Representations of ordered doppelsemigroups by binary relations
We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. We construct the ordered doppelsemigroup of binary relations on an arbitrary set and prove that every ordered doppelsemigroup is isomorphic to some ordered doppelsemigroup of binary relations. In particul...
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1884282023-03-01T01:27:05Z Representations of ordered doppelsemigroups by binary relations Zhuchok, Y.V. Koppitz, J. We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. We construct the ordered doppelsemigroup of binary relations on an arbitrary set and prove that every ordered doppelsemigroup is isomorphic to some ordered doppelsemigroup of binary relations. In particular, we obtain an analogue of Cayley’s theorem for semigroups in the class of doppelsemigroups. We also describe the representations of ordered doppelsemigroups by binary transitive relations. 2019 Article Representations of ordered doppelsemigroups by binary relations / Y.V. Zhuchok, J. Koppitz // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 144–154. — Бібліогр.: 20 назв. — англ. 1726-3255 2010 MSC: 17A30, 06F05, 43A65. http://dspace.nbuv.gov.ua/handle/123456789/188428 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. We construct the ordered doppelsemigroup of binary relations on an arbitrary set and prove that every ordered doppelsemigroup is isomorphic to some ordered doppelsemigroup of binary relations. In particular, we obtain an analogue of Cayley’s theorem for semigroups in the class of doppelsemigroups. We also describe the representations of ordered doppelsemigroups by binary transitive relations. |
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Article |
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Zhuchok, Y.V. Koppitz, J. |
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Zhuchok, Y.V. Koppitz, J. Representations of ordered doppelsemigroups by binary relations Algebra and Discrete Mathematics |
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Zhuchok, Y.V. Koppitz, J. |
| author_sort |
Zhuchok, Y.V. |
| title |
Representations of ordered doppelsemigroups by binary relations |
| title_short |
Representations of ordered doppelsemigroups by binary relations |
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Representations of ordered doppelsemigroups by binary relations |
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Representations of ordered doppelsemigroups by binary relations |
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Representations of ordered doppelsemigroups by binary relations |
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representations of ordered doppelsemigroups by binary relations |
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Інститут прикладної математики і механіки НАН України |
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2019 |
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http://dspace.nbuv.gov.ua/handle/123456789/188428 |
| citation_txt |
Representations of ordered doppelsemigroups by binary relations / Y.V. Zhuchok, J. Koppitz // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 144–154. — Бібліогр.: 20 назв. — англ. |
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| fulltext |
“adm-n1” — 2019/3/22 — 12:03 — page 144 — #152
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 1, pp. 144–154
c© Journal “Algebra and Discrete Mathematics”
Representations of ordered doppelsemigroups
by binary relations∗
Yurii V. Zhuchok, Jörg Koppitz
Communicated by V. A. Artamonov
Abstract. We extend the study of doppelsemigroups and
introduce the notion of an ordered doppelsemigroup. We construct
the ordered doppelsemigroup of binary relations on an arbitrary
set and prove that every ordered doppelsemigroup is isomorphic to
some ordered doppelsemigroup of binary relations. In particular,
we obtain an analogue of Cayley’s theorem for semigroups in the
class of doppelsemigroups. We also describe the representations of
ordered doppelsemigroups by binary transitive relations.
1. Introduction
The notion of a doppelalgebra was introduced by B. Richter in [7] in
the context of algebraic K-theory. A doppelalgebra is a vector space over
a field with two binary linear associative operations ⊣ and ⊢ satisfying
the following identities
(x ⊣ y) ⊢ z = x ⊣ (y ⊢ z), (D1)
(x ⊢ y) ⊣ z = x ⊢ (y ⊣ z) (D2)
∗The paper was written during the research stay of the first author at the Potsdam
University in the frame of the Programme of German Academic Exchange Service
(DAAD).
2010 MSC: 17A30, 06F05, 43A65.
Key words and phrases: doppelsemigroup, ordered doppelsemigroup, semigroup,
binary relation, representation.
