Problems of Physics, Mathematics and Technics, № 4 (17), 2013
МАТЕМАТИКА
УДК 512.579
CВОБОДНЫЕ n -ДИНИЛЬПОТЕНТНЫЕ ДИМОНОИДЫ
А.В. Жучок
Луганский национальный университет имени Тараса Шевченко, Луганск, Украина
FREE n -DINILPOTENT DIMONOIDS
Anatolii V. Zhuchok
Luhansk Taras Shevchenko National Univesity, Luhansk, Ukraine
Построен свободный n-динильпотентный димоноид и охарактеризована наименьшая n-динильпотентная конгруэнция
на свободном димоноиде.
Ключевые слова: n-динильпотентный димоноид, свободный n-динильпотентный димоноид, димоноид, полугруппа,
конгруэнция.
We construct a free n-dinilpotent dimonoid and characterize the least n-dinilpotent congruence on a free dimonoid.
Keywords: n-dinilpotent dimonoid, free n-dinilpotent dimonoid, dimonoid, semigroup, congruence.
Introduction and preliminaries
In the author’s preceding papers [1]–[4] the
problem of the construction of some relatively free
dimonoids was solved. The structure of the corresponding algebras was described and some least
congruences on a free dimonoid were characterized.
The present paper continues this trend of research by considering the so-called n -dinilpotent
dimonoids, that is dimonoids with two n -nilpotent
semigroups. It turns out that the class of such
dimonoids is a subvariety of the variety of all
dimonoids. For the indicated variety a free object is
constructed and the least n -dinilpotent congruence
on a free dimonoid is characterized.
Let us recall that a nonempty set D equipped
with two binary operations E and A satisfying the
following axioms:
( x E y ) E z = x E ( y E z ),
(D1)
( x E y ) E z = x E ( y A z ),
(D2)
( x A y ) E z = x A ( y E z ),
(D3)
( x E y ) A z = x A ( y A z ),
(D4)
( x A y) A z = x A ( y A z)
(D5)
for all x, y, z ∈ D, is called a dimonoid.
An element 0 of a dimonoid ( D, E, A) will be
called zero, if
x∗0 = 0 = 0∗ x
for all x ∈ D and ∗∈ {E, A}.
As usual, N denotes the set of all positive integers.
We call a semigroup S nilpotent, if S n +1 = 0
for some n ∈ N. The least such n we shall call the
nilpotency index of S. For k ∈ N a nilpotent semigroup of nilpotency index ≤ k is said to be k -nilpotent.
© Anatolii V. Zhuchok, 2013
A dimonoid ( D, E, A ) with zero will be called
dinilpotent, if ( D,E ) and ( D,A ) are nilpotent semigroups.
A dinilpotent dimonoid ( D, E, A ) will be called
n -dinilpotent, if ( D,E ) and ( D,A ) are n -nilpotent
semigroups. If ρ is a congruence on a dimonoid
( D, E, A)
such that ( D, E, A ) / ρ is an n -dinilpotent
dimonoid, then we say that ρ is an n -dinilpotent
congruence.
Note that operations of any 1 -dinilpotent
dimonoid coincide and it is a zero semigroup.
Lemma 0.1. The class of all n -dinilpotent
dimonoids is a subvariety of the variety of all dimonoids.
Proof. Indeed, the class of all n -dinilpotent
dimonoids is a subclass of the variety of all dimonoids which is closed under taking of homomorphic
images, subdimonoids and Cartesian products, and
consequently, it is a variety.
□
A dimonoid which is free in the variety of
n -dinilpotent dimonoids will be called a free n -dinilpotent dimonoid.
The necessary information about varieties of
dimonoids can be found in [1].
J.-L. Loday described a free dimonoid [5]. We
constructed the dimonoid isomorphic to the free
dimonoid in [6]. Recall this construction.
Let F [ X ] be the free semigroup in the alphabet X . We denote the length of a word w ∈ F [ X ]
by lw . Define operations E and A on
F = {( w, m) ∈ F [ X ] × N | lw ≥ m}
by
( w1 , m1 ) E ( w2 , m2 ) = ( w1 w2 , m1 ),
43
Anatolii V. Zhuchok
( w1 , m1 ) A ( w2 , m2 ) = ( w1 w2 , lw1 + m2 )
Let
for all ( w1 , m1 ), ( w2 , m2 ) ∈ F . Denote the algebra
( F, E, A) by F [ X ]. By Lemma 3 from [6] F [ X ] is
the free dimonoid over X .
