Free (ℓr, rr)-dibands
We prove that varieties of (ℓr, rr)-dibands and (ℓn, rn)-dibands coincide and describe the structure of free (ℓr, rr)-dibands. We also show that operations of an idempotent dimonoid with left (right) regular bands coincide, construct a new class of dimonoids and for such dimonoids give an example of...
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nasplib_isofts_kiev_ua-123456789-1522972025-02-10T01:06:43Z Free (ℓr, rr)-dibands Zhuchok, A.V. We prove that varieties of (ℓr, rr)-dibands and (ℓn, rn)-dibands coincide and describe the structure of free (ℓr, rr)-dibands. We also show that operations of an idempotent dimonoid with left (right) regular bands coincide, construct a new class of dimonoids and for such dimonoids give an example of a semiretraction. 2013 Article Free (ℓr, rr)-dibands / A. V. Zhuchok // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 2. — С. 295–304. — Бібліогр.: 12 назв. — англ. 1726-3255 2010 MSC:08B20, 20M10, 20M50, 17A30, 17A32. https://nasplib.isofts.kiev.ua/handle/123456789/152297 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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We prove that varieties of (ℓr, rr)-dibands and (ℓn, rn)-dibands coincide and describe the structure of free (ℓr, rr)-dibands. We also show that operations of an idempotent dimonoid with left (right) regular bands coincide, construct a new class of dimonoids and for such dimonoids give an example of a semiretraction. |
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Free (ℓr, rr)-dibands / A. V. Zhuchok // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 2. — С. 295–304. — Бібліогр.: 12 назв. — англ. |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 15 (2013). Number 2. pp. 295 – 304
c© Journal “Algebra and Discrete Mathematics”
Free (ℓr, rr)-dibands
Anatolii V. Zhuchok
Communicated by V. I. Sushchansky
Abstract. We prove that varieties of (ℓr, rr)-dibands and
(ℓn, rn)-dibands coincide and describe the structure of free (ℓr, rr)-
dibands. We also show that operations of an idempotent dimonoid
with left (right) regular bands coincide, construct a new class of
dimonoids and for such dimonoids give an example of a semiretrac-
tion.
1. Introduction
The notions of a dialgebra and a dimonoid were introduced
by J.-L. Loday [1]. For further details and background see [1], [4].
Varieties of algebras are classes of algebras which can be given with the
help of a set of identities. The special position of varieties in general algebra
is defined by a circumstance that many structures in classical algebra
such as groups, rings, lattices, Boolean algebras and etc. form a variety.
In our time the most deep results and problems of the variety theory are
connected with the investigation of concrete varieties and construction
of relatively free algebras. A variety of dimonoids is a class of dimonoids
which is characterized by some five identities. By the language of these
identities we can express many properties of dimonoids and their classes.
The first result about dimonoids is a Loday’s description [1] of an absolute
free dimonoid generated by a given set. Free commutative dimonoids and
free rectangular dimonoids were construcred in [5] and [6] respectively. Free
2010 MSC: 08B20, 20M10, 20M50, 17A30, 17A32.
Key words and phrases: left (right) regular band, (ℓr, rr)-diband, diband of
subdimonoids, dimonoid, semigroup.
296 Free (ℓr, rr)-dibands
normal dibands and some other relatively free dimonoids were constructed
in [7]. T. Pirashvili [2] considered sets with two associative operations
(so-called duplexes) and constructed a free duplex. In Pirashvili’s paper
it is also considered duplexes with some additional conditions. For such
duplexes the structure of free objects is known (see [2]).
In this paper we find necessary and sufficient conditions under which
an arbitrary dimonoid is a (ℓr, rr)-diband and, as a consequence, es-
tablish that the variety of (ℓr, rr)-dibands coincides with the variety of
(ℓn, rn)-dibands. In terms of dibands of subdimonoids we also describe
the structure of free (ℓr, rr)-dibands. It turns out that operations of an
idempotent dimonoid with left (right) regular bands coincide and it is a
left (right) regular band. Moreover, we construct a new class of dimonoids
and for such dimonoids give an example of a semiretraction.
We refer to [6] and [7] for the terminology and notations.
2. A new class of dimonoids
In this section we construct a new class of dimonoids via semire-
tractions [8] and for dimonoids from this class give an example of a
semiretraction.