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Yu. V. Zhuchok, J. Koppitz 145
An algebraic system which consists of a nonempty set with two binary
associative operations ⊣ and ⊢ satisfying the identities (D1) and (D2) is of
independent interest. These algebras were first considered in [6] and later
they were called as doppelsemigroups in [11]. Thus, doppelalgebras are lin-
ear analogs of doppelsemigroups. The first results about doppelsemigroups
are the descriptions of the free product of doppelsemigroups and such free
objects as the free doppelsemigroup, the free commutative doppelsemi-
group and the free n-nilpotent doppelsemigroup, and the characterization
of the least commutative congruence (the least n-nilpotent congruence) on
a free doppelsemigroup [11]. Doppelalgebras and doppelsemigroups have
relationships with such algebraic structures as semigroups, duplexes [6]
(sets with two associative operations), interassociative semigroups [1, 2],
restrictive bisemigroups [8, 9], n-tuple semigroups and n-tuple algebras
of associative type [3], dimonoids and dialgebras [4, 12, 17], trioids and
trialgebras [5, 13, 18], and other related systems (see, e.g., [19]). If the
operations of a doppelsemigroup coincide, then the doppelsemigroup be-
comes a semigroup. Commutative dimonoids (i.e., dimonoids with both
commutative operations) are examples of doppelsemigroups [11]. On the
other hand, doppelsemigroups are examples of n-tuple semigroups [3], for
n = 2. More information on doppelsemigroups can be found, for instance,
in [14, 15].
It is well-known that according to Cayley’s theorem for semigroups,
every semigroup is isomorphic to a semigroup of transformations of some
set. For ordered semigroups, a similar statement was proved by K. A. Zaret-
skiy [10], where in particular, it was shown that every ordered semigroup
can be embedded into the ordered semigroup of all binary relations on
a suitable set. Also necessary and sufficient conditions under which an
ordered semigroup is isomorphic to some ordered semigroup of reflexive (or
transitive) binary relations were found. An analogue of Cayley’s theorem
for the class of dimonoids was obtained in [12] by means the notion of
an α-consistent subsemigroup. The concept of an ordered dimonoid was
introduced in [20], where the ordered dimonoid of binary relations was
defined and an analogue of Zaretskiy’s theorem in the class of ordered
dimonoids was obtained.
In the present paper, we introduce the concept of an ordered dop-
pelsemigroup and consider the mentioned above problems for it. We give
examples of ordered doppelsemigroups and construct the ordered dop-
pelsemigroup of binary relations on an arbitrary nonempty set using the
semigroup of binary relations and a particular variant of it (i.e., a sandwich
semigroup). The main result of this paper is a representation theorem
“adm-n1” — 2019/3/22 — 12:03 — page 146 — #154
146 Representations of ordered doppelsemigroups
which shows that every ordered doppelsemigroup can be embedded into
the constructed ordered doppelsemigroup of binary relations on a suitable
set. As a consequence, we define the transformation doppelsemigroup on
an arbitrary nonempty set and show that Cayley’s theorem for semigroups
has an analogue in the class of doppelsemigroups. Finally, we study the
representations of ordered doppelsemigroups by binary transitive relations.
2. Ordered doppelsemigroups and their examples
Let (D,⊣,⊢) be an arbitrary doppelsemigroup and let 6 be a partial
order relation on D. The algebraic system (D,⊣,⊢,6) is called an ordered
doppelsemigroup if the order relation 6 is stable with respect to both
operations ⊣ and ⊢, that is, x 6 y implies
z ∗ x 6 z ∗ y and x ∗ z 6 y ∗ z
for all x, y, z ∈ D and ∗ ∈ {⊣,⊢}.
Now, we give several examples of ordered doppelsemigroups.
Obviously, every doppelsemigroup can be considered as an ordered
doppelsemigroup with respect to the diagonal relation.
