Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid
We determine all isomorphisms between the endomorphism semigroups of free commutative g-dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative g-dimonoid are quasi-inner.
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| Cite this: | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid / Y.V. Zhuchok // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 309-324. — Бібліогр.: 19 назв. — англ. |
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| citation_txt | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid / Y.V. Zhuchok // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 309-324. — Бібліогр.: 19 назв. — англ. |
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| description | We determine all isomorphisms between the endomorphism semigroups of free commutative g-dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative g-dimonoid are quasi-inner.
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 2, pp. 309–324
© Journal “Algebra and Discrete Mathematics”
Automorphisms of the endomorphism
semigroup of a free commutative g-dimonoid
Yurii V. Zhuchok
Communicated by V. I. Sushchansky
Abstract. We determine all isomorphisms between the
endomorphism semigroups of free commutative g-dimonoids and
prove that all automorphisms of the endomorphism semigroup of a
free commutative g-dimonoid are quasi-inner.
1. Introduction
A dimonoid is an algebra (D,⊣,⊢) with two binary associative op-
erations ⊣ and ⊢ such that for all x, y, z ∈ D the following conditions
hold:
(D1) (x ⊣ y) ⊣ z = x ⊣ (y ⊢ z),
(D2) (x ⊢ y) ⊣ z = x ⊢ (y ⊣ z),
(D3) (x ⊣ y) ⊢ z = x ⊢ (y ⊢ z).
This notion was introduced by Jean-Louis Loday in [1] and now it plays a
prominent role in problems from the theory of Leibniz algebras. A vector
space equipped with the structure of a dimonoid is called a dialgebra.
Thus, a dialgebra is a linear analog of a dimonoid. It is known that Leibniz
algebras are a non-commutative variation of Lie algebras and dialgebras
are a variation of associative algebras.
2010 MSC: 08A35, 08B20, 17A30.
Key words and phrases: g-dimonoid, free commutative g-dimonoid, endomor-
phism semigroup, automorphism group.
310 Endomorphisms of free commutative g-dimonoids
There exist some generalizations of dimonoids, for example, 0-dial-
gebras and duplexes (see, e.g., [2], [3]), g-dimonoids etc. Omitting the
axiom (D2) of an inner associativity in the definition of a dimonoid, we
obtain the notion of a g-dimonoid. An associative 0-dialgebra, that is, a
vector space equipped with two binary associative operations ⊣ and ⊢
satisfying the axioms (D1) and (D3), is a linear analog of a g-dimonoid.
Free g-dimonoids and free n-nilpotent g-dimonoids were constructed in
[4], [5] and [5], respectively. The construction of a free commutative g-
dimonoid and the least commutative congruence on a free g-dimonoid
were described in [6]. Defining identities of a g-dimonoid appear also in
axioms of trialgebras and of trioids [7–9].
Endomorphism semigroups of algebraic systems have been studied by
numerous authors. The problem of studying the endomorphism semigroup
for free algebras in a certain variety was raised by B.I. Plotkin in his papers
on universal algebraic geometry (see, e.g., [10], [11]). In this direction there
are many papers devoted to describing automorphisms of endomorphism
semigroups of free finitely generated universal algebras of some varieties:
groups [12], semigroups [13], associative algebras [14], inverse semigroups
[15], modules and semimodules [16], Lie algebras [17] and other algebras
(see also [18]). In this paper we solve the similar problem for the variety
of commutative g-dimonoids.
The paper is organized in the following way. In Section 2, we give
necessary definitions and statements. In Section 3, we define the notion
of a crossed isomorphism of g-dimonoids and prove auxiliary lemmas.
In Section 4, we describe all isomorphisms between the endomorphism
monoids of free commutative g-dimonoids of rank 1. In Section 5, we prove
that automorphisms of the endomorphism semigroup of a free commutative
g-dimonoid of a non-unity rank are inner or "mirror inner". We show
also that the automorphism group of the endomorphism semigroup of
a free commutative g-dimonoid is isomorphic to the direct product of a
symmetric group and a 2-element group.
2. Preliminaries
Let D1 = (D1,⊣1,⊢1) and D2 = (D2,⊣2,⊢2) be arbitrary g-dimonoids.
A mapping ϕ : D1 → D2 is called a homomorphism of D1 into D2 if
(x ⊣1 y)ϕ = xϕ ⊣2 yϕ, (x ⊢1 y)ϕ = xϕ ⊢2 yϕ
for all x, y ∈ D1.
Yu. V. Zhuchok 311
A bijective homomorphism ϕ : D1 → D2 is called an isomorphism of
D1 onto D2. In this case g-dimonoids D1 and D2 are called isomorphic.
A g-dimonoid (D,⊣,⊢) is called commutative if for all x, y ∈ D,
x ⊣ y = y ⊣ x, x ⊢ y = y ⊢ x.
Firstly we give an example of a g-dimonoid which is not a dimonoid.
Let A be an arbitrary nonempty set and A = {x | x ∈ A}. For every
x ∈ A assume x̃ = x and introduce a mapping α = αA : A ∪ A → A by
the following rule:
yα =
{
y, y ∈ A,
ỹ, y ∈ A.
