a3 | a4 =
a3}. There are three interassociates of this semigroup: (M3,1, ∗k), where
ax ∗k a
y = ax+y+k−2 for every ax, ay ∈ M3,1 and k ∈ {1, 2, 3}, see [7,
Theorem 1.1]. It easy to check that (M3,1, ∗1) = {a2, a3}+1 ∼= O+1
2 ,
(M3,1, ∗2) = (M3,1, ∗) and (M3,1, ∗3) ∼= O3. So, in this case we have
three doppelsemigroups: M3,1, M3,1 ≬O
+1
2 and M3,1 ≬O3. Since all three
interassociates of M3,1 are pairwise non-isomorphic, according to Propo-
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V. Gavrylkiv, D. Rendziak 241
sition 2.2 we conclude that up to isomorphism there are no other dop-
pelsemigroups (D,⊣,⊢) such that (D,⊣) ∼= M3,1. Since Aut(M3,1) ∼= C1,
Aut(M3,1 ≬O
+1
2 ) ∼= C1 and Aut(M3,1 ≬O3) ∼= C1.
Case O+1
2 . Because the semigroup O+1
2 is a monoid, all of its interassoci-
ates are variants. Since there are only three variants of a three-element
semigroup, previous cases imply that these interassociates isomorphic to
O+1
2 , M3,1 and O3. Taking into account that O+1
2 is a commutative monoid,
we conclude that in this case there are three pairwise non-isomorphic strong
doppelsemigroups: O+1
2 , O+1
2 ≬M3,1 and O+1
2 ≬O3. Since Aut(O+1
2 ) ∼= C1,
Aut(O+1
2 ≬M3,1) ∼= C1 and Aut(O+1
2 ≬O3) ∼= C1.
Case O+0
2 . Proposition 3.1 implies that there are three interassociates
of semigroup O+0
2 , and all these interassociates are strong. They are
isomorphic to O+0
2 , L3 and O3. So, in this case there are three strong
doppelsemigroups O+0
2 , O+0
2 ≬ L3
∼= (O2 ≬ L2)
+0 and O+0
2 ≬O3. Since all
three interassociates of O+0
2 are pairwise non-isomorphic, according to
Proposition 2.2 we conclude that up to isomorphism there are no other
doppelsemigroups (D,⊣,⊢) such that (D,⊣) ∼= O+0
2 . Since Aut(O+0
2 ) ∼=
C1, Aut(O+0
2 ≬ L3) ∼= C1 and Aut(O+0
2 ≬O3) ∼= C1.
Case L3. Since the linear semilattice L3 is a monoid, all of its interasso-
ciates are variants. Three-element semigroup has only three variants, thus
previous cases imply that these interassociates isomorphic to L3, O
+0
2 and
O3. Therefore, in this case we have the trivial doppelsemigroup L3 and
two (strong) doppelsemigroups L3 ≬O
+0
2
∼= (L2 ≬O2)
+0 and L3 ≬O3. Since
Aut(L3) ∼= C1, Aut(L3 ≬O
+0
2 ) ∼= C1 and Aut(L3 ≬O3) ∼= C1.
Case C+0
2 . Consider the semigroup C+0
2 isomorphic to a commuta-
tive monoid ({−1, 1, 0}, ·) with zero 0. Except a null semigroup O3, this
monoid has two isomorphic variants ({−1, 1, 0}, ·) and ({−1, 1, 0}, ·−1). In
this case there are three (strong) doppelsemigroups: C+0
2 ≬O3, C
+0
2 and
({−1, 1, 0}, ·, ·−1). These doppelsemigroups are not isomorphic. Indeed, let
ψ is an isomorphism from ({−1, 1, 0}, ·, ·−1) to ({−1, 1, 0}, ·, ·). Taking into
account that −1 is a neutral element of the semigroup ({−1, 1, 0}, ·−1) and
ψ must preserve the neutral elements of both semigroups ({−1, 1, 0}, ·) and
({−1, 1, 0}, ·−1) of the doppelsemigroup ({−1, 1, 0}, ·, ·−1), we conclude
that ψ(−1) = 1 and ψ(1) = 1, which contradicts the assertion that ψ is an
isomorphism. Taking into account that ({−1, 1, 0}, ·−1) ∼= (C−12 )+0, where
C−12 = ({−1, 1}, ·−1), we denote by C+0
2 ≬ (C−12 )+0 the doppelsemigroup
({−1, 1, 0}, ·, ·−1). It is easy to see that C+0
2 ≬(C−12 )+0 ∼= (C2≬C
−1
2 )+0, and
hence Aut(C+0
2 ≬ (C−12 )+0) ∼= Aut((C2 ≬ C
−1
2 )+0) ∼= Aut(C2 ≬ C
−1
2 ) ∼= C1.
