Construction of free g-dimonoids
In this paper, the concept of a g-dimonoid is introduced and the construction of a free g-dimonoid is described. (A g-dimonoid is a duplex satisfying two additional identities.)
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Construction of free g-dimonoids / Yu. Movsisyan, S. Davidov, M. Safaryan // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 138–148. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1533512019-06-16T01:27:10Z Construction of free g-dimonoids Movsisyan, Yu. Davidov, S. Safaryan, M. In this paper, the concept of a g-dimonoid is introduced and the construction of a free g-dimonoid is described. (A g-dimonoid is a duplex satisfying two additional identities.) 2014 Article Construction of free g-dimonoids / Yu. Movsisyan, S. Davidov, M. Safaryan // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 138–148. — Бібліогр.: 16 назв. — англ. 1726-3255 2010 MSC:03C05, 08B20, 20M05. http://dspace.nbuv.gov.ua/handle/123456789/153351 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper, the concept of a g-dimonoid is introduced and the construction of a free g-dimonoid is described. (A g-dimonoid is a duplex satisfying two additional identities.) |
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Movsisyan, Yu. Davidov, S. Safaryan, M. |
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Movsisyan, Yu. Davidov, S. Safaryan, M. Construction of free g-dimonoids Algebra and Discrete Mathematics |
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Movsisyan, Yu. Davidov, S. Safaryan, M. |
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Movsisyan, Yu. |
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Construction of free g-dimonoids |
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Construction of free g-dimonoids |
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Construction of free g-dimonoids |
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Construction of free g-dimonoids |
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Construction of free g-dimonoids |
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construction of free g-dimonoids |
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Інститут прикладної математики і механіки НАН України |
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2014 |
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Construction of free g-dimonoids / Yu. Movsisyan, S. Davidov, M. Safaryan // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 138–148. — Бібліогр.: 16 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT movsisyanyu constructionoffreegdimonoids AT davidovs constructionoffreegdimonoids AT safaryanm constructionoffreegdimonoids |
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2025-07-14T04:34:32Z |
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1837595543623172096 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 18 (2014). Number 1, pp. 138 – 148
© Journal “Algebra and Discrete Mathematics”
Construction of free g-dimonoids
Yuri Movsisyan, Sergey Davidov and Mher Safaryan
Communicated by A. V. Zhuchok
Abstract. In this paper, the concept of a g-dimonoid is
introduced and the construction of a free g-dimonoid is described.
(A g-dimonoid is a duplex satisfying two additional identities.)
Introduction
The concepts of a dimonoid and a dialgebra were introduced by
Loday [1]. Dimonoids are a tool to study Leibniz algebras [1]. A dimonoid
is a set with two binary associative operations satisfying the additional
identities. A dialgebra is a linear analogue of the dimonoid. One of the first
results about dimonoids is the description of the free dimonoid generated
by the given set. Using properties of the free dimonoid, the free dialgebras
were described and the cohomologies of dialgebras were studied in [1].
In [2], using the concept of a dimonoid, the concept of a unileteral diring
was introduced and the basic properties of dirings were studied. In [4, 5]
free dimonoids and free commutative dimonoids were described. In [6]
the concept of a duplex (which generalizes the concept of a dimonoid)
and the construction of a free duplex were introduced. A duplex is a
set equipped with two associative operations. In [7, 8] the concept of a
Boolean bisemigroup (which generalizes the concept of a Boolean algebra)
was introduced and a Stone-type representation theorem was proved
(cf. [9–13]).
2010 MSC: 03C05, 08B20, 20M05.
Key words and phrases: dimonoid, g-dimonoid, free algebra, canonical form.
Yu. Movsisyan, S. Davidov, M. Safaryan 139
The concept of a 0-dialgebra was introduced in [14]. The 0-dialgebra
under the field F is a vector space under F with two binary operations,
⊣ and ⊢, such that the following two identities are satisfied:
(x ⊣ y) ⊢ z = (x ⊢ y) ⊢ z, z ⊣ (x ⊢ y) = z ⊣ (x ⊣ y).
In this paper the concept of a g-dimonoid (a generalized dimonoid)
is introduced and the construction of a free g-dimonoid is described.
(A g-dimonoid is a duplex satisfying two additional identities.)