Give an arbitrary semigroup S and define operations ≺ and ≻ on
S ∪ S as follows:
a ≺ b = (aαS)(bαS), a ≻ b = (aαS)(bαS)
for all a, b ∈ S ∪ S. The algebra (S ∪ S,≺,≻) is denoted by S(α).
Proposition 1 ([6]). S(α) is a g-dimonoid but not a dimonoid.
We note that if X is a generating set of a semigroup S, then S(α) \X
is a g-subdimonoid of S(α) generated by X.
For an arbitrary commutative semigroup S, obviously, S(α) is a com-
mutative g-dimonoid.
Recall the construction of a free commutative g-dimonoid. Let F [A]
be the free commutative semigroup generated by a set A.
Theorem 1 ([6]). F [A](α) \A is the free commutative g-dimonoid.
Observe that A is a generating set of F [A](α) \A, the cardinality of
A is the rank of F [A](α) \A and this g-dimonoid is uniquely determined
up to an isomorphism by |A|.
Further the free commutative g-dimonoid generated by A will be
denoted by FCD
g
A.
In particular, we consider the free commutative g-dimonoid of rank 1.
Let N be the set of all natural numbers and N
∗ = (N ∪ N) \ {1}. Define
operations ≺ and ≻ on N
∗ by
m ≺ n = m+ n, q ≺ r = q + r,
m ≺ r = m+ r, q ≺ n = q + n,
a ≻ b = a ≺ b,
for all m,n ∈ N, q, r ∈ N \ {1} and a, b ∈ N
∗.
312 Endomorphisms of free commutative g-dimonoids
Proposition 2 ([6]). The free commutative g-dimonoid FCD
g
A of rank 1
is isomorphic to (N∗,≺,≻).
Recall that the content of ω = x1x2 . . . xn ∈ F [A] is the set c(ω) =
{x1, x2, . . . , xn} and the length of ω is the number l(ω) = n.
For every ω ∈ FCD
g
A, the set c(ωα) and the number l(ωα) we call the
content and the length of ω, respectively, and denote it by c(ω) and l(ω).
For example, for w = bacda we have c(w) = {a, b, c, d} and l(w) = 5.
3. Auxiliary statements
We start this section with the following lemma.
Lemma 1. Let FCD
g
X and FCD
g
Y be free commutative g-dimonoids ge-
nerated by X and Y , respectively. Every bijection ϕ : X → Y induces an
isomorphism εϕ : FCDg
X → FCD
g
Y such that
ωεϕ =
{
x1ϕ ≺ x2ϕ ≺ . . . ≺ xmϕ, ω = x1x2 . . . xm, m > 1,
x1ϕ ≻ x2ϕ ≻ . . . ≻ xmϕ, ω = x1x2 . . . xm, m > 1
for all ω ∈ FCD
g
X .
Proof. The proof of this statement is obvious.
Now we introduce the notion of a crossed isomorphism of g-dimonoids.
A mapping ϕ : D1 → D2 we call a crossed homomorphism of a g-dimonoid
D1 = (D1,⊣1,⊢1) into a g-dimonoid D2 = (D2,⊣2,⊢2) if for all x, y ∈ D1,
(x ⊣1 y)ϕ = xϕ ⊢2 yϕ, (x ⊢1 y)ϕ = xϕ ⊣2 yϕ.
A bijective crossed homomorphism ϕ : D1 → D2 will be called a
crossed isomorphism of D1 onto D2. In such case g-dimonoids D1 and D2
we call crossed isomorphic.
An example of crossed isomorphic g-dimonoids gives the next lemma.
Lemma 2. Let FCD
g
X and FCD
g
Y be free commutative g-dimonoids ge-
nerated by X and Y , respectively. Every bijection ϕ : X → Y induces a
crossed isomorphism ε∗
ϕ : FCDg
X → FCD
g
Y such that
ωε∗
ϕ =
{
x1ϕ ≻ x2ϕ ≻ . . . ≻ xmϕ, ω = x1x2 . . . xm, m > 1,
x1ϕ ≺ x2ϕ ≺ . . . ≺ xmϕ, ω = x1x2 . . . xm, m > 1
for all ω ∈ FCD
g
X .
Yu. V. Zhuchok 313
Proof. It is clear that ε∗
ϕ is a bijection. Take arbitrary u, v ∈ FCD
g
X and
consider the following cases.
Case 1. u = u1u2 . . . um, v = v1v2 . . . vn ∈ F [X], then
(u ≺ v)ε∗
ϕ = (uαvα)ε∗
ϕ = (uv)ε∗
ϕ
= u1ϕ ≻ . . . ≻ umϕ ≻ v1ϕ ≻ . . . ≻ vnϕ = uε∗
ϕ ≻ vε∗
ϕ,
(u ≻ v)ε∗
ϕ = (uαvα)ε∗
ϕ = (uv)ε∗
ϕ
= u1ϕ ≺ . . . ≺ umϕ ≺ v1ϕ ≺ . . . ≺ vnϕ
= u1ϕ . . . umϕ ≺ v1ϕ . . . vnϕ = uε∗
ϕ ≺ vε∗
ϕ.