Since Aut(C+0
2 ) ∼= C1, Aut(C
+0
2 ≬O3) ∼= C1.
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242 Interassociativity and 3-element doppelsemigroups
Case O2
3. Consider the non-linear semilattice O2
3 isomorphic to the semi-
group {a, b, 0} with the operation ⊣:
x ⊣ y =
{
x if y = x ∈ {a, b},
0 otherwise.
According to Proposition 3.2, this semigroup has four (strong) interas-
sociates: O2
3, O3, ({a, b, 0},⊢a) and ({a, b, 0},⊢b), where for i ∈ {a, b}
x ⊢i y =
{
x if y = x = i,
0 otherwise.
It is easy to check that the map ψ : {a, b, 0} → {a, b, 0}, ψ(a) = b, ψ(b) = a
and ψ(0) = 0, is a doppelsemigroup isomorphism from ({a, b, 0},⊣,⊢a) to
({a, b, 0},⊣,⊢b).
Therefore, in this case there are three pairwise non-isomorphic
(strong) doppelsemigroups: O2
3, O
2
3 ≬O3 and O2
3 ≬O
1
3. Since Aut(O1
3)
∼= C1,
Aut(O2
3 ≬O
1
3)
∼= C1. By Proposition 2.3, Aut(O2
3 ≬O3) ∼= Aut(O3 ≬O
2
3)
∼=
Aut(O2
3)
∼= C2.
Case O1
3. Consider the last commutative semigroup O1
3 isomorphic to
the semigroup ({a, b, 0},⊢a) from the previous case. By Proposition 3.2,
this semigroup has the same four (strong) interassociates as O2
3. Show
that the doppelsemigroups ({a, b, 0},⊢a,⊢a) and ({a, b, 0},⊢a,⊢b) are not
isomorphic. Suppose that ψ is an isomorphism from ({a, b, 0},⊢a,⊢b) to
({a, b, 0},⊢a,⊢a). Then ψ must preserve a unique non-zero idempotent
of these doppelsemigroups. Therefore, ψ(a) = a and ψ(b) = a, which
contradicts the assertion that ψ is an isomorphism. Denote by Oa3 ≬O
b
3 the
doppelsemigroup ({a, b, 0},⊢a,⊢b). Thus, in this case we have four non-
isomorphic (strong) doppelsemigroups: O1
3, O
a
3 ≬O
b
3, O
1
3 ≬O
2
3 and O1
3 ≬O3.
Since Aut(Oa3)
∼= Aut(O1
3)
∼= C1, Aut(O
a
3 ≬O
b
3)
∼= C1, Aut(O
1
3 ≬O
2
3)
∼= C1
and Aut(O1
3 ≬O3) ∼= C1.
Let (D,⊣,⊢) be a doppelsemigroup. Denote by (D,⊣,⊢)d its dual
doppelsemigroup (D,⊣d,⊢d), where x ⊣d y = y ⊣ x and x ⊢d y = y ⊢ x.
In fact, (D,⊣,⊢)d is a (strong) doppelsemigroup if and only if (D,⊣,⊢)
is a (strong) doppelsemigroup. So, non-commutative doppelsemigroups
are divided into the pairs of dual doppelsemigroups. A map ψ : D1 → D2
is a isomorphism from a doppelsemigroup (D1,⊣1,⊢1) to (D2⊣2,⊢2) if
and only if ψ is a isomorphism from a doppelsemigroup (D1,⊣1,⊢1)
d to
(D2⊣2,⊢2)
d. Thus, Aut((D,⊣,⊢)d) = Aut(D,⊣,⊢).