1. Auxiliary results
Definition 1. An algebra (A; ⊣,⊢) is called a g-dimonoid if it satisfies
the following identities:
(A1) (x ⊣ y) ⊣ z = x ⊣ (y ⊣ z),
(A2) (x ⊣ y) ⊣ z = x ⊣ (y ⊢ z),
(A3) (x ⊣ y) ⊢ z = x ⊢ (y ⊢ z),
(A4) (x ⊢ y) ⊢ z = x ⊢ (y ⊢ z).
An g-dimonoid (A; ⊣,⊢) is called a dimonoid, if it satisfies the following
additional identity:
(x ⊢ y) ⊣ z = x ⊢ (y ⊣ z).
Let us give an example of a g-dimonoid, which is not a dimonoid ([3]).
Let X be an arbitrary nonempty set, |X| > 1 and let X∗ be the set of all
finite nonempty words in the alphabet X. Denote the first (respectively,
the last) letter of a word ω ∈ X∗ by ω(0) (respectively, by ω(1)). Define
the following operations ⊣,⊢ on X∗ by
ω ⊣ u = ω(0), ω ⊢ u = u(1)
for all ω, u ∈ X∗. It is easy to check that the binary algebra (X∗,⊣,⊢) is
a g-dimonoid, but is not a dimonoid.
Definition 2. A map f : A1 → A2 between g-dimonoids A1 and A2 is
called a homomorphism if f(x ⊣ y) = f(x) ⊣ f(y) and f(x ⊢ y) = f(x) ⊢
f(y) for all x, y ∈ A1. A bijective homomorphism between g-dimonoids is
called an isomorphism.
140 Free g-dimonoids
A g-dimonoid F is called a free g-dimonoid if there exists a subset
X ⊆ F such that F is generated by X and for any g-dimonoid D and for
any map f : X → D there exists a unique homomorphism of g-dimonoids
g : F → D such that g(x) = f(x) for all x ∈ X. If this holds, then we
say that F is a free g-dimonoid with the system of free generators X
([15,16]).
We recall the concept of a term of a g-dimonoid A by the following:
any element x ∈ A is a term of A; if t1, t2 are terms of A, then t1 ⊣ t2
and t1 ⊢ t2 also are the terms of A; and there are no other terms. By
t(x1, . . . , xn) we mean a term with the elements x1, . . . , xn each of which
meets once and in the mentioned order.
Lemma 1. Let (A; ⊣,⊢) be a g-dimonoid. Then for any term t =
t(x1, . . . , xn), x ∈ A the following equalities hold:
(i) x ⊣ t = x ⊣ x1 ⊣ . . . ⊣ xn,
(ii) t ⊢ x = x1 ⊢ . . . ⊢ xn ⊢ x.
Proof. Prove (i) by induction on n. If n = 1, 2 then the statement is
obvious. Let it be true for n < k, where k > 2. For t = t1 ∗ t2, where
∗ ∈ {⊣,⊢}, t1 = t1(x1, . . . , xk1
), t2 = t2(xk1+1, . . . , xk), 0 < k1 < k, let us
consider the following cases. If ∗ =⊣, then
x ⊣ t = x ⊣ (t1 ⊣ t2)
(A1)
= (x ⊣ t1) ⊣ t2
= (x ⊣ x1 ⊣ . . . ⊣ xk1
) ⊣ t2
= (x ⊣ x1 ⊣ . . . ⊣ xk1
) ⊣ xk1+1 ⊣ . . . ⊣ xk
= x ⊣ x1 ⊣ . . . ⊣ xk.
If ∗ =⊢, then
x ⊣ t = x ⊣ (t1 ⊢ t2)
(A2)
= (x ⊣ t1) ⊣ t2
= (x ⊣ x1 ⊣ . . . ⊣ xk1
) ⊣ t2
= (x ⊣ x1 ⊣ . . . ⊣ xk1
) ⊣ xk1+1 ⊣ . . . ⊣ xk
= x ⊣ x1 ⊣ . . . ⊣ xk.
Hence, (i) holds for n = k. (ii) is proved analogously.
Let e be an arbitrary symbol; introduce the following sets:
I1 = {e}, In = {0, 1}n−1 =
Yu. Movsisyan, S. Davidov, M. Safaryan 141
= {ε = (ε1, . . . , εn−1) : εk ∈ {0, 1}, k = 1, n− 1}, n > 1,
I =
⋃
n>1
In.
If l = 0 then the sequence ε1, . . . , εl without brackets we consider empty,
and the sequence (ε1, . . . , εl) with brackets we consider e. For example, if
n = 1, then the sequence (ε1, . . . , εn−1,
m
︷ ︸︸ ︷
1, 1, . . . , 1) is (
m
︷ ︸︸ ︷
1, 1, . . . , 1).