Case 2. u = u1u2 . . . um ∈ F [X], v = v1v2 . . . vn ∈ F [X] \X, then
(u ≺ v)ε∗
ϕ = (uv)ε∗
ϕ = u1ϕ . . . umϕv1ϕ . . . vnϕ
= u1ϕ . . . umϕ ≻ (v1ϕ . . . vnϕ) = uε∗
ϕ ≻ vε∗
ϕ,
(u ≻ v)ε∗
ϕ = (uv)ε∗
ϕ = u1ϕ . . . umϕv1ϕ . . . vnϕ
= u1ϕ . . . umϕ ≺ (v1ϕ . . . vnϕ) = uε∗
ϕ ≺ vε∗
ϕ.
Case 3, where u = u1u2 . . . um ∈ F [X] \ X, v = v1v2 . . . vn ∈ F [X], can
be omited since operations ≺ and ≻ are commutative.
Case 4, where u = u1u2 . . . um, v = v1v2 . . . vn ∈ F [X] \ X, is analogous
to the case 1.
From cases 1–4 it follows that ε∗
ϕ is a crossed homomorphism which
completes the proof of this statement.
For an arbitrary algebra A, we denote the endomorphism semigroup
and the automorphism group of A by End(A) and Aut(A), respectively.
Anywhere the composition of mappings is defined from left to right.
Lemma 3. Let D1 = (D1,⊣1,⊢1) and D2 = (D2,⊣2,⊢2) be arbitrary
g-dimonoids, and ϕ be any isomorphism or a crossed isomorphism of D1
onto D2. The mapping
Φ : f 7→ fΦ = ϕ−1fϕ, f ∈ End(D1),
is an isomorphism of End(D1) onto End(D2).
314 Endomorphisms of free commutative g-dimonoids
Proof. Let ϕ be a crossed isomorphism of D1 onto D2. Clearly, ϕ−1 is a
crossed isomorphism of D2 onto D1. For all u, v ∈ D2 and f ∈ End(D1),
(u ⊣2 v)ϕ−1fϕ = (uϕ−1 ⊢1 vϕ
−1)fϕ
= (uϕ−1f ⊢1 vϕ
−1f)ϕ = u(ϕ−1fϕ) ⊣2 v(ϕ−1fϕ).
In similar way, ϕ−1fϕ ∈ End(D2,⊢2) and so fΦ ∈ End(D2) for all
f ∈ End(D1). The remaining part of the proof is trivial.
We call Φ from Lemma 3 as the isomorphism induced by the isomor-
phism or the crossed isomorphism ϕ.
For an arbitrary nonempty set X the identity transformation of X is
denoted by idX . By Lemma 2, ε∗
idX
is a crossed automorphism of the free
commutative g-dimonoid FCD
g
X .
By Lemma 3, a transformation Φ1 of the endomorphism monoid
End(FCDg
X) defined by ηΦ1 = (ε∗
idX
)−1ηε∗
idX
for all η ∈ End(FCDg
X), is
an automorphism. Obviously, (ε∗
idX
)−1 = ε∗
idX
.
The automorphism Φ1 we will call the mirror automorphism of
the endomorphism monoid End(FCDg
X). By Φ0 we denote the identity
automorphism of End(FCDg
X). It is clear that {Φ0,Φ1} is a group with
respect to the composition of permutations.
Let FCDg
X be the free commutative g-dimonoid generated by X. Each
endomorphism ξ of FCDg
X is uniquely determined by a mapping ϕ : X →
FCD
g
X . Really, to define ξ, it suffices to put
ωξ =
{
x1ϕ ≺ x2ϕ ≺ . . . ≺ xmϕ, ω = x1x2 . . . xm, m > 1,
x1ϕ ≻ x2ϕ ≻ . . . ≻ xmϕ, ω = x1x2 . . . xm, m > 1
for all ω ∈ FCD
g
X .
In particular, an endomorphism ξ of FCDg
X is an automorphism if
and only if a restriction ξ on X belong to the symmetric group S(X).
Therefore, the group Aut(FCDg
X) is isomorphic to S(X) (see [6]).
Let u ∈ FCD
g
X . An endomorphism θu ∈ End(FCDg
X) is called constant
if xθu = u for all x ∈ X.
Lemma 4. (i) Let u ∈ FCD
g
X , ξ ∈ End(FCDg
X). Then θuξ = θuξ.
(ii) An endomorphism ξ of FCD
g
X is constant if and only if ψξ = ξ for
all ψ ∈ Aut(FCDg
X).
(iii) An endomorphism ξ of FCD
g
X is constant idempotent if and only if
ξ = θx for some x ∈ X.
Yu. V. Zhuchok 315
Proof. (i) It is obvious.
(ii) Take a constant θu ∈ End(FCDg
X) for some u ∈ FCD
g
X , and let
ψ ∈ Aut(FCDg
X). Then x(ψθu) = (xψ) θu = u = xθu for all x ∈ X.