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V. Gavrylkiv, D. Rendziak 243
It follows that it is sufficient to consider non-commutative three-element
semigroups LO3, LO
+0
2 , LO∼01←2, LO
+1
2 , LOB3, LO2←3. The cases of semi-
groups RO3, RO
+0
2 , RO∼01←2, RO
+1
2 , ROB3, RO2←3 we shall get using the
duality.
Case LO3. Since LO3 is a rectangular band, all its interassociates coincide
with LO3, and therefore, in this case there is a unique doppelsemigroup
LO3.
Case LO+0
2 . Consider the semigroup LO+0
2 isomorphic to {a, b, 0} with
the operation ⊣:
x ⊣ y =
{
x if y ∈ {a, b},
0 if y = 0.
According to Proposition 3.4, this semigroup has four interassociates:
LO+0
2 , O3, ({a, b, 0},⊢a) and ({a, b, 0},⊢b), where for i ∈ {a, b}
x ⊢i y =
{
x if y = i,
0 if y 6= i.
It is easy to check that the map ψ : {a, b, 0} → {a, b, 0}, ψ(a) = b, ψ(b) = a
and ψ(0) = 0, is a doppelsemigroup isomorphism from ({a, b, 0},⊣,⊢a) to
({a, b, 0},⊣,⊢b). Since ({a, b, 0},⊢a) ∼= ({a, b, 0},⊢b) ∼= LO∼01←2, denote by
LO+0
2 ≬ LO∼01←2 the doppelsemigroup ({a, b, 0},⊣,⊢a) ∼= ({a, b, 0},⊣,⊢b).
Thus, in this case we have three pairwise non-isomorphic doppelsemi-
groups: LO+0
2 , LO+0
2 ≬O3 and LO+0
2 ≬LO∼01←2. Consequently, up to isomor-
phism there are no other doppelsemigroups (D,⊣,⊢) such that (D,⊣) ∼=
LO+0
2 . By Proposition 3.4, the doppelsemigroups LO+0
2 and LO+0
2 ≬O3
are strong while LO+0
2 ≬ LO∼01←2 is not strong.
According to Proposition 2.4,
Aut(LO+0
2 ≬ LO∼01←2)
∼= Aut(LO∼01←2)
∼= C1.
By Proposition 2.3,
Aut(LO+0
2 ≬O3) ∼= Aut(O3 ≬ LO
+0
2 ) ∼= Aut(LO+0
2 ) ∼= C2.
Case LO∼01←2. Consider the semigroup LO∼01←2 isomorphic to the semi-
group ({a, b, 0},⊢a) from the previous case. Since this semigroup is the
last semigroup with zero, the previous cases imply that it has the fol-
lowing interassociates: O3, LO
+0
2 , and interassociates that isomorphic to
({a, b, 0},⊢a).
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244 Interassociativity and 3-element doppelsemigroups
Consider interassociates of ({a,b,0},⊢a) that isomorphic to ({a,b,0},⊢a).
Since an isomorphism ψ must preserve a unique right identity a and
zero 0, we conclude that ψ(a) must be a right identity and ψ(0) = 0. Thus,
({a, b, 0},⊢b) is a unique different from ({a, b, 0},⊢a) interassociate isomor-
phic to ({a, b, 0},⊢a). Show that the doppelsemigroups ({a, b, 0},⊢a,⊢a)
and ({a, b, 0},⊢a,⊢b) are not isomorphic. Suppose that ψ is an isomorphism
from ({a, b, 0},⊢a,⊢b) to ({a, b, 0},⊢a,⊢a). Then ψ must preserve right
identities a and b of the semigroups ({a, b, 0},⊢a) and ({a, b, 0},⊢b), respec-
tively. Therefore, ψ(a) = a and ψ(b) = a, which contradicts the assertion
that ψ is an isomorphism. Denote by LO∼0a←2 ≬ LO
∼0
b←2 the doppelsemi-
group ({a, b, 0},⊢a,⊢b). Thus, up to isomorphism, LO∼01←2, LO
∼0
1←2 ≬ O3,
LO∼01←2≬LO
+0
2 and LO∼0a←2≬LO
∼0
b←2 are the last four doppelsemigroups with
zero. Since Aut(LO∼0a←2)
∼= Aut(LO∼01←2)
∼= C1, Aut(LO
∼0
1←2≬LO
+0
2 ) ∼= C1,
Aut(LO∼01←2 ≬ O3) ∼= C1 and Aut(LO∼0a←2 ≬ LO∼0b←2)
∼= C1. By Proposi-
tion 3.4, the doppelsemigroups LO∼01←2 and LO∼01←2 ≬O3 are strong while
LO∼01←2 ≬ LO
+0
2 and LO∼0a←2 ≬ LO
∼0
b←2 are not strong.