Let (D,⊣,⊢) be an arbitrary doppelsemigroup and let P(D) be the
set of all subsets of D. Define on P(D) two binary operations ⊣′ and ⊢′
by the following rule:
A ⊣′ B = {a ⊣ b | a ∈ A, b ∈ B} and A ⊢′ B = {a ⊢ b | a ∈ A, b ∈ B}.
Proposition 1. Let (D,⊣,⊢) be a doppelsemigroup. The algebraic system
(P(D),⊣′,⊢′,⊆) is an ordered doppelsemigroup with respect to the set-
theoretic inclusion ⊆.
Proof. It is obvious.
Let (S, ∗) be an arbitrary semigroup and a ∈ S. Define on S a new
binary operation ∗a by
x ∗a y = x ∗ a ∗ y
for all x, y ∈ S. Clearly, (S, ∗a) is a semigroup, it is called a variant of
(S, ∗), or a sandwich semigroup of (S, ∗) with respect to the element a.
Proposition 2. Let (S, ∗,6) be an ordered monoid and a, b ∈ S. The
algebraic system (S, ∗a, ∗b,6) is an ordered doppelsemigroup.
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Yu. V. Zhuchok, J. Koppitz 147
Proof. By [11, Lemma 2.1], (S, ∗a, ∗b) is a doppelsemigroup. The proof of
stability of 6 with respect to ∗a and ∗b is obvious.
Note that if at least one of the elements a or b from Proposition 2 is
the identity of (S, ∗), we obtain the ordered doppelsemigroups (S, ∗, ∗,6),
(S, ∗, ∗b,6), and (S, ∗a, ∗,6).
Let X be a nonempty set, F[X] be the free semigroup on X, and T be
the free monoid on a fixed two-element set {a, b} with the empty word θ.
The length of a word w ∈ F[X] ∪ T is denoted by lw.
Define two binary operations ⊣ and ⊢ on the set
FDS(X) = {(u, v) ∈ F[X]× T | lu − lv = 1}
by
(u1, v1) ⊣ (u2, v2) = (u1u2, v1av2),
(u1, v1) ⊢ (u2, v2) = (u1u2, v1bv2)
for all (u1, v1), (u2, v2) ∈ FDS(X).
We fix linear orders �1 and �2 in X and on {θ, a, b}, respectively,
where θ is the least element with respect to �2. The lexicographic order
relations on the free semigroup F[X] and the free monoid T that are
defined by �1 and �2, we will denote as �F[X] and �T , respectively.
Now we define a binary relation � on FDS(X) in such way:
(u1, v1) � (u2, v2) ⇐⇒ u1 �F[X] u2 & v1 �T v2.
Proposition 3. The algebraic system (FDS(X),⊣,⊢,�) is an ordered
doppelsemigroup.
Proof. According to [11, Theorem 3.5], (FDS(X),⊣,⊢) is the free dop-
pelsemigroup on X. The rest of the proof is obvious.
Let (N,+) be the additive semigroup of all natural numbers and
N2̃ = N ∪ {2̃}, where 2̃ /∈ N . Define binary operations ⊣ and ⊢ on N2̃ by
2̃ ⊣ 2̃ = 2̃ ⊢ 2̃ = 4,
m ⊣ n = m+ n,
m ⊣ 2̃ = 2̃ ⊣m = m ⊢ 2̃ = 2̃ ⊢m = m+ 2,
m ⊢ n =
{
2̃, m = n = 1,
m+ n, otherwise
for all m,n ∈ N .
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148 Representations of ordered doppelsemigroups
We consider the ordinary arithmetic order relation 6 on N and extend
this relation to N2̃ as follows:
1 6 2̃ 6 3 6 4 6 ...,
in addition, the elements 2 and 2̃ are not related by 6.
Proposition 4. The algebraic system (N2̃,⊣,⊢,6) is an ordered doppel-
semigroup.
Proof. From [16, Proposition 3] it follows that (N2̃,⊣,⊢) is a commutative
dimonoid. As already mentioned in the introduction, any commutative
dimonoid is a doppelsemigroup. Therefore, (N2̃,⊣,⊢) is a doppelsemigroup.