Now let ψξ = ξ for all ψ ∈ Aut(FCDg
X) and some ξ ∈ End(FCDg
X).
Fixing x ∈ X, we obtain xξ = x (ψξ) = (xψ) ξ = yξ, where y = xψ. Since
{xψ | ψ ∈ Aut(FCDg
X)} = X, then xξ = yξ for all y ∈ X. Consequently,
ξ = θu for u = xξ.
(iii) Let ξ ∈ End(FCDg
X) be a constant idempotent. Then ξ = θu, u ∈
FCD
g
X , and by (i) of this lemma, θu = θuθu = θuθu
. This implies u = uθu
and, therefore, l(u) = 1 and u ∈ X. Converse is obvious.
4. The automorphism group of End(FCDg
X
), |X| = 1
The free commutative g-dimonoid FCD
g
X on an n-element set X we
denote by FCDg
n. Recall that the g-dimonoid FCD
g
1 is isomorphic to
(N∗,≺,≻) (see Proposition 2). Therefore, we will identify elements of
FCD
g
1 with corresponding elements of (N∗,≺,≻).
Define a binary operation ⊙ on N
∗ = (N ∪ N) \ {1} by
m⊙ n = m⊙ n = m · n, m⊙ n = m⊙ n = m · n,
1 ⊙ x = x⊙ 1 = x
for all m,n ∈ N \ {1}, m,n ∈ N \ {1} and x ∈ N
∗.
Lemma 5. (i) The operation ⊙ is associative.
(ii) The operation ⊙ is distributive with respect to ≺ and ≻.
Proof. It can be verified directly.
From Lemma 5 (i) it follows that (N∗,⊙) is a semigroup.
Lemma 6. The semigroups End(FCDg
1) and (N∗,⊙) are isomorphic.
Proof. Let ϕ be an arbitrary endomorphism of (N∗,≺,≻) and 1ϕ = k for
some k ∈ N
∗. For all a ∈ N and b ∈ N \ {1} we obtain
aϕ = (1 ≺ 1 ≺ . . . ≺ 1︸ ︷︷ ︸
a
)ϕ = a⊙ k, bϕ = (1 ≻ 1 ≻ . . . ≻ 1︸ ︷︷ ︸
b
)ϕ = b⊙ k.
Converse, any transformation ϕk : N∗ → N
∗, k ∈ N
∗, defined by
aϕk = a⊙ k
316 Endomorphisms of free commutative g-dimonoids
is an endomorphism of (N∗,≺,≻). Indeed, using the condition (ii) of
Lemma 5, for all a, b ∈ N
∗ and ⋆ ∈ {≺,≻} we obtain
(a ⋆ b)ϕk = (a ⋆ b) ⊙ k = (a⊙ k) ⋆ (b⊙ k) = aϕk ⋆ bϕk.
Consequently,
End(N∗,≺,≻) = {ϕk | k ∈ N
∗}.
Define a mapping Θ of End(N∗,≺,≻) into (N∗,⊙) by ϕkΘ = k for
all ϕk ∈ End(N∗,≺,≻). An immediate verification shows that Θ is an
isomorphism.
Remark 1. Note that all endomorphisms of a g-dimonoid (N∗,≺,≻) are
injective but they are not surjective (except an identity automorphism).
So that the automorphism group of (N∗,≺,≻) is singleton.
Let P be the set of all prime numbers, P = {x | x ∈ P} and P
∗ = P∪P.
For any mapping f : A → B and a nonempty subset C ⊆ A, we denote
the restriction of f to C by f |C .
Further let A,B ⊆ N \ {1}, C ⊆ N \ {1} be nonempty subsets and
ϕ : A → B, ψ : B → C be arbitrary mappings. Denote by ϕ and
−→
ψ the
mappings A → B and, respectively, B → Cα (the mapping α was defined
in Section 2) such that
a ϕ = b if aϕ = b and b
−→
ψ = c if bψ = c.
Proposition 3. Let End(FCDg
X) ∼= End(FCDg
Y ), where X is a single-
ton set, Y is an arbitrary set. Then |Y | = 1 and the isomorphisms of
End(FCDg
X) onto End(FCDg
Y ) are in a natural one-to-one correspondence
with permutations f : P∗ → P
∗ such that
Pf = P, f |
P
= f |P or Pf = P, f |
P
=
−→
f |P.
Proof. According to Lemma 6, End(FCDg
1) ∼= (N∗,⊙). Let |Y | > 2 and
a, b ∈ Y, a 6= b. Define a binary relation ρ on N
∗ by
(a; b) ∈ ρ ⇔ a = b = 1 or a 6= 1 6= b, a⊙ b = b⊙ a.
Obviously, ρ is an equivalence and N
∗/ρ = {N \ {1},N \ {1}, {1}}. Since
End(FCDg
Y ) ∼= (N∗,⊙), we will use the relation ρ for End(FCDg
Y ) too.