Definition 3. Let (A; ⊣,⊢) be a g-dimonoid. For any x1, x2, . . . , xn ∈ A
and for any ε ∈ In define the element
x1x2 . . . xnε ∈ A (⋆)
by induction on n > 1 in the following way:
1. x1e = x1,
2. x1x2 . . . xn(ε1, ε2, . . . , εn−2, 0) = x1 ⊢ x2 . . . xn(ε1, ε2, . . . , εn−2),
x1 . . . xn−1xn(ε1, ε2, . . . , εn−2, 1)=x1 . . . xn−1(ε1, ε2, . . . , εn−2)⊣ xn,
if n > 1.
In particular, if ε = (
n−1
︷ ︸︸ ︷
1, 1, . . . , 1), then x1 . . . xnε = x1 ⊣ · · · ⊣ xn;
if ε = (
n−1
︷ ︸︸ ︷
0, 0, . . . , 0), then x1 . . . xnε = x1 ⊢ · · · ⊢ xn; and if n = 1, then
x1x2 . . . xn(ε1, . . . , εn−1) = x1e (according to above agreement).
Lemma 2. In any g-dimonoid (A; ⊣,⊢) the following identities hold:
x1x2 . . . xn(ε1, . . . , εn−1) ⊣ y1y2 . . . ym(θ1, . . . , θm−1)
= x1x2 . . . xny1y2 . . . ym(ε1, . . . , εn−1,
m
︷ ︸︸ ︷
1, 1, . . . , 1),
x1x2 . . . xn(ε1, . . . , εn−1) ⊢ y1y2 . . . ym(θ1, . . . , θm−1)
= x1x2 . . . xny1y2 . . . ym(θ1, . . . , θm−1,
n
︷ ︸︸ ︷
0, 0, . . . , 0).
Proof.
x1x2 . . .xn(ε1, . . . , εn−1) ⊣ y1y2 . . . ym(θ1, . . . , θm−1)
(i)
= x1x2 . . . xn(ε1, . . . , εn−1) ⊣ y1 ⊣ y2 ⊣ · · · ⊣ ym
= x1x2 . . . xny1y2 . . . ym(ε1, . . . , εn−1,
m
︷ ︸︸ ︷
1, 1, . . . , 1),
142 Free g-dimonoids
x1x2 . . .xn(ε1, . . . , εn−1) ⊢ y1y2 . . . ym(θ1, . . . , θm−1)
(ii)
= x1 ⊢ x2 ⊢ · · · ⊢ xn ⊢ y1y2 . . . ym(θ1, . . . , θm−1)
= x1x2 . . . xny1y2 . . . ym(θ1, . . . , θm−1,
n
︷ ︸︸ ︷
0, 0, . . . , 0).
All terms of a given g-dimonoid can be described by the elements of
the form (⋆); namely, we can show that any term of a given g-dimonoid
can be reduced to (⋆).
Theorem 1. Let t = t(x1, . . . , xn) be a term of a given g-dimonoid. Then
there is such ε ∈ In that
t = x1x2 . . . xnε.
Proof. Prove the theorem by induction on n. For n = 1, 2, the statement is
obvious. Let it be true for n < k, where k > 2. Suppose t = t1(x1, . . . , xk1
)∗
t2(xk1+1, . . . , xk), where ∗ ∈ {⊣,⊢}, 0 < k1 < k.
Since k1 < k, k − k1 < k, then for the terms t1(x1, . . . , xk1
) and
t2(xk1+1, . . . , xk) there are such ε = (ε1, . . . , εk1−1), θ = (θ1, . . . , θk−k1−1)
that
t1(x1, . . . , xk1
) = x1x2 . . . xk1
ε,
t2(xk1+1, . . . , xk) = xk1+1xk1+2 . . . xkθ.
If ∗ =⊣, then
t = t1(x1, . . . , xk1
) ⊣ t2(xk1+1, . . . , xk)
(i)
= t1(x1, . . . , xk1
) ⊣ xk1+1 ⊣ · · · ⊣ xk
= x1 . . . xk1
(ε1, . . . , εk1−1) ⊣ xk1+1 ⊣ · · · ⊣ xk
2.
= x1 . . . xk(ε1, . . . , εk1−1,
k−k1
︷ ︸︸ ︷
1, 1, . . . , 1).