The proof of the fact that 6 is a stable order relation on N2̃ with respect
to ⊣ and ⊢ is obvious.
One can generalize Proposition 4 in the following way. Let (I,�) be a
linearly ordered set of indexes and let
S = {xi,1, xi,2, xi,3, xi,4 | i ∈ I}.
We define binary operations ⊣ and ⊢ on S by
xi,j ⊣ xk,l =
xk,4 if k � i, k 6= i,
xi,2 if i = k, j = l = 1,
xi,4 otherwise
and
xi,j ⊢ xk,l =
xk,4 if k � i, k 6= i,
xi,3 if i = k, j = l = 1,
xi,4 otherwise.
Further, we define a binary relation 6 on S by xi,j 6 xk,l if i � k,
i 6= k, or i = k and (j, l) ∈ ρ, where ρ = {(1, 2), (1, 3), (1, 4), (2, 4), (3, 4)}∪
{(a, a) | a ∈ {1, 2, 3, 4}}.
Proposition 5. The algebraic system (S,⊣,⊢,6) is an ordered doppel-
semigroup.
Proof. In fact, we can easy verify that, for all xi,j , xu,v, xp,q ∈ S and
∗1, ∗2 ∈ {⊣,⊢},
(xi,j ∗1 xu,v) ∗2 xp,q = xm,4 = xi,j ∗1 (xu,v ∗2 xp,q),
where m is the least element of i, u, and p with respect to �.
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Yu. V. Zhuchok, J. Koppitz 149
We have still to show that 6 is stable with respect to ⊣ and ⊢. For
this, let xi1,j1 6 xi2,j2 and xi3,j3 ∈ S. By k (and l, respectively), we
denote the least element of i1 and i3 (of i2 and i3, respectively) with
respect to �. Further let ∗ ∈ {⊣,⊢}. Then there exist r, s ∈ {2, 3, 4} such
that xi1,j1 ∗ xi3,j3 = xk,r and xi2,j2 ∗ xi3,j3 = xl,s. Since xi1,j1 6 xi2,j2 ,
we have i1 � i2 and thus k � l. If k 6= l then xk,r 6 xl,s. Suppose
that k = l. If s = 2 or s = 3 then we can calculate that i2 = i3 and
j2 = j3 = 1. In particular, i2 = i3 = l = k � i1 � i2 provides i1 = i2,
where (j1, j2) = (j1, 1) ∈ ρ. The latter implies j1 = 1, i.e. xi1,j1 = xi2,j2 ,
and thus xk,r = xl,s. In the case s = 4, we have xk,r 6 xl,s since (2, 4),
(3, 4), and (4, 4) belong to ρ.
Dually, we obtain xi3,j3 ∗ xi1,j1 6 xi3,j3 ∗ xi2,j2 .
For any nonempty set X, let X = {x |x ∈ X} be a disjoint copy of X.
In particular, we have x1 = x2 if and only if x1 = x2 for all x1, x2 ∈ X.
Further, we put X = X ∪X, and let
△X = {(x, x) |x ∈ X} and △X = {(x, x) |x ∈ X}.
Define two binary operations ◦1 and ◦2 on the set B(X) of all binary
relations on X in such way:
α ◦1 β = △X ◦ α ◦ △X ◦ β,
α ◦2 β = △X ◦ α ◦ △X ◦ β,
where ◦ is the ordinary composition of binary relations.
Proposition 6. The algebraic system (B(X), ◦1, ◦2,⊆) is an ordered
doppelsemigroup.
Proof. Taking into account the equalities △X ◦△X = △X and △X ◦△X =
△X , we immediately obtain associativity of the operations ◦1 and ◦2.
Moreover, for these operations doppelsemigroup axioms (D1) and (D2)
hold since △X ◦ △X = △X ◦ △X = ∅. The stability of ⊆ with respect to
the both operations ◦1 and ◦2 follows from the stability of ⊆ with respect
to the composition ◦.