Case LO+1
2 . Consider a monoid LO+1
2 with operation ⊣ and identity 1,
where LO2 = {a, b} is a two-element left zero semigroup. Since each
interassociate of LO+1
2 is a variant, we conclude that except LO+1
2 there
two interassociates: ({a, b, 1},⊢a) and ({a, b, 1},⊢b) isomorphic to LO2←3,
where for i ∈ {a, b}
x ⊢i y =
{
x, x 6= 1,
i, x = 1.
It is easy to check that the map ψ : {a, b, 1} → {a, b, 1}, ψ(a) = b, ψ(b) = a
and ψ(0) = 0, is a doppelsemigroup isomorphism from ({a, b, 1},⊣,⊢a) to
({a, b, 1},⊣,⊢b). Since 1⊣(b⊢ab) = 1⊣b = b while 1⊢a(b⊣b) = 1⊢ab = a 6= b,
the doppelsemigroup ({a, b, 1},⊣,⊢a) is not strong. Therefore, in this
case there are two pairwise non-isomorphic doppelsemigroups: LO+1
2 and
LO+1
2 ≬LO2←3. The semigroup LO+1
2 is strong while LO+1
2 ≬LO2←3 is not
strong. By Proposition 2.4, Aut(LO+1
2 ≬ LO2←3)∼=Aut(LO2←3) ∼= C2.
Case LOB3. Consider a non-commutative band LOB3 isomorphic to the
semigroup {a, b, c} with the operation ⊣ac , where
x ⊣ac y =
x if x 6= c,
a if x = c and y 6= c,
c if x = y = c.
By Proposition 3.6, LOB3 has two interassociates isomorphic to LOB3
and LO2←3. According to Proposition 2.1, up to isomorphism there are
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V. Gavrylkiv, D. Rendziak 245
no other doppelsemigroups (D,⊣,⊢) such that (D,⊣) ∼= LOB3. Thus,
in this case there are two non-isomorphic doppelsemigroups: LOB3 and
LOB3 ≬ LO2←3. By Proposition 3.6, these doppelsemigroups are strong.
Since Aut(LOB3) ∼= C1, Aut(LOB3 ≬ LO2←3) ∼= C1.
Case LO2←3. Finally, consider the last three-element semigroup LO2←3
isomorphic to the semigroup {a, b, c} with operation ⊣ defined as follows:
x ⊣ y =
{
x, x 6= c,
a, x = c.
Since this semigroup is the last semigroup, the previous cases imply
that it has the following interassociates: LO+1
2 , LOB3, and interassociates
that isomorphic to ({a, b, c},⊣). Consider interassociates of ({a, b, c},⊣)
that isomorphic to ({a, b, c},⊣). Since a and b are left zeros of ({a, b, c},⊣),
they must be left zeros of each interassociate of ({a, b, c},⊣). It is clear
that there exists only one different from ({a, b, c},⊣) its interassociate
({a, b, c},⊢) ∼= ({a, b, c},⊣), where
x ⊢ y =
{
x, x 6= c,
b, x = c.
It is easy to check that the map ψ : {a, b, c} → {a, b, c}, ψ(a) = b,
ψ(b) = a and ψ(c) = c, is a doppelsemigroup isomorphism from
({a, b, c},⊣,⊣) to ({a, b, c},⊣,⊢). Consequently, LO2←3, LO2←3 ≬ LO
+1
2
and LO2←3 ≬ LOB3 are the last three doppelsemigroups of order 3.