If ∗ =⊢, then
t = t1(x1, . . . , xk1
) ⊢ t2(xk1+1, . . . , xk)
(ii)
= x1 ⊢ . . . xk1
⊢ t2(xk1+1, . . . , xk)
= x1 ⊢ · · · ⊢ xk1
⊢ xk1+1 . . . xk(θ1, . . . , θk−k1−1)
2.
= x1 . . . xk(θ1, . . . , θk−k1−1,
k1
︷ ︸︸ ︷
0, 0, . . . , 0).
Therefore the theorem is valid for n = k.
Yu. Movsisyan, S. Davidov, M. Safaryan 143
By virtue of Theorem 1, any term of a given g-dimonoid can be reduced
to the form (⋆) which we call the canonical form of a given term. For
example, for the term ((x1 ⊣ x2) ⊢ (x3 ⊣ x4)) ⊣ (x5 ⊢ x6) the canonical
form is:
((x1 ⊣ x2) ⊢ (x3 ⊣ x4)) ⊣ (x5 ⊢ x6)
= (x1x2(1) ⊢ x3x4(1)) ⊣ x5x6(0)
= x1x2x3x4(1, 0, 0) ⊣ x5x6(0)
= x1x2x3x4x5x6(1, 0, 0, 1, 1).
Define operations ⊣ and ⊢ on I in the following way:
(ε1, . . . , εn−1) ⊣ (θ1, . . . , θm−1) = (ε1, . . . , εn−1,
m
︷ ︸︸ ︷
1, 1, . . . , 1),
(ε1, . . . , εn−1) ⊢ (θ1, . . . , θm−1) = (θ1, . . . , θm−1,
n
︷ ︸︸ ︷
0, 0, . . . , 0).
Lemma 3. The algebra (I; ⊣,⊢) is a g-dimonoid.
Proof. The axioms (A1), (A2), (A3), (A4) are checked directly.
Note that (I; ⊣,⊢) is not a dimonoid.
Lemma 4. In the algebra (I; ⊣,⊢) we have
ee . . . e
︸ ︷︷ ︸
n
ε = ε
for any ε ∈ In.
Proof. Prove by induction on n. If n = 1, 2, then the statement is clear.
Let it be true for n = k, k > 1 and let ε ∈ Ik+1.
If ε = (ε1, . . . , εk−1, 0) = e ⊢ ε′, where ε′ = (ε1, . . . , εk−1), then
ee . . . e
︸ ︷︷ ︸
k+1
ε = e ⊢ ee . . . e
︸ ︷︷ ︸
k
ε′ = e ⊢ ε′ = ε.
If ε = (ε1, . . . , εk−1, 1) = ε′ ⊣ e, where ε′ = (ε1, . . . , εk−1), then
ee . . . e
︸ ︷︷ ︸
k+1
ε = ee . . . e
︸ ︷︷ ︸
k
ε′ ⊣ e = ε′ ⊣ e = ε.
From definitions of operations ⊣,⊢ it follows:
144 Free g-dimonoids
Lemma 5. If α ∈ In, θ ∈ Im, then α ⊣ θ, α ⊢ θ ∈ In+m.
Now we can prove the uniqueness of the canonical form for the g-
dimonoid (I; ⊣,⊢) .
Theorem 2. The canonical form is unique for any term of the g-dimonoid
(I; ⊣,⊢).
Proof. Assume that for some term t there are two canonical forms:
x1x2 . . . xnε = y1y2 . . . ymθ (⋆⋆)
for some ε ∈ In and θ ∈ Im.
Replacing all variables by e ∈ I, we get:
ee . . . e
︸ ︷︷ ︸
n
ε = ee . . . e
︸ ︷︷ ︸
m
θ,
whence and from Lemma 4 it follows ε = θ. Hence, (⋆⋆) has the following
form:
x1x2 . . . xnε = y1y2 . . . ynε.
Let the variables xk and yk be different for some 1 6 k 6 n. In the last
equality, replacing all variables except yk by e and replacing the variable
yk by (1) we get:
ε = e . . . e(1)e . . . eε,
which is contradiction, because ε ∈ In and e . . . e(1)e . . . eε ∈ In+1 due
to Lemma 5. Therefore, the canonical forms x1x2 . . . xnε and y1y2 . . . ymθ
graphically coincide, which proves the uniqueness of the canonical form
of the term t.