If f : D1 → D2 is a homomorphism of dimonoids, then the corresponding congruence on D1
will be denoted by Δ f .
1 Free objects
In this section we construct a free n -dinilpotent dimonoid of an arbitrary rank and consider
separately free n -dinilpotent dimonoids of rank 1.
Fix n ∈ N and define operations E and A on
FDn = {( w, m) ∈ F [ X ] × N |
m ≤ lw , m ≤ n, lw − m + 1 ≤ n} ∪ {0}
by
( w1 , m1 ) E ( w2 , m2 ) =
⎧⎪( w1 w2 , m1 ) , if lw1w2 − m1 + 1 ≤ n,
=⎨
0, if lw1w2 − m1 + 1 > n,
⎪⎩
( w1 , m1 ) A ( w2 , m2 ) =
⎧⎪⎛⎜ w1 w2 , lw + m2 ⎞⎟ , if lw + m2 ≤ n,
1
1
⎠
= ⎨⎝
0, if lw1 + m2 > n,
⎪⎩
( w1 , m1 ) ∗ 0 = 0 ∗ ( w1 , m1 ) = 0 ∗ 0 = 0
( w1 , m1 ) , ( w2 , m2 ) ∈ FDn \ {0} and ∗∈ {E, A}.
algebra ( FDn , E, A ) will be denoted by
for all
The
FDn ( X ).
Theorem 1.1. FDn ( X ) is the free n -dinilpotent dimonoid.
Proof. First prove that FDn ( X ) is a dimonoid.
Let ( w1 , m1 ), ( w2 , m2 ), ( w3 , m3 ) ∈ FDn \ {0} and let
(1.1)
lw1 + lw2 + lw3 − m1 + 1 ≤ n.
From (1.1) it follows
lw1 + lw2 − m1 + 1 < n,
(1.2)
lw2 + lw3 − m2 + 1 < n,
(1.3)
lw2 + m3 < n.
(1.4)
Using (1.1)–(1.4), we get
(( w1 , m1 ) E ( w2 , m2 )) E ( w3 , m3 ) =
= ( w1 w2 , m1 ) E ( w3 , m3 ) = ( w1 w2 w3 , m1 ) =
= ( w1 , m1 ) E ( w2 w3 , m2 ) =
= ( w1 , m1 ) E (( w2 , m2 ) E ( w3 , m3 )),
( w1 , m1 ) E (( w2 , m2 ) A ( w3 , m3 )) =
= ( w1 , m1 ) E ( w2 w3 , lw2 + m3 ) = ( w1 w2 w3 , m1 )
and so, the axioms ( D1) and ( D 2) of a dimonoid
hold. If
lw1 + lw2 + lw3 − m1 + 1 > n,
then, obviously, the axioms ( D1) and ( D 2) hold too.
44
lw1 + m2 ≤ n,
lw2 + lw3 − m2 + 1 ≤ n.
(1.5)
(1.6)
Using (1.5), (1.6), we obtain
(( w1 , m1 ) A ( w2 , m2 )) E ( w3 , m3 ) =
= ( w1 w2 , lw1 + m2 ) E ( w3 , m3 ) =
= ( w1 w2 w3 , lw1 + m2 ) =
= ( w1 , m1 ) A ( w2 w3 , m2 ) =
= ( w1 , m1 ) A (( w2 , m2 ) E ( w3 , m3 )).
If
lw2 + m2 > n or
lw2 + lw3 − m2 + 1 > n,
then, obviously,
(( w1 , m1 ) A ( w2 , m2 )) E ( w3 , m3 ) = 0 =
= ( w1 , m1 ) A (( w2 , m2 ) E ( w3 , m3 )).
Thus, the axiom ( D3) holds.
Let
lw1 + lw2 + m3 ≤ n.
(1.7)
From (1.7) it follows
lw1 + m2 < n,
(1.8)
lw2 + m3 < n,
(1.9)
lw1 + lw2 − m1 + 1 < n.
(1.10)
According to (1.7)–(1.10), we have
(( w1 , m1 ) A ( w2 , m2 )) A ( w3 , m3 ) =
= ( w1 w2 , lw1 + m2 ) A ( w3 , m3 ) =
= ( w1 w2 w3 , lw1w2 + m3 ) =
= ( w1 w2 w3 , lw1 + lw2 + m3 ) =
= ( w1 , m1 ) A ( w2 w3 , lw2 + m3 ) =
= ( w1 , m1 ) A (( w2 , m2 ) A ( w3 , m3 )),
(( w1 , m1 ) E ( w2 , m2 )) A ( w3 , m3 ) =
= ( w1 w2 , m1 ) A ( w3 , m3 ) =
= ( w1 w2 w3 , lw1w2 + m3 )
and so, the axioms ( D5) and ( D 4) of a dimonoid
hold.