For constants θ
ab
, θa, θab ∈ End(FCDg
Y ) and some y ∈ Y we have
y(θ
ab
θa) = abθa = aa 6= ab = aθ
ab
= y(θaθab),
y(θ
ab
θab) = abθab = abab 6= abab = abθ
ab
= y(θabθab),
Yu. V. Zhuchok 317
therefore (θ
ab
, θa) /∈ ρ and (θ
ab
, θab) /∈ ρ. From here it follows that
(θab, θa) ∈ ρ which contradicts the fact that θabθa 6= θaθab. Then |Y | = 1.
It is clear that the semigroup (N∗ \ {1},⊙) is generated by P
∗ and
P
∗f = P
∗ for all f ∈ Aut(N∗,⊙). Assume that there exist p, q ∈ P such
that pf = p′ ∈ P and qf = q′ ∈ P for some f ∈ Aut(N∗,⊙). Then
p′ · q′ = p′ ⊙ q′ = (p · q)f = (q · p)f = q′ ⊙ p′ = p′ · q′.
It means that Pf = P and so Pf = P, or Pf = P and then Pf = P.
If Pf = P, then for all p ∈ P we have (pf)2 = p2f = (p⊙p)f = pf⊙pf,
whence pf = pf . Thus, f |
P
= f |P. In a similar way it can be shown that
in the case Pf = P we obtain f |
P
=
−→
f |P.
On the other hand, as it is not hard to check, every permutation
f : P∗ → P
∗ such that Pf = P, f |
P
= f |P, or Pf = P, f |
P
=
−→
f |P, uniquely
determines an automorphism of (N∗,⊙). These permutations and hence
the isomorphisms End(FCDg
X) → End(FCDg
Y ), are in a natural one-to-one
correspondence.
An automorphism Φ : End(FCDg
X) → End(FCDg
X) is called quasi-
inner if there exists a permutation α of FCDg
X such that βΦ = α−1βα
for all β ∈ End(FCDg
X). If α turns out to be an automorphism of FCDg
X ,
Φ is an inner automorphism of End(FCDg
X).
We denote the symmetric group on a set X by S(X). A 2-element
group with identity e is denoted by C2 = {e, a}.
Proposition 4. Automorphisms of the monoid End(FCDg
1) are quasi-
inner. In addition, the automorphism group of End(FCDg
1) is isomorphic
to the direct product S(P) × C2.
Proof. Let Ψ : End(FCDg
1) → End(FCDg
1) be an arbitrary automorphism.
Define a bijection ψ : N∗ → N∗ putting xψ = y if ϕxΨ = ϕy. It
is clear that ψ ∈ Aut(N∗,⊙), however ψ /∈ Aut(N∗,≺,≻) except the
identity permutation (see Remark 1). Then for all x ∈ N∗ and some
ϕi ∈ End(FCDg
1), i ∈ N∗, we have
x(ψ−1ϕiψ) = (xψ−1)ϕiψ = ((xψ−1) ⊙ i)ψ
= (xψ−1)ψ ⊙ iψ = x⊙ iψ = xϕiψ.
Thus, ψ−1ϕiψ = ϕiΨ and Ψ is a quasi-inner automorphism.
318 Endomorphisms of free commutative g-dimonoids
The immediate check shows that a mapping Θ of Aut(N∗,⊙) onto
S(P) × C2 defined as follows:
ξΘ =
{
(ξ|P , e), P ξ = P,
(ξ|P , a), P ξ = P
for all ξ ∈ Aut(N∗,⊙), is an isomorphism.
By Lemma 6, End(FCDg
1) ∼= (N∗,⊙), therefore Aut(End(FCDg
1)) and
S(P) × C2 are isomorphic.
5. The automorphism group of End(FCDg
X
), |X| > 2
An automorphism Ψ of the endomorphism monoid End(FCDg
X) of
the free commutative g-dimonoid FCD
g
X is called stable if Ψ induces the
identity permutation of X, that is, θxΨ = θx for all x ∈ X.
Lemma 7. For all u, v ∈ F [X] \X the following equalities hold:
θuθv = θuθv and θuθv = θuθv.
Proof. It is obvious.
Lemma 8. Let Ψ be a stable automorphism of End(FCDg
X),
u, v ∈ F [X] \X, x ∈ X and ξ ∈ End(FCDg
X). Then
(i) θxξΨ = θx(ξΨ);
(ii) θuΨ = θv implies θuΨ = θv;
(iii) θuΨ = θv implies θuΨ = θv.
Proof. (i) By Lemma 4 (i), θxξΨ = (θxξ)Ψ = θx(ξΨ) = θx(ξΨ).
(ii) Let θuΨ = θv. By (i) of this lemma, θuΨ = θw for some w ∈ FCD
g
X .
Using Lemma 7, we obtain
θvl(v) = θ2
v = (θuΨ)2 = (θ2
u)Ψ
= (θuθu)Ψ = θuΨθuΨ = θvθw = θwl(v) ,
where wl(v) = w ≺ w ≺ . . . ≺ w︸ ︷︷ ︸
l(v)
. From here w = v or w = v. In the first
case we have θuΨ = θv which contradicts to injectivity of Ψ, therefore
θuΨ = θv.