2. Free g-dimonoids
Let us turn to the construction of a free g-dimonoid. Let X be an
arbitrary and nonempty set. Denote:
Yn = Xn × In, n ∈ N ,
where Xn = X ×X × . . .×X
︸ ︷︷ ︸
n
= {(x1, x2, . . . , xn) : xk ∈ X, k = 1, n},
G(X) =
⋃
n>1
Yn.
Yu. Movsisyan, S. Davidov, M. Safaryan 145
For convenience the elements of G(X) are denoted by (x1, x2, . . . , xn)ε
instead of ((x1, x2, . . . , xn), ε), where ε ∈ In; we consider the sets X × I1
and X being the same, that is, we identify the symbol x ∈ X with the
element xe ∈ G(X). Define operations ⊣,⊢ on G(X) in the following way:
(x1, x2, . . . , xk)ε ⊣ (xk+1, xk+2, . . . , xl)θ = (x1, x2, . . . , xl)(ε ⊣ θ),
(x1, x2, . . . , xk)ε ⊢ (xk+1, xk+2, . . . , xl)θ = (x1, x2, . . . , xl)(ε ⊢ θ).
Theorem 3. The binary algebra (G(X); ⊣,⊢) is a free g-dimonoid with
the system of free generators X.
Proof. The fact that the algebra G(X) is a g-dimonoid follows from
Lemma 3. From the definition of operations ⊣,⊢ it follows that G(X) is gen-
erated by X. Namely, if (x1, . . . , xn)ε ∈ G(X), where ε = (ε′, 1) = ε′ ⊣ e,
then
(x1, . . . , xn)ε = (x1, . . . , xn)(ε′ ⊣ e)
= (x1, . . . , xn−1)ε′ ⊣ xne = (x1, . . . , xn−1)ε′ ⊣ xn.
Analogously, if ε = (ε′, 0) = e ⊢ ε′, then
(x1, . . . , xn)ε = (x1, . . . , xn)(e ⊢ ε′)
= x1e ⊢ (x2, . . . , xn)ε′ = x1 ⊢ (x2, . . . , xn)ε′.
Hence, using induction, we can prove that any element (x1, . . . , xn)ε ∈
G(X) can be written as a word in the alphabet x1, . . . , xn and ⊣,⊢.
Let us prove that it is a free g-dimonoid. Let (D; ⊣,⊢) be an arbitrary
g-dimonoid and ϕ : X → D be an arbitrary map. Define the map
ψ0 : (G(X); ⊣,⊢) → (D; ⊣,⊢) in the following way:
ψ0((x1, x2, . . . , xn)ε) = ϕ(x1)ϕ(x2) . . . ϕ(xn)ε.
The map ψ0 matches with ϕ on X:
ψ0(x) = ψ0(xe) = ϕ(x)e = ϕ(x), x ∈ X.
Show that ψ0 is a homomorphism. From Lemmas 2 and 3 it follows:
ψ0((x1,x2, . . . , xk)ε ⊣ (xk+1, xk+2, . . . , xl)θ)
= ψ0((x1, x2, . . . , xl)(ε ⊣ θ)) = ϕ(x1)ϕ(x2) . . . ϕ(xl)(ε ⊣ θ)
= ϕ(x1)ϕ(x2) . . . ϕ(xk)ε ⊣ ϕ(xk+1)ϕ(xk+2) . . . ϕ(xl)θ
= ψ0((x1, x2, . . . , xk)ε) ⊣ ψ0((xk+1, xk+2, . . . , xl)θ),
146 Free g-dimonoids
ψ0((x1,x2, . . . , xk)ε ⊢ (xk+1, xk+2, . . . , xl)θ)
= ψ0((x1, x2, . . . , xl)(ε ⊢ θ)) = ϕ(x1)ϕ(x2) . . . ϕ(xl)(ε ⊢ θ)
= ϕ(x1)ϕ(x2) . . . ϕ(xk)ε ⊢ ϕ(xk+1)ϕ(xk+2) . . . ϕ(xl)θ
= ψ0((x1, x2, . . . , xk)ε) ⊢ ψ0((xk+1, xk+2, . . . , xl)θ).
Prove that if ψ : (G(X); ⊣,⊢) → (D; ⊣,⊢) is a homomorphism coincid-
ing with ϕ on X, then ψ ≡ ψ0.