If lw1 + lw2 + m3 > n, then, obviously, the axioms ( D 4) and ( D5) hold too.
The proofs of the remaining cases are obvious.
Consequently, FDn ( X ) is a dimonoid.
Take arbitrary elements ( wi , mi ) ∈ FDn \ {0},
1 ≤ i ≤ n + 1. It is clear that lw1w2 …wn+1 − m1 + 1 > n.
From here
( w1 , m1 ) E ( w2 , m2 ) E …E ( wn +1 , mn +1 ) = 0.
Besides, ( x1 ,1) E ( x2 ,1) E …E ( xn ,1) ≠ 0 for any
( xi ,1) ∈ FDn \ {0}, where xi ∈ X , 1 ≤ i ≤ n. From the
last arguments we conclude that ( FDn , E) is a nilpotent semigroup of nilpotency index n.
Проблемы физики, математики и техники, № 4 (17), 2013
Free n-dinilpotent dimonoids
Further note that lw1w2 …wn + mn +1 > n. From the
= ( x1γ A′ … A′ xl γ E′ …E′ xs γ ) A′
above it follows that
( w1 , m1 ) A ( w2 , m2 ) A … A ( wn +1 , mn +1 ) = 0.
Moreover,
( x1 ,1) A ( x2 ,1) A … A ( xn ,1) ≠ 0
for any ( xi ,1) ∈ FDn \ {0}, where xi ∈ X , 1 ≤ i ≤ n.
According to the above remarks we deduce that
( FDn , A) is a nilpotent semigroup of nilpotency index n.
Thus, by the definition, FDn ( X ) is an n -dinilpotent dimonoid.
Let us show that FDn ( X ) is free.
A′ ( y1γ A′ … A′ yt γ E′ …E′ yr γ ) =
Let
(T , E′, A′ )
be an arbitrary n -dinilpotent
dimonoid and γ : X → T be an arbitrary map. Define a map
λ : FDn ( X ) → (T , E′, A′) : u 6 uλ ,
assuming
u = ( x1 …xs , l ),
⎧
⎪ x1γ A′ …A′ xl γ E′ …E′ xs γ , if
uλ = ⎨
xi ∈ X ,1 ≤ i ≤ s,
⎪
0, if u = 0.
⎩
Show that λ is a homomorphism. We will use
the axioms of a dimonoid.
For arbitrary elements
( x1 …xi …xs , l ) , ( y1 …y j …yr , t ) ∈ FDn \ {0},
where xi , y j ∈ X , 1 ≤ i ≤ s, 1 ≤ j ≤ r , we obtain
(( x …x …x , l ) E ( y …y …y , t )) λ =
1
i
s
1
j
r
⎧( x …x y …y , l ) λ , if s + r − l + 1 ≤ n,
=⎨ 1 s 1 r
0λ , if s + r − l + 1 > n,
⎩
(
)
( x1 …xi …xs , l ) A ( y1 …y j …yr , t ) λ =
⎧( x …x y …y , s + t ) λ , if s + t ≤ n,
=⎨ 1 s 1 r
0λ , if s + t > n.
⎩
If s + r − l + 1 ≤ n, then we have
( x1 …xs y1 …yr , l ) λ =
= x1γ A′ … A′ xl γ E′ …E′ xs γ E′ y1γ E′ …E′ yr γ =
= ( x1γ A′ … A′ xl γ E′ …E′ xs γ ) E′
E′ ( y1γ A′ … A′ yt γ E′ …E′ yr γ ) =
= ( x1 …xi …xs , l )λ E′ ( y1 …y j …yr , t ) λ .
If s + r − l + 1 > n, then
0λ = 0 =
= x1γ A′ … A′ xl γ E′ …E′ xs γ E′ y1γ E′ …E′ yr γ =
= ( x1γ A′ … A′ xl γ E′ …E′ xs γ ) E′
E′ ( y1γ A′ … A′ yt γ E′ …E′ yr γ ) =
= ( x1 …xi …xs , l )λ E′ ( y1 …y j …yr , t )λ .