It follows that Aut(LO2←3 ≬LO
+1
2 ) ∼= Aut(LO+1
2 ≬LO2←3) ∼= C2 and
Aut(LO2←3 ≬LOB3) ∼= Aut(LOB3 ≬LO2←3) ∼= C1. Since LOB3 ≬LO2←3
is strong, LO2←3 ≬ LOB3 is strong as well. By analogy, LO2←3 ≬ LO
+1
2 is
not strong.
We summarize the obtained results on the pairwise non-isomorphic non-
trivial three-element (strong) doppelsemigroups and their automorphism
groups in the following Tables 4, 5 and 6.
It follows that we have proved the following theorem.
Theorem 4.1. There exist 75 pairwise non-isomorphic three-element
doppelsemigroups among which 41 doppelsemigroups are commutative.
Non-commutative doppelsemigroups are divided into 17 pairs of dual dop-
pelsemigroups. Also up to isomorphism there are 65 strong doppelsemi-
groups of order 3, and all non-strong doppelsemigroups are not commuta-
tive. There exist exactly 24 pairwise non-isomorphic three-element trivial
doppelsemigroups.
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246 Interassociativity and 3-element doppelsemigroups
D C3 ≬ C
−1
3 O3 ≬M3,1 O3 ≬O
+1
2 O3 ≬O
+0
2 O3 ≬ L3 O3 ≬ C
+0
2
Aut(D) C1 C1 C1 C1 C1 C1
D O3 ≬O
2
3 O3 ≬O
1
3 M2,2 ≬ C
+1
2 M2,2 ≬ C
1̃
2 C+1
2 ≬ C 1̃
2 C+1
2 ≬M2,2
Aut(D) C2 C1 C1 C1 C1 C1
D C 1̃
2 ≬M2,2 C 1̃
2 ≬ C+1
2 M3,1 ≬O
+1
2 M3,1 ≬O3 O+1
2 ≬M3,1 O+1
2 ≬O3
Aut(D) C1 C1 C1 C1 C1 C1
D (O2 ≬ L2)
+0 O+0
2 ≬O3 L3 ≬O3 (L2 ≬O2)
+0 (C2 ≬ C
−1
2 )+0 C+0
2 ≬O3
Aut(D) C1 C1 C1 C1 C1 C1
D O2
3 ≬O
1
3 O2
3 ≬O3 Oa3 ≬O
b
3 O1
3 ≬O
2
3 O1
3 ≬O3
Aut(D) C1 C2 C1 C1 C1
Table 4. Three-element (strong) non-trivial commutative doppelsemigroups
and their automorphism groups
D O3 ≬ LO
+0
2 O3 ≬ LO
∼0
1←2 LO+0
2 ≬O3 LO∼01←2 ≬O3 LOB3 ≬ LO2←3 LO2←3 ≬ LOB3
O3 ≬RO
+0
2 O3 ≬RO
∼0
1←2 RO+0
2 ≬O3 RO∼01←2 ≬O3 ROB3 ≬RO2←3 RO2←3 ≬ROB3
Aut(D) C2 C1 C2 C1 C1 C1
Table 5. Three-element non-trivial non-commutative strong doppelsemigroups
and their automorphism groups
D LO+0
2 ≬ LO∼01←2 LO∼01←2 ≬ LO
+0
2 LO∼0a←2 ≬ LO
∼0
b←2 LO+1
2 ≬ LO2←3 LO2←3 ≬ LO
+1
2
RO+0
2 ≬RO∼01←2 RO∼01←2 ≬RO
+0
2 RO∼0a←2 ≬RO
∼0
b←2 RO+1
2 ≬RO2←3 RO2←3 ≬RO
+1
2
Aut(D) C1 C1 C1 C2 C2
Table 6. Three-element (non-commutative) non-strong doppelsemigroups and
their automorphism groups
5. Acknowledgment
The authors would like to express their sincere thanks to the anony-
mous referee for a very careful reading of the paper and for all its insightful
comments and valuable suggestions, which improve considerably the pre-
sentation of this paper.
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Contact information
V. Gavrylkiv,
D. Rendziak
Vasyl Stefanyk Precarpathian National
University, Ivano-Frankivsk, Ukraine
E-Mail(s): vgavrylkiv@gmail.com,
rendziakdiana007@gmail.com
Received by the editors: 02.08.2019
and in final form 12.11.2019.
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