A transformation τ of a dimonoid (D, ⊣, ⊢) is called a left semiretrac-
tion, if
(x ⊣ y)τ = (xτ ⊣ y)τ, (1)
(x ⊢ y)τ = (xτ ⊢ y)τ (2)
for all x, y ∈ D. If instead of (1), (2) the identities
(x ⊣ y)τ = (x ⊣ yτ)τ, (3)
(x ⊢ y)τ = (x ⊢ yτ)τ (4)
hold, then we say about a right semiretraction. If for τ the identities
(1) − (4) hold, then τ is called a (symmetric) semiretraction of (D, ⊣, ⊢).
If operations of a dimonoid coincide, then from the definition of a left
(right, symmetric) semiretraction of a dimonoid we obtain the notion of a
left (right, symmetric) semiretraction of a semigroup (see [3], [9]).
For any transformation π of a dimonoid (D, ⊣, ⊢) let
∇π = {(x, y) ∈ D × D | xπ = yπ}.
The following statement gives general characteristic of (symmetric)
semiretractions.
A. V. Zhuchok 297
Proposition 1 ([8], Sect. 3.2, Proposition). For an idempotent transfor-
mation π of a dimonoid (D, ⊣, ⊢) the following statements are equivalent:
1. π is a (symmetric) semiretraction;
2. π is a left semiretraction and ∇π is a congruence on (D, ⊣, ⊢);
3. π is a right semiretraction and ∇π is a congruence on (D, ⊣, ⊢);
4. for all x, y ∈ D the identities
(x ⊣ y)π = (xπ ⊣ yπ)π, (x ⊢ y)π = (xπ ⊢ yπ)π
hold.
Thus, the problem of the description of congruences on dimonoids from
a given class is reduced to the description of semiretractons of dimonoids.
That is, if we know the action of a semiretraction π on a dimonoid, then
we can construct the unique congruence ∇π which corresponds to π.
Conversely, if we know the structure of a congruence on a dimonoid, then
we can give the class of semiretractions π such that relations ∇π coincide
with a given congruence.
Examples of semiretractions of dimonoids can be found in [8].
Let S be an arbitrary semigroup and τ be an idempotent semiretraction
of S. Define operations ⊣ and ⊢ on S × S by
(a, b) ⊣ (c, d) = (a, (bcd)τ) ,
(a, b) ⊢ (c, d) = ((abc)τ, d)
for all (a, b) , (c, d) ∈ S × S. The algebra (S × S, ⊣, ⊢) will be denoted
by S[τ ].
The following statement gives the opportunity to construct new di-
monoids with the help of semiretractions.
Proposition 2. For every idempotent semiretraction τ of a semigroup
S the algebra S[τ ] is a dimonoid.
Proof. For all (a, b) , (c, d), (x, y) ∈ S × S we obtain
((a, b) ⊣ (c, d)) ⊣ (x, y) = (a, (bcd)τ) ⊣ (x, y) =
= (a, ((bcd)τ xy)τ) = (a, (bcd xy)τ),
(a, b) ⊣ ( (c, d) ⊣ (x, y)) = (a, b) ⊣ (c, (dxy)τ) =
298 Free (ℓr, rr)-dibands
= (a, (bc(dxy)τ )τ) = (a, (bcd xy)τ),
(a, b) ⊣ ( (c, d) ⊢ (x, y)) = (a, b) ⊣ ((cdx)τ, y) =
= (a, (b(cdx)τ y )τ) = (a, ((b(cdx)τ)τ y )τ) =
= (a, ((bcdx)τ y )τ) = (a, (bcd xy)τ),
((a, b) ⊢ (c, d)) ⊢ (x, y) = ((abc)τ, d) ⊢ (x, y) =
= (((abc)τ dx)τ, y) = ((abcdx)τ, y),
(a, b) ⊢ ((c, d) ⊢ (x, y)) = (a, b) ⊢ ((cdx)τ, y) =
= ((ab(cdx)τ)τ , y) = ((abcdx)τ, y),
((a, b) ⊣ (c, d)) ⊢ (x, y) = (a, (bcd)τ) ⊢ (x, y) =
= ((a(bcd)τ x)τ, y) = (((a(bcd)τ)τ x)τ, y) =
= (((abcd)τ x)τ, y) = ((abcdx)τ, y),
((a, b) ⊢ (c, d)) ⊣ (x, y) =
= ((abc)τ, d) ⊣ (x, y) = ((abc)τ, (dxy)τ),
(a, b) ⊢ ( (c, d) ⊣ (x, y)) =
= (a, b) ⊢ (c, (dxy)τ) = ((abc)τ, (dxy)τ)
according to Proposition 1. Comparing these expressions, we conclude
that S[τ ] is a dimonoid.