3. Representations of ordered doppelsemigroups
by binary relations
In this section, we show that any ordered doppelsemigroup can be
embedded to a suitable ordered doppelsemigroup consisting of binary
relations on some set.
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150 Representations of ordered doppelsemigroups
Let (D,⊣,⊢,6) and (D′,⊣′,⊢′,6′) be arbitrary ordered doppelsemi-
groups. A bijective mapping ϕ : D → D′ is called an isomorphism of the
ordered doppelsemigroups if for all x, y ∈ D and ∗ ∈ {⊣,⊢} the following
conditions hold:
ϕ(x ∗ y) = ϕ(x) ∗′ ϕ(y), (H1)
x 6 y ⇐⇒ ϕ(x) 6′ ϕ(y). (H2)
We use the notations from Proposition 6. For any nonempty set X,
we put X = X ∪X and let
ρX = {(x, x), (x, x) |x ∈ X}.
It is clear that (B(X), ◦ρX ) is the sandwich semigroup of the semigroup
(B(X), ◦) with respect to the element ρX . Moreover, by the remark after
Proposition 2, (B(X), ◦, ◦ρX ,⊆) is an ordered doppelsemigroup.
We call the doppelsemigroup (B(X), ◦, ◦ρX ,⊆) as the ordered
doppelsemigroup of all binary relations on X. Subdoppelsemigroups of
(B(X), ◦, ◦ρX ,⊆), we will call ordered doppelsemigroups of binary relations
on X.
The main result of this paper is the following theorem.
Theorem 1. Every ordered doppelsemigroup is isomorphic to an ordered
doppelsemigroup of binary relations on some set.
Proof. Let (D,⊣,⊢,6) be an arbitrary ordered doppelsemigroup and
D1 be the set D with externally adjoined element 1 /∈ D such that
1 ⊣ s = 1 ⊢ s = s for all s ∈ D.
Further for every s ∈ D, we put
fs = {(s1, s2) ∈ D1 ×D | s2 6 s1 ⊣ s} ∪ {(s1, s2) ∈ D1 ×D | s2 6 s1 ⊢ s},
and let
FD = {fs | s ∈ D}.
It is clear that FD ⊆ B(D1). We will prove that
f : s 7→ fs
is an isomorphism of (D,⊣,⊢,6) into (FD, ◦, ◦ρ
D1
,⊆). Clearly, by defini-
tion of f , the mapping f is surjective. The mapping f is also injective.
In fact, let s1, s2 ∈ D with fs1 = fs2 . Because of s1 6 s1 = 1 ⊣ s1, we
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Yu. V. Zhuchok, J. Koppitz 151
obtain (1, s1) ∈ fs1 = fs2 , i.e. s1 6 1 ⊣ s2 = s2. Dually, we get s2 6 s1,
thus s1 = s2.
Show that f satisfies (H1). First, we consider the operation ⊣. Let
(x, y) ∈ fs1 ◦ fs2 for some s1, s2 ∈ D. Then there exists z ∈ D such that
(x, z) ∈ fs1 and (z, y) ∈ fs2 . Clearly, y ∈ D. Then y 6 z ⊣ s2. On the
other hand, we have x ∈ D1 ∪D1. If x ∈ D1 then z 6 x ⊣ s1 and using
stability and transitivity of 6 we obtain y 6 (x⊣s1)⊣s2 = x⊣(s1⊣s2), i.e.
(x, y) ∈ fs1⊣s2 . If x ∈ D1 then there exists u ∈ D such that u = x and we
get z 6 u⊢s1. Similarly as above, this provides y 6 (u⊢s1)⊣s2 = u⊢(s1⊣s2)
and therefore (u, y) ∈ fs1⊣s2 .