(iii) This statement is anologous to the case (ii).
An endomorphism θ of the free commutative g-dimonoid FCD
g
X is
called linear if xθ ∈ X for all x ∈ X.
Yu. V. Zhuchok 319
Lemma 9. Let Ψ be a stable automorphism of End(FCDg
X), u, v∈FCD
g
X ,
x ∈ X and ξ ∈ End(FCDg
X). The following conditions hold:
(i) ξΨ = ξ, if ξ is linear;
(ii) c(u) = c(v), if θuΨ = θv;
(iii) l(xξ) = l(x(ξΨ)).
Proof. (i) If ξ is linear, then xξ ∈ X for all x ∈ X. Hence by stability of
Ψ, θx(ξΨ) = θxξΨ = θxξ. From here, ξΨ = ξ.
(ii) Let θuΨ = θv and c(u) \ c(v) 6= ∅. We take z ∈ c(u) \ c(v), and
x ∈ X,x 6= z, and ξ ∈ End(FCDg
X) such that zξ = x and yξ = y for all
y ∈ X, y 6= z. Then ξ is linear, vξ = v and
θuΨ = θv = θvξ = θvξ = (θuΨ)(ξΨ) = (θuξ)Ψ = θuξΨ.
From here θu = θuξ and then u = uξ which contradicts to the definition
of ξ, so c(u)\ c(v) = ∅. If z ∈ c(v)\ c(u) 6= ∅, z 6= x and ξ ∈ End(FCDg
X)
the same as above, then
θv = θuΨ = θuξΨ = (θuξ)Ψ = (θuΨ)(ξΨ) = θvξ = θvξ,
whence v=vξ which contradicts to the definition of ξ. Thus, c(v) \ c(u)=∅
and therefore, c(u) = c(v).
(iii) Let ξ1, ξ2 ∈ End(FCDg
X) such that l(xξ1) = l(xξ2) = m and
l(x(ξ1Ψ)) = r, l(x(ξ2Ψ)) = s. For all t ∈ X we obtain
t(θxξ1θx) = (xξ1)θx =
{
xm = tθxm , xξ1 ∈ F [X],
xm = tθxm , xξ1 ∈ F [X] \X.
Analogously it is proved that θxξ2θx =
{
θxm , xξ2 ∈ F [X],
θxm , xξ2 ∈ F [X] \X.
Consider following four cases.
Case 1. xξ1, xξ2 ∈ F [X]. Using that Ψ is stable, we have
θxmΨ = (θxξ1θx)Ψ = θx(ξ1Ψ)θx =
{
θxr , x(ξ1Ψ) ∈ F [X],
θxr , x(ξ1Ψ) ∈ F [X] \X,
θxmΨ = (θxξ2θx)Ψ = θx(ξ2Ψ)θx =
{
θxs , x(ξ2Ψ) ∈ F [X],
θxs , x(ξ2Ψ) ∈ F [X] \X.
If x(ξ1Ψ) ∈ F [X], x(ξ2Ψ) ∈ F [X]\X or x(ξ1Ψ) ∈ F [X]\X, x(ξ2Ψ) ∈
F [X], then we obtain θxr = θxs or θxr = θxs which is false. Otherwise, we
have r = s.
320 Endomorphisms of free commutative g-dimonoids
Case 2. xξ1, xξ2 ∈ F [X] \X. It is similar to the case 1.
Case 3. xξ1 ∈ F [X], xξ2 ∈ F [X] \X. Assume that θxmΨ = θxr , then by
(ii) of Lemma 8 we have θxmΨ = θxr . On the other hand,
θxmΨ = (θxξ2θx)Ψ = θx(ξ2Ψ)θx =
{
θxs , x(ξ2Ψ) ∈ F [X],
θxs , x(ξ2Ψ) ∈ F [X] \X.
For x(ξ2Ψ) ∈ F [X] we obtain xr = xs which is false. If x(ξ2Ψ) ∈
F [X] \X, then θxr = θxs , whence r = s.
In similar way we can show that r = s if θxmΨ = θxr .
Case 4. xξ1 ∈ F [X] \X, xξ2 ∈ F [X]. It is analogous to the case 3.
Thus, cases 1–4 imply that r and s coincides.
Further, let A be a nonempty finite subset of X and
EndmA (x) = {ξ ∈ End(FCDg
X) | l(xξ) = m, c(xξ) = A}.
For θxξ ∈ EndmA (x) by (i) of Lemma 8 we have θxξΨ = θx(ξΨ). By (ii)
of given lemma, c(xξ) = c(x(ξΨ)). Taking into account the previous
arguments, there exists k such that EndmA (x)Ψ ⊆ EndkA(x). Since Ψ is
bijective, k = m. Thus, l(xξ) = l(x(ξΨ)) for all ξ ∈ End(FCDg
X) and
x ∈ X.
Corollary 1. Let Ψ be a stable automorphism of End(FCDg
X) and
x1, x2 ∈ X are distinct. Then
θx1x2Ψ = θx1x2 or θx1x2Ψ = θx1x2 .