Observe that a dimonoid which was constructed by J.-L. Loday in [1]
is a particular case of the dimonoid S[τ ]. Indeed, if S is a monoid and τ
is an identity transformation of S, then S[τ ] coincides with a dimonoid
from [1], see p.12.
We finish this section with the consideration of a semiretraction of
the dimonoid S [τ ].
Let π : S [τ ] → S [τ ] : (a, b) 7→ (aτ, bτ).
Lemma 1. π is a semiretraction of the dimonoid S [τ ].
A. V. Zhuchok 299
Proof. For all (a, b), (c, d) ∈ S [τ ] we have
((a, b) ⊣ (c, d)) π = (a, (bcd) τ) π = (aτ, (bcd) τ) =
= (aτ, ((bc) τdτ) τ) = (aτ, ((bτcτ) τdτ) τ) =
= (aτ, (bτcτdτ) τ) = (aτ, (bτcτdτ) τ) π =
= ((aτ, bτ) ⊣ (cτ, dτ)) π = ((a, b) π⊣ (c, d) π) π,
((a, b) ⊢ (c, d)) π = ((abc) τ, d) π = ((abc) τ, dτ) =
= (((ab) τcτ) τ, dτ) = (((aτbτ) τcτ) τ, dτ) =
= ((aτbτcτ) τ, dτ) = ((aτbτcτ) τ, dτ) π =
= ((aτ, bτ) ⊢ (cτ, dτ)) π = ((a, b) π⊢ (c, d) π) π.
Hence, by Proposition 1, π is a semiretraction.
3. Dimonoids and left (right) regular bands
In this section we show that operations of an idempotent dimonoid
(D, ⊣, ⊢) with a left (respectively, right) regular band (D, ⊢) (respectively,
(D, ⊣)) coincide. We also find necessary and sufficient conditions under
which an arbitrary dimonoid is a (ℓr, rr)-diband and, as a consequence,
establish that the variety of (ℓr, rr)-dibands coincides with the variety of
(ℓn, rn)-dibands. At the end of the section we describe the structure of
free (ℓr, rr)-dibands.
Recall that an idempotent semigroup S is called a left regular band, if
aba = ab (5)
for all a, b ∈ S. If instead of (5) the identity
aba = ba (6)
holds, then S is a right regular band. An idempotent semigroup S is
called a left normal band, if
axy = ayx (7)
for all a, x, y ∈ S. If instead of (7) the identity
xya = yxa (8)
holds, then S is a right normal band. A semigroup S is called left (re-
spectively, right) commutative, if it satisfies the identity xya = yxa
(respectively, axy = ayx).
300 Free (ℓr, rr)-dibands
Lemma 2. Operations of a dimonoid (D, ⊣, ⊢) with an idempotent oper-
ation ⊣ (respectively, ⊢) coincide, if (D, ⊢) (respectively, (D, ⊣)) is a left
(respectively, right) regular band.
Proof. For all x, y ∈ D we have
x ⊣ y = (x ⊣ y) ⊢ (x ⊣ y) = x ⊢ (y ⊢ (x ⊣ y)) =
= x ⊢ ((y ⊢ x) ⊣ y) = (x ⊢ (y ⊢ x)) ⊣ y =
= (x ⊢ y) ⊣ y = x ⊢ (y ⊣ y) = x ⊢ y
according to the idempotent property of operations, the axioms of a
dimonoid and the identity (5).
For all x, y ∈ D we have
x ⊢ y = (x ⊢ y) ⊣ (x ⊢ y) = ((x ⊢ y) ⊣ x) ⊣ y =
= (x ⊢ (y ⊣ x)) ⊣ y = x ⊢ ((y ⊣ x) ⊣ y) =
= x ⊢ (x ⊣ y) = (x ⊢ x) ⊣ y = x ⊣ y
according to the idempotent property of operations, the axioms of a
dimonoid and the identity (6).
From Lemma 2 it follows that an idempotent dimonoid (D, ⊣, ⊢)
with left (respectively, right) regular bands (D, ⊣) and (D, ⊢) is a left
(respectively, right) regular band.