We have that the maps ψ and ψ0 match on Y1. Let they match on
the sets Y1, . . . , Yn. Then
ψ((x1,x2, . . . , xn+1)(ε1, . . . , εn−1, 0))
= ψ(x1 ⊢ (x2, . . . , xn+1)(ε1, . . . , εn−1))
= ψ(x1) ⊢ ψ((x2, . . . , xn+1)(ε1, . . . , εn−1))
= ψ0(x1) ⊢ ψ0((x2, . . . , xn+1)(ε1, . . . , εn−1))
= ψ0(x1 ⊢ (x2, . . . , xn+1)(ε1, . . . , εn−1))
= ψ0((x1, x2, . . . , xn+1)(ε1, . . . , εn−1, 0)),
ψ((x1,x2, . . . , xn+1)(ε1, . . . , εn−1, 1))
= ψ((x1, x2, . . . , xn)(ε1, . . . , εn−1) ⊣ xn+1)
= ψ((x1, x2, . . . , xn)(ε1, . . . , εn−1)) ⊣ ψ(xn+1)
= ψ0((x1, x2, . . . , xn)(ε1, . . . , εn−1)) ⊣ ψ0(xn+1)
= ψ0((x1, x2, . . . , xn)(ε1, . . . , εn−1) ⊣ xn+1)
= ψ0((x1, x2, . . . , xn+1)(ε1, . . . , εn−1, 1)).
Hence
ψ((x1, x2, . . . , xn+1)(ε1, . . . , εn)) = ψ0((x1, x2, . . . , xn+1)(ε1, . . . , εn))
for any (x1, x2, . . . , xn+1)(ε1, . . . , εn) ∈ Yn+1. So, the maps ψ and ψ0
coincide on Yn+1. Therefore, ψ ≡ ψ0.
Let us give another description of a free g-dimonoid. Let F [X] be
the free semigroup with the system of free generators X. For any word
ω ∈ F [X] we denote the length of ω by |ω|. Define operations ⊣,⊢ on the
set
FG = {(ω, ε) : ω ∈ F [X], ε ∈ I |ω|}
in the following way:
(ω1, ε) ⊣ (ω2, θ) = (ω1ω2, ε ⊣ θ),
Yu. Movsisyan, S. Davidov, M. Safaryan 147
(ω1, ε) ⊢ (ω2, θ) = (ω1ω2, ε ⊢ θ),
where (ω1, ε), (ω2, θ) ∈ FG. It is easy to verify that the binary algebra
(FG; ⊣,⊢) is a g-dimonoid, which we denote by FG[X].
Theorem 4. The g-dimonoids (G(X); ⊣,⊢) and FG[X] are isomorphic.
Proof. Define the map σ : G(X) → FG[X] in the following way:
σ : (x1, x2, . . . , xk)ε 7→ (x1x2 . . . xk, ε), (x1, x2, . . . , xk)ε ∈ G(X).
From the definition it follows that σ is a bijection and a homomorphism.
Hence, the binary algebra FG[X] is also a free g-dimonoid with the
system of free generators X.
Lemma 6. The g-dimonoid (I; ⊣,⊢) is a free g-dimonoid which is iso-
morphic to the g-dimonoid FG[X], where |X| = 1.
Proof. Let X = {a}. Define the map
τ : (I; ⊣,⊢) → FG[X]
in the following way: τ(ε) = (an, ε) ∈ FG[X] for all ε ∈ In, n > 1. From
the definition it follows that the map τ is a bijection. Prove that it is a
homomorphism. Indeed, by Lemma 5, ε ⊣ θ, ε ⊢ θ ∈ In+m for any ε ∈ In,
θ ∈ Im, hence
τ(ε ⊣ θ) = (an+m, ε ⊣ θ) = (an, ε) ⊣ (am, θ) = τ(ε) ⊣ τ(θ),
τ(ε ⊢ θ) = (an+m, ε ⊢ θ) = (an, ε) ⊢ (am, θ) = τ(ε) ⊢ τ(θ).
Therefore, τ is an isomorphism.
Thus, the free g-dimonoid of rank 1 coincides with the g-dimonoid
(I; ⊣,⊢) up to isomorphism.
Acknowledgement
Thanks to the referees for useful remarks.
148 Free g-dimonoids
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Contact information
Yu. Movsisyan,
S. Davidov,
M. Safaryan
Yerevan State University, Alex Manoogian 1,
Yerevan 0025, Armenia
E-Mail: yurimovsisyan@yahoo.com,
davidov@ysu.am,
mher.safaryan@gmail.com
URL: www.ysu.am
Received by the editors: 20.06.2013
and in final form 01.04.2014.
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