Conversely, let (x, y) ∈ fs1⊣s2 , i.e. y 6 x ⊣ (s1 ⊣ s2) = (x ⊣ s1) ⊣ s2 if
x ∈ D1 and y 6 u ⊢ (s1 ⊣ s2) = (u ⊢ s1) ⊣ s2 if u ∈ D1 with u = x. Then
we can conclude that (x ⊣ s1, y) ∈ fs2 and (u ⊢ s1, y) ∈ fs2 , respectively,
where (x, x ⊣ s1) ∈ fs1 and (u, u ⊢ s1) ∈ fs1 follow from x ⊣ s1 6 x ⊣ s1
and u ⊢ s1 6 u ⊢ s1, respectively. Hence, (x, y) ∈ fs1 ◦ fs2 and we have
shown that fs1 ◦ fs2 = fs1⊣s2 .
Now we consider the operation ⊢. Let (x, y)∈fs1◦ρD1
fs2 =fs1◦ρD1◦fs2
for some s1, s2 ∈ D. Then there exists z ∈ D such that (x, z) ∈ fs1 and
(z, y) ∈ fs2 . Clearly, y ∈ D and y 6 z ⊢ s2. If x ∈ D1 then z 6 x ⊣ s1 and
we obtain y 6 (x ⊣ s1) ⊢ s2 = x ⊣ (s1 ⊢ s2), i.e. (x, y) ∈ fs1⊢s2 . If x ∈ D1
then there exists u ∈ D1 with u = x and y 6 (u ⊢ s1) ⊢ s2 = u ⊢ (s1 ⊢ s2),
i.e. (x, y) ∈ fs1⊢s2 .
Conversely, let (x, y) ∈ fs1⊢s2 . If x ∈ D1 then y 6 x ⊣ (s1 ⊢ s2)
= (x ⊣ s1) ⊢ s2 and x ⊣ s1 6 x ⊣ s1. This provides (x ⊣ s1, y) ∈ fs2 and
(x, x ⊣ s1) ∈ fs1 , i.e. (x, y) ∈ fs1 ◦ ρD1 ◦ fs2 = fs1 ◦ρD1
fs2 . If x ∈ D1 then
there exists u ∈ D1 such that u = x and y 6 u ⊢ (s1 ⊢ s2) = (u ⊢ s1) ⊢ s2
with u ⊢ s1 6 u ⊢ s1. It means that (u ⊢ s1, y) ∈ fs2 and (u, u ⊢ s1) ∈ fs1 ,
and thus (x, y) ∈ fs1 ◦ ρD1 ◦ fs2 = fs1 ◦ρD1
fs2 . Consequently, we have
shown that fs1 ◦ρD1
fs2 = fs1⊢s2 .
Finally, we show that f satisfies (H2). Let s1, s2 ∈ D such that s1 6 s2.
Clearly, 1⊣s1 6 1⊣s2 and by stability of 6, x⊣s1 6 x⊣s2 and x⊢s1 6 x⊢s2
for all x ∈ D. If (x, y) ∈ fs1 with x ∈ D1 then y 6 x ⊣ s1 6 x ⊣ s2, i.e.
(x, y) ∈ fs2 . For (x, y) ∈ fs1 with x ∈ D1, we have y 6 x ⊢ s1 6 x ⊢ s2, i.e.
(x, y) ∈ fs2 . This shows fs1 ⊆ fs2 . Conversely, let s1, s2 ∈ D with fs1 ⊆ fs2 .
Since s1 6 s1 = 1⊣s1, we have (1, s1) ∈ fs1 ⊆ fs2 , i.e. s1 6 1⊣s2 = s2.
Remark 1. From Theorem 1, it follows that every ordered semigroup
is isomorphic to some ordered semigroup of binary relations, that is,
Zaretskiy’s theorem [10, Theorem of Sect. 5] is a corollary of Theorem 1.
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152 Representations of ordered doppelsemigroups
In particular, the representations from Theorem 1 and the mentioned
Zaretskiy’s theorem are different.
For any nonempty set X, we denote by T (X) the set of all nonempty
functional relations (i.e., transformations) on the set X = X ∪ X. It
is clear that ρX ∈ T (X) and (T (X), ◦, ◦ρX ) is a subdoppelsemigroup
of (B(X), ◦, ◦ρX ). We call this doppelsemigroup (T (X), ◦, ◦ρX ) as the
doppelsemigroup of all transformations of X. Subdoppelsemigroups of
(T (X), ◦, ◦ρX ), we call doppelsemigroups of transformations of X.