If (D, ⊣) is a semigroup, then D with an operation ⊢, defined by
x ⊢ y = y ⊣ x for all x, y ∈ D, is a semigroup. The semigroup (D, ⊢) is
called dual to the semigroup (D, ⊣).
Lemma 3. Let (D, ⊣) be an arbitrary semigroup and (D, ⊢) be a dual
semigroup to (D, ⊣). Then an algebra (D, ⊣, ⊢) is a dimonoid if and only
if (D, ⊣) is a right commutative semigroup.
Proof. Let (D, ⊣, ⊢) be a dimonoid and (D, ⊢) be dual to (D, ⊣). Then
for all x, y, z ∈ D,
x ⊣ y ⊣ z = x ⊣ (y ⊢ z) = x ⊣ (z ⊣ y)
according to the axiom of a dimonoid and the definition of the operation
⊢. Hence (D, ⊣) is a right commutative semigroup.
Conversely, let (D, ⊣) be a right commutative semigroup and x, y, z ∈
D. Then
(x ⊣ y) ⊣ z = x ⊣ (z ⊣ y) = x ⊣ (y ⊢ z),
A. V. Zhuchok 301
(x ⊢ y) ⊣ z = (y ⊣ x) ⊣ z = y ⊣ (x ⊣ z) =
= y ⊣ (z ⊣ x) = (y ⊣ z) ⊣ x = x ⊢ (y ⊣ z),
x ⊢ (y ⊢ z) = (y ⊢ z) ⊣ x = z ⊣ (y ⊣ x) =
= z ⊣ (x ⊣ y) = (x ⊣ y) ⊢ z
according to the definition of the operation ⊢ and the right commutativity
of (D, ⊣). So, (D, ⊣, ⊢) is a dimonoid.
Dually, the following lemma can be proved.
Lemma 4. Let (D, ⊢) be an arbitrary semigroup and (D, ⊣) be a dual
semigroup to (D, ⊢). Then an algebra (D, ⊣, ⊢) is a dimonoid if and only
if (D, ⊢) is a left commutative semigroup.
A dimonoid (D, ⊣, ⊢) will be called a (ℓr, rr)-diband, if (D, ⊣) is a left
regular band and (D, ⊢) is a right regular band. Recall that a dimonoid
(D, ⊣, ⊢) is called a (ℓn, rn)-diband, if (D, ⊣) is a left normal band and
(D, ⊢) is a right normal band [7].
Note that every left (right) normal band is left (right) regular. The
converse statement is not true. It is natural to consider the similar question
for (ℓr, rr)-dibands and (ℓn, rn)-dibands.
The following theorem gives necessary and sufficient conditions under
which an arbitrary dimonoid is a (ℓr, rr)-diband.
Theorem 1. A dimonoid (D, ⊣, ⊢) is a (ℓr, rr)-diband if and only if
(D, ⊣, ⊢) is a (ℓn, rn)-diband.
Proof. Let (D, ⊣, ⊢) be an arbitrary (ℓr, rr)-diband. For all x, y ∈ D we
have
x ⊢ y ⊣ x = (y ⊢ (x ⊢ y)) ⊣ x = y ⊢ ((x ⊢ y) ⊣ x) =
= y ⊢ (x ⊢ (y ⊣ x)) = (y ⊣ x) ⊢ (y ⊣ x) = y ⊣ x,
x ⊢ y ⊣ x = x ⊢ ((y ⊣ x) ⊣ y) = (x ⊢ (y ⊣ x)) ⊣ y =
= ((x ⊢ y) ⊣ x) ⊣ y = (x ⊢ y) ⊣ (x ⊢ y) = x ⊢ y
according to the idempotent property of operations, the axioms of a
dimonoid and the identities (6) and (5). Hence
x ⊢ y = y ⊣ x (9)
for all x, y ∈ D.
302 Free (ℓr, rr)-dibands
Further by Lemma 3 (D, ⊣) is a right commutative semigroup. As
(D, ⊣) is also idempotent, then (D, ⊣) is a left normal band.
Let x, y, z ∈ D. From
x ⊣ y ⊣ z = x ⊣ z ⊣ y
we have
z ⊢ y ⊢ x = y ⊢ z ⊢ x
by (9). So, (D, ⊢) is a left commutative semigroup. As (D, ⊢) is also
idempotent, then (D, ⊢) is a right normal band.
Thus, (D, ⊣, ⊢) is a (ℓn, rn)-diband.