Proof. By Lemma 8 (i), θx1x2Ψ = θu for some u ∈ FCD
g
X , and by (ii) of
Lemma 9, c(u) = {x1, x2}. By (iii) of Lemma 9, l(u) = 2. Thus, u = x1x2
or u = x1x2.
Lemma 10. Let Ψ be a stable automorphism of End(FCDg
X) and
x1, x2 ∈ X are distinct. Then
(i) θx1x2Ψ = θx1x2 implies Ψ = Φ0;
(ii) θx1x2Ψ = θx1x2 implies Ψ = Φ1.
Proof. (i) By induction on the length of u we show that θuΨ = θu for
all u ∈ F [X]. By stability of Ψ, θxΨ = θx for all x ∈ X. Assume that
θvΨ = θv for all v ∈ F [X] with l(v) < n, and let u = u1 . . . un ∈ F [X],
where n > 2. Let v1 = u1 . . . un−1, v2 = un and f ∈ End(FCDg
X) is such
that x1f = v1, x2f = v2 and yf = y for all y ∈ X \ {x1, x2}. Then
x(θx1x2f) = (x1x2)f = x1fx2f = u = xθu for all x ∈ X.
Yu. V. Zhuchok 321
By Lemma 8 (i) and the induction hypothesis, we have
θxi(fΨ) = θxifΨ = θvi
Ψ = θvi
= θxif , i ∈ {1, 2},
θx(fΨ) = θxfΨ = θxΨ = θx = θxf , x ∈ X \ {x1, x2}.
So, fΨ = f and then for all u ∈ F [X] with l(u) > 2,
θuΨ = (θx1x2f)Ψ = (θx1x2Ψ)(fΨ) = θx1x2f = θu.
By (ii) of Lemma 8, θuΨ = θu for all u ∈ F [X] \X, so that θuΨ = θu
for all u ∈ FCD
g
X . Now for all x ∈ X and ϕ ∈ End(FCDg
X),
θx(ϕΨ) = θxϕΨ = θxϕ.
This implies ϕΨ = ϕ for all ϕ ∈ End(FCDg
X), that is, Ψ = Φ0.
(ii) Take the crossed automorphism ε∗
idX
of FCD
g
X (see Lemma 2).
For all u ∈ FCD
g
X and f ∈ End(FCDg
X) we use denotations u∗ = uε∗
idX
and f∗ = (ε∗
idX
)−1fε∗
idX
.
By induction on l(u) we show that θuΨ = θu∗ for all u ∈ F [X]. The
induction base follows from the fact that Ψ is stable.
Let us suppose that θvΨ = θv∗ for all v ∈ F [X] such that l(v) < n,
and let u = u1 . . . un ∈ F [X], n > 2. We put v1 = u1, v2 = u2 . . . un, and
take the endomorphism f of FCDg
X such that x1f = v1, x2f = v2, and
yf = y for all y ∈ X \ {x1, x2}.
Similarly as in (i) of this lemma, we can show that θx1x2f = θu. By
Lemma 8 (i) and the induction hypothesis,
θxi(fΨ) = θxifΨ = θvi
Ψ = θv∗
i
= θxif∗ , i ∈ {1, 2},
θx(fΨ) = θxfΨ = θxΨ = θx∗ = θxf∗ , x ∈ X \ {x1, x2}.
From here, fΨ = f∗. Then for all u ∈ F [X] with l(u) > 2,
θuΨ = (θx1x2f)Ψ = (θx1x2Ψ)(fΨ) = θx1x2f
∗ = θu = θu∗ .
Taking into account Lemma 8 (iii), θuΨ = θu for all u ∈ F [X] \X. It
means that θuΨ = θu∗ for all u ∈ FCD
g
X .
Finally, for all x ∈ X and ϕ ∈ End(FCDg
X) we have
θx(ϕΨ) = θxϕΨ = θ(xϕ)∗ = θxϕ∗ .
Hence, ϕΨ = ϕ∗ for all ϕ ∈ End(FCDg
X), that is, Ψ = Φ1.
322 Endomorphisms of free commutative g-dimonoids
Theorem 2. Let X be an arbitrary set with |X| > 2. Every isomorphism
Φ : End(FCDg
X) → End(FCDg
Y ) is induced either by the isomorphism εf
or by the crossed isomorphism ε∗
f of FCD
g
X onto FCD
g
Y for a uniquely
determined bijection f : X → Y .
Proof. Let Φ : End(FCDg
X) → End(FCDg
Y ) be an arbitrary isomorphism.
In similar way as in the case of free abelian dimonoids (see [19, Theorem 3]),
using Lemma 4 can be shown that for every x ∈ X there exists y ∈ Y
such that θxΦ = θy. Define a bijection f : X → Y putting xf = y if
θxΦ = θy. In this case we say that f is induced by Φ.