Finally from Theorem 1, we immediately obtain an analogue of Cayley’s
theorem for semigroups in the class of doppelsemigroups.
Corollary 1. Every doppelsemigroup is isomorphic to a doppelsemigroup
of transformations of some set.
Proof. Let (D,⊣,⊢) be an arbitrary doppelsemigroup and D1 be the set
D with an adjoined element 1 /∈ D such that for all s ∈ D,
1 ⊣ s = s and 1 ⊢ s = s.
Take the diagonal relation
∆D = {(s, s) | s ∈ D}
as a partial order on D. In this case, we can consider (D,⊣,⊢) as an
ordered doppelsemigroup (D,⊣,⊢,∆D). In addition,
fs = {(s1, s2) ∈ D1 ×D | s2 = s1 ⊣ s} ∪ {(s1, s2) ∈ D1 ×D | s2 = s1 ⊢ s}
is a functional relation on D1 for all s ∈ D. By Theorem 1, the ordered dop-
pelsemigroup (D,⊣,⊢,∆D) is isomorphic to the ordered doppelsemigroup
(FD, ◦, ◦ρ
D1
,⊆). In particular, (D,⊣,⊢) is isomorphic to (FD, ◦, ◦ρ
D1
),
where FD is a subset of T (D1).
Remark 2. On the one hand, Cayley’s theorem for semigroups follows
from Corollary 1 immediately. However on the other hand, this theorem
is not a consequence of Zaretskiy’s theorem [10, Theorem of Sect. 5].
At the end of the paper, we will study conditions under which an
arbitrary ordered doppelsemigroup is isomorphic to some ordered dop-
pelsemigroup of transitive relations.
Let X be a nonempty set. A doppelsemigroup of binary relations on
X = X ∪X we call as a doppelsemigroup of binary transitive relations if
it consists entirely of binary transitive relations.
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Yu. V. Zhuchok, J. Koppitz 153
Theorem 2. An arbitrary ordered doppelsemigroup (D,⊣,⊢,6) is iso-
morphic to some ordered doppelsemigroup of binary transitive relations if
and only if x ⊣ x 6 x for all x ∈ D.
Proof. Let (D,⊣,⊢,6) be an ordered doppelsemigroup isomorphic to a
subdoppelsemigroup T of (B(X), ◦, ◦ρX ,⊆) consisting entirely of binary
transitive relations defined on a set X = X ∪X. In particular, (D,⊣,6)
and (T, ◦,⊆) are isomorphic as ordered semigroups. By [10, Theorem of
Sect. 6], we obtain x ⊣ x 6 x for all x ∈ D.
Conversely, suppose that x⊣x 6 x for every element x of an ordered dop-
pelsemigroup (D,⊣,⊢,6). By Theorem 1, (D,⊣,⊢,6) and (FD, ◦, ◦ρ
D1
,⊆)
are isomorphic with respect to the isomorphism f : s 7→ fs. Moreover for
all x ∈ D, we get
x ⊣ x 6 x ⇐⇒ f(x ⊣ x) ⊆ f(x) ⇐⇒ f(x) ◦ f(x) ⊆ f(x).
Thus, all relations of FD are transitive.
Observe that Zaretskiy’s theorem on representations of ordered semi-
groups of binary transitive relations [10, Theorem of Sect. 6] follows from
Theorem 2.
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Contact information
Yurii V. Zhuchok Luhansk Taras Shevchenko National University,
Gogol square 1, Starobilsk, Ukraine, 92703
E-Mail(s): zhuchok.yu@gmail.com
Jörg Koppitz Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences, Acad. G.
Bonchev St, Bl. 8, 1113 Sofia, Bulgaria
E-Mail(s): koppitz@math.bas.bg
Received by the editors: 27.11.2018.
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