Conversely, let (D, ⊣, ⊢) be a (ℓn, rn)-diband. Then
x ⊣ y ⊣ z = x ⊣ z ⊣ y,
z ⊢ y ⊢ x = y ⊢ z ⊢ x
for all x, y, z ∈ D. Substituting z = x in the last two equalities and using
the idempotent property of operations, we obtain
x ⊣ y ⊣ x = x ⊣ y, x ⊢ y ⊢ x = y ⊢ x.
Thus, (D, ⊣, ⊢) is a (ℓr, rr)-diband.
From Theorem 1 we obtain
Corollary 1. The variety of (ℓr, rr)-dibands coincides with the variety
of (ℓn, rn)-dibands.
We call a dimonoid which is free in the variety of (ℓr, rr)-dibands
(respectively, (ℓn, rn)-dibands) a free (ℓr, rr)-diband (respectively, free
(ℓn, rn)-diband).
Let X be an arbitrary nonempty set. Assume (X, ⊣) and (X, ⊢) be a
left zero semigroup and a right zero semigroup respectively. Recall that
we call the dimonoid Xℓz,rz = (X, ⊣, ⊢) as a left and right diband (see [6]).
Let B(X) be the semilattice of all nonempty finite subsets of X with
respect to the operation of the set theoretical union, Xℓz,rz be a left and
right diband and
Bℓz,rz(X) = {(x, A) ∈ Xℓz,rz × B(X) | x ∈ A}.
By [7] Bℓz,rz(X) is the free (ℓn, rn)-diband.
From Theorem 1 we obtain
A. V. Zhuchok 303
Corollary 2. Bℓz,rz(X) is the free (ℓr, rr)-diband.
For all Y ∈ B(X), i ∈ X put
T Y = {(x, A) ∈ Bℓz,rz(X) | A = Y },
T(i) = {(x, A) ∈ Bℓz,rz(X) | x = i},
Bi(X) = {A ∈ B(X) | i ∈ A}.
The notion of a diband of subdimonoids (see [10], [11]) is effective
to describe structural properties of dimonoids. In terms of dibands of
subdimonoids (see also [12]), similar to Theorems 4 (vii) and 3 (iv) from
[7], the following theorem can be proved.
Theorem 2. Let Bℓz,rz(X) be the free (ℓr, rr)-diband. Then
(i) Bℓz,rz(X) is the free semilattice B(X) of subdimonoids T Y ,
Y ∈ B(X), such that T Y ∼= Yℓz,rz for every Y ∈ B(X);
(ii) Bℓz,rz(X) is a left and right diband Xℓz,rz of subsemigroups T(i),
i ∈ Xℓz,rz, such that T(i)
∼= Bi(X) for every i ∈ Xℓz,rz.
If ϕ : D1 → D2 is a homomorphism of dimonoids, then the corre-
sponding congruence on D1 will be denoted by ∆ϕ.
Let
µ : Bℓz,rz(X) → B(X) : (x, A) 7→ (x, A)µ = A,
q : Bℓz,rz(X) → Xℓz,rz : (x, A) 7→ (x, A)q = x.
It is evident that µ and q are homomorphisms.
If ρ is a congruence on a dimonoid (D, ⊣, ⊢) such that operations
of (D, ⊣, ⊢)/ρ coincide and it is a semilattice, then we say that ρ is a
semilattice congruence. If ρ is a congruence on a dimonoid (D, ⊣, ⊢) such
that (D, ⊣, ⊢)/ρ is a left and right diband, then we say that ρ is a left zero
and right zero congruence.
From Theorem 2 we obtain
Corollary 3. Let Bℓz,rz(X) be the free (ℓr, rr)-diband. Then
(i) ∆µ is the least semilattice congruence on Bℓz,rz(X);
(ii) ∆q is the least left zero and right zero congruence on Bℓz,rz(X).
304 Free (ℓr, rr)-dibands
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[12] A.V. Zhuchok, Semilattices of subdimonoids, Asian-Eur. J. Math. 4 (2011), no. 2,
359–371.
Contact information
A. V. Zhuchok Department of Mathematical Analysis and Algebra,
Luhansk Taras Shevchenko National University,
Oboronna str., 2, Luhansk, 91011, Ukraine
E-Mail: zhuchok_a@mail.ru
Received by the editors: 03.03.2013
and in final form 03.04.2013.
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