In the case s + t ≤ n,
( x1 …xs y1 …yr , s + t ) λ =
= x1γ A′ … A′ xs γ A′ y1γ A′ … A′ yt γ E′ …E′ yr γ =
Problems of Physics, Mathematics and Technics, № 4 (17), 2013
= ( x1 …xi …xs , l ) λ A′ ( y1 …y j …yr , t ) λ .
If s + t > n, then
0λ = 0 =
= x1γ A′ … A′ xs γ A′ y1γ A′ … A′ yt γ E′ …E′ yr γ =
= ( x1γ A′ … A′ xl γ E′ …E′ xs γ ) A′
A′ ( y1γ A′ … A′ yt γ E′ …E′ yr γ ) =
= ( x1 …xi …xs , l ) λ A′ ( y1 …y j …yr , t ) λ .
The proofs of the remaining cases are obvious.
Thus, λ is a homomorphism. This completes the
proof of Theorem 1.1.
□
Now we construct a dimonoid which is isomorphic to the free n -dinilpotent dimonoid of rank 1.
Fix n ∈ N and define operations E and A on
N n = {(m, t ) ∈ N × N |
t ≤ m, t ≤ n, m − t + 1 ≤ n} ∪ {0}
by
( m1 , t1 ) E ( m2 , t2 ) =
⎧( m1 + m2 , t1 ) , if m1 + m2 − t1 + 1 ≤ n,
=⎨
⎩
0, if m1 + m2 − t1 + 1 > n,
( m1 , t1 ) A ( m2 , t2 ) =
⎧( m1 + m2 , m1 + t2 ) , if m1 + t2 ≤ n,
=
⎨
0, if m1 + t2 > n,
⎩
( m1 , t1 ) ∗ 0 = 0 ∗ ( m1 , t1 ) = 0 ∗ 0 = 0
for all ( m1 , t1 ) , ( m2 , t2 ) ∈ N n \ {0} and ∗∈ {E, A}. An
immediate verification shows that axioms of a
dimonoid hold concerning operations E and A . So,
(N n , E, A) is a dimonoid. Denote it by N ( n ) .
Lemma 1.1. If | X |= 1, then FDn ( X ) ≅ N ( n ) .
Proof. Assume X = {a} and define a map
η : FDn ( X ) → N ( n )
by the rule
⎧(k , l ), u = (a k , l ),
uη = ⎨
⎩ 0, u = 0.
An easy verification shows that η is an isomorphism.
□
2 The least n -dinilpotent congruence on a
free dimonoid
In this section we present the least n -dinilpotent congruence on a free dimonoid.
Let F [ X ] be the free dimonoid over X (see
introduction and preliminaries). Fix n ∈ N and let
I ( n ) = {( w, m) ∈ F [ X ] | m > n or
lw − m + 1 > n}.
45
Anatolii V. Zhuchok
Define a relation ξ ( n ) on F [ X ] by
Let lw2 − m2 + 1 > n. Then
( w1 , m1 ) A ( w2 , m2 ) ⎞⎟⎠ ϕ = ⎛⎜⎝ w1w2 , lw + m2 ⎞⎟⎠ ϕ =
= 0 = ( w1 , m1 ) ϕ A 0 = ( w1 , m1 ) ϕ A ( w2 , m2 ) ϕ .
( w1 , m1 )ξ ( n ) ( w2 , m2 ) if and only if
⎛
⎜
⎝
( w1 , m1 ) = ( w2 , m2 ) or
( w1 , m1 ), ( w2 , m2 ) ∈ I ( n ) .
Theorem 2.1. The relation ξ ( n ) on the free di
monoid F [ X ] is the least n -dinilpotent congruence.
Proof. Let ( w1 , m1 ), ( w2 , m2 ) ∈ F [ X ] . It is not
difficult to see that
lw1 + lw2 − m1 + 1 ≤ n
1
In the case lw2 − m2 + 1 ≤ n we consider the following two cases. If (2.5) holds, then, using (2.2),
(2.3), (2.6), we have
⎛
⎞
⎛
⎞
⎜ ( w1 , m1 ) A ( w2 , m2 ) ⎟ ϕ = ⎜ w1 w2 , lw + m2 ⎟ ϕ =
⎝
⎠
1
⎝
⎠
= ⎛⎜⎝ w1 w2 , lw1 + m2 ⎞⎟⎠ = ( w1 , m1 ) A ( w2 , m2 ) =
= ( w1 , m1 ) ϕ A ( w2 , m2 ) ϕ .