By Lemma 1, f induces the isomorphism εf : FCDg
X → FCD
g
Y . Ac-
cording to Lemma 3, Ef : η 7→ ε−1
f ηεf is an isomorphism of End(FCDg
X)
onto End(FCDg
Y ). From this it follows that the composition ΦE−1
f is an
automorphism of End(FCDg
X).
Further for all x ∈ X we have
θx(ΦE−1
f ) = (θxΦ)E−1
f = θxfE
−1
f = θ(xf)f−1 = θx,
which implies stability of ΦE−1
f .
Using Corollary 1 and Lemma 10, we obtain ΦE−1
f is either the identity
automorphism Φ0 or the mirror automorphism Φ1. Assume, ΦE−1
f = Φ0,
then Φ = Ef which means that Φ is an isomorphism induced by εf . If
ΦE−1
f = Φ1, then Φ = Φ1Ef which means that Φ is an isomorphism
induced by ε∗
f .
The following statement gives the positive solution of the definability
problem of free commutative g-dimonoids by its endomorphism semi-
groups.
Corollary 2. Let FCD
g
X and FCD
g
Y be free commutative g-dimonoids
such that End(FCDg
X) ∼= End(FCDg
Y ). Then FCD
g
X and FCD
g
Y are iso-
morphic.
Proof. As shown in the proof of Theorem 2, every isomorphism Φ :
End(FCDg
X) → End(FCDg
Y ) induces a bijection X → Y , therefore obvi-
ously we obtain FCD
g
X
∼= FCD
g
Y .
We recall that an automorphism Φ : End(FCDg
X) → End(FCDg
X) is
quasi-inner if there exists α ∈ S(FCDg
X) such that βΦ = α−1βα for all
β ∈ End(FCDg
X).
At the end we consider the automorphism group of End(FCDg
X).
Yu. V. Zhuchok 323
Theorem 3. Let X be an arbitrary set with |X| > 2. Then
(i) all automorphisms of End(FCDg
X) are quasi-inner;
(ii) the automorphism group Aut(End(FCDg
X)) is isomorphic to the
direct product S(X) × C2.
Proof. (i) Let X = Y in Theorem 2, then it will be the part of Theorem 3.
It is not hard to see that every automorphism Φ of End(FCDg
X) is either
an inner automorphism or the product of a mirror automorphism and an
inner automorphism. Namely, we have Φ = Eϕ or Φ = Φ1Eϕ for a suitable
bijection ϕ : X → X. It means that all automorphisms of End(FCDg
X)
are quasi-inner.
(ii) It is clear that the automorphism group {Φ0,Φ1} of End(FCDg
X) is
isomorphic to C2. Define a mapping ζ : Aut(End(FCDg
X)) → S(X) × C2
as follows:
Φζ =
{
(ϕ,Φ0), Φ = Eϕ,
(ϕ,Φ1), Φ = Φ1Eϕ
for all Φ ∈ Aut(End(FCDg
X)).
It is easy to see that ζ is a bijection. Since for all ϕ,ψ ∈ S(X) and
f ∈ End(FCDg
X),
f(EϕEψ) = (ε−1
ϕ fεϕ)Eψ = (εϕεψ)−1f(εϕεψ)
= ε−1
ϕψfεϕψ = fEϕψ
and
f(EϕΦ1) = (ε−1
ϕ fεϕ)Φ1 = (ε∗
idX
ε−1
ϕ )f(εϕε
∗
idX
)
= (ε−1
ϕ ε∗
idX
)f(ε∗
idX
εϕ) = (ε∗
idX
fε∗
idX
)Eϕ = f(Φ1Eϕ),
we obtain EϕEψ = Eϕψ and EϕΦ1 = Φ1Eϕ.
The immediate check shows that ζ is a homomorphism.
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Contact information
Yurii V. Zhuchok Kyiv National Taras Shevchenko University,
Faculty of Mechanics and Mathematics, 64,
Volodymyrska Street, Kyiv, Ukraine, 01601
E-Mail(s): zhuchok.yu@gmail.com
Received by the editors: 12.04.2016
and in final form 30.05.2016.
|
| id | nasplib_isofts_kiev_ua-123456789-155257 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T15:22:31Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Zhuchok, Y.V. 2019-06-16T14:53:31Z 2019-06-16T14:53:31Z 2016 Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid / Y.V. Zhuchok // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 309-324. — Бібліогр.: 19 назв. — англ. 1726-3255 2010 MSC:08A35, 08B20, 17A30. https://nasplib.isofts.kiev.ua/handle/123456789/155257 We determine all isomorphisms between the endomorphism semigroups of free commutative g-dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative g-dimonoid are quasi-inner. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid Article published earlier |
| spellingShingle | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid Zhuchok, Y.V. |
| title | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid |
| title_full | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid |
| title_fullStr | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid |
| title_full_unstemmed | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid |
| title_short | Automorphisms of the endomorphism semigroup of a free commutative g-dimonoid |
| title_sort | automorphisms of the endomorphism semigroup of a free commutative g-dimonoid |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155257 |
| work_keys_str_mv | AT zhuchokyv automorphismsoftheendomorphismsemigroupofafreecommutativegdimonoid |