(2.1)
implies
m2 < n,
lw1 − m1 + 1 < n,
(2.2)
(2.3)
lw2 − m2 + 1 < n
(2.4)
lw1 + m2 ≤ n
(2.5)
If lw1 + m2 > n, then
⎛
⎜
⎝
( w1 , m1 ) A ( w2 , m2 ) ⎞⎟⎠ ϕ = ⎛⎜⎝ w1w2 , lw + m2 ⎞⎟⎠ ϕ =
= 0 = ( w1 , m1 ) ϕ A ( w2 , m2 ) ϕ .
Thus,
and
(2.6)
( w, m ) ϕ =
⎧ ( w, m ) , if m ≤ n, lw − m + 1 ≤ n,
=⎨
⎩ 0, if m > n or lw − m + 1 > n
( (w, m ) ∈ F [ X ]). Show that ϕ is a homomorphism.
Let m1 > n. Then
⎛
⎜
⎝
( w1 , m1 ) E ( w2 , m2 ) ⎞⎟⎠ ϕ = ( w1w2 , m1 ) ϕ = 0 =
= 0 E ( w2 , m2 ) ϕ = ( w1 , m1 ) ϕ E ( w2 , m2 ) ϕ .
In the case m1 ≤ n we consider the following
two cases. If (2.1) holds, then, using (2.2)–(2.4), we
have
⎛
⎞
⎜ ( w1 , m1 ) E ( w2 , m2 ) ⎟ ϕ = ( w1 w2 , m1 ) ϕ =
⎝
⎠
= ( w1 w2 , m1 ) = ( w1 , m1 ) E ( w2 , m2 ) =
= ( w1 , m1 ) ϕ E ( w2 , m2 ) ϕ .
If
lw1 + lw2 − m1 + 1 > n,
then
⎛
⎜
⎝
Thus,
( w1 , m1 ) E ( w2 , m2 ) ⎞⎟⎠ ϕ = ( w1w2 , m1 ) ϕ =
= 0 = ( w1 , m1 ) ϕ E ( w2 , m2 ) ϕ .
( w1 , m1 ) E ( w2 , m2 ) ⎞⎟⎠ ϕ =
= ( w1 , m1 ) ϕ E ( w2 , m2 ) ϕ
( w1 , m1 ), ( w2 , m2 ) ∈ F [ X ] .
⎛
⎜
⎝
for all
46
( w1 , m1 ) A ( w2 , m2 ) ⎞⎟⎠ ϕ =
= ( w1 , m1 ) ϕ A ( w2 , m2 ) ϕ
( w1 , m1 ), ( w2 , m2 ) ∈ F [ X ] . Consequently, ϕ
⎛
⎜
⎝
implies (2.2), (2.3) and
m1 < n.
Define a map ϕ : F [ X ] → FDn ( X ) by
1
for all
is a surjective homomorphism.
By Theorem 1.1 FDn ( X ) is the free n -dinilpotent dimonoid. Then Δϕ is the least n -dinilpotent
congruence on F [ X ] . From the definition of ϕ it
follows that Δϕ = ξ ( n ) .
□
REFERENCES
1. Zhuchok, A.V. Free commutative dimonoids
/ A.V. Zhuchok // Algebra and Discrete Math. –
2010. – Vol. 9, № 1. – P. 109–119.
2. Zhuchok, A.V. Free rectangular dibands and
free dimonoids / A.V. Zhuchok // Algebra and Discrete Math. – 2011. – Vol. 11, № 2. – P. 92–111.
3. Zhuchok, A.V. Free normal dibands /
A.V. Zhuchok // Algebra and Discrete Math. – 2011.
– Vol. 12, № 2. – P. 112–127.
4. Zhuchok, A.V. Free (Ar , rr ) -dibands /
A.V. Zhuchok // Algebra and Discrete Math. – 2013.
– Vol. 15, № 2. – P. 295–304.
5. Loday, J.-L. Dialgebras, In: Dialgebras and
related operads, Lect. Notes Math. / J.-L. Loday //
Springer-Verlag, Berlin, 2001. – Vol. 1763. – P. 7–66.
6. Zhuchok, A.V. Free dimonoids / A.V. Zhuchok // Ukr. Math. J. – 2011. – Vol. 63, № 2. –
P. 196–208.
Поступила в редакцию 18.09.13.
Проблемы физики, математики и техники, № 4 (17), 2013