Pseudo-nearrings and quasi-modules over them
In this paper we start to investigate a new notion of pseudo-nearrings and a generalization of linear spaces to quasi-modules over pseudo-nearrings. Pseudo-nearrings can be treated as ringoids in the sense of J. Hion (see [6]). The idea of pseudo-nearings is based on the notion of a ∗-associative qu...
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| description | In this paper we start to investigate a new notion of pseudo-nearrings and a generalization of linear spaces to quasi-modules over pseudo-nearrings. Pseudo-nearrings can be treated as ringoids in the sense of J. Hion (see [6]). The idea of pseudo-nearings is based on the notion of a ∗-associative quasigroup, i.e. on an involutive groupoid (A;+,* ) in which the following identities hold: (x*)* = x, (x + y)* = y* + x*, (x + y)* + z = x + (y + z)*. We assume also commutativity and quasigroup properties of (A;+).
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Український математичний вiсник
Том 1 (2004), № 1, 129 – 139
Pseudo-nearrings and quasi-modules over them
Anna Chwastyk and Kazimierz G lazek
(Presented by V.M. Usenko 10.01.2003 )
Abstract. In this paper we start to investigate a new notion of pseudo-
nearrings and a generalization of linear spaces to quasi-modules over
pseudo-nearrings. Pseudo-nearrings can be treated as ringoids in the
sense of J. Hion (see [6]). The idea of pseudo-nearings is based on the no-
tion of a ∗-associative quasigroup, i.e. on an involutive groupoid (A; +,∗ )
in which the following identities hold:
(x∗)∗ = x, (x + y)∗ = y
∗ + x
∗
, (x + y)∗ + z = x + (y + z)∗.
We assume also commutativity and quasigroup properties of (A; +).
2000 MSC. 20N02, 20N05, 16Y30, 16W10, 16D99.
1. Introduction
An algebra (A; +,∗ ) is said to be an involutive groupoid if the following
identities hold:
(x∗)∗ = x, (x+ y)∗ = y∗ + x∗.
We call an involutive groupoid ∗-associative if it satisfies the equation:
(x+ y)∗ + z = x+ (y + z)∗.
A ∗-associative groupoid (A; +,∗ ) is a ∗-associative quasigroup
if (A; +) is a quasigroup.
The concepts of ∗- associative groupoid and quasigroup were intro-
duced in [2]. For the standard terminology of semigroups, quasigroups
and near-rings, see [1], [7], [9] and [10].
Examples 1 and 2. Define the following operations in the set Z5:
x∗ = 4x(mod5), x⊕ y = 4x+ 4y + 2x2y2(x+ y)(mod5);
Received 10.01.2003
Key words and phrases. Groupoid with involution, ∗-associative groupoid, term oper-
ation, quasigroup, ∗–associative quasigroup, pseudo-nearring, quasi-module.
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
130 Pseudo-nearrings and quasi-modules over them
and in the set Z7:
x∗ = 6x(mod7), x⊕ y = 6x+ 6y+ 2x2y2(x3 + y3) + 4x3y3(x+ y)(mod7).
Then (Z5;⊕,∗ ) and (Z7;⊕,∗ ) are ∗-associative quasigroups.
Example 3. The algebra (Z;⊕,∗ ), where Z is the set of integers, x⊕y =
−(x+y)+3a (for some fixed a ∈ Z), and x∗ = −x+2a, is a ∗-associative
quasigroup.
For more examples, we refer the reader to [2] and [5].
In our investigation, we use the following proposition (see [3]):
Proposition 1. Let (A; +,∗ ) be a commutative ∗-associative groupoid.
Then
(a) (a+ b) + c = (a+ c∗) + b∗;
(b) (a+ b) + (c+ d) = (a+ c) + (b+ d), for all a, b, c ∈ A.
Therefore the considered groupoid (A; +) is medial.
In the following we need to have a description of term operations for
the considered algebras.
For a fixed algebra A = (A; F) and n = 0, 1, 2, . . . , we denote by
T
(n)(A) (or T
(n) for short) the class of all n-ary term operations of A,
i.e. the smallest class of operations satisfying the conditions:
(i) eni ∈ T
(n), (eni (x1, x2, . . . , xn) = xi for i = 1, 2, . . . , n)
(ii) if g1, g2, . . . , gk ∈ T
(n), f ∈ F
(k), then
f(g1, g2, . . . , gk)(x1, x2, . . . , xn) =
= f(g1(x1, x2, . . . , xn), . . . , gk(x1, x2, . . . , xn))
belongs to T
(n).
For completeness, we recall the description of term operations in com-
mutative ∗-associative groupoids (see [3]).
Denote g(k,l)(x) = (. . . (xi1 +xi2)+. . .)+xil)+. . .)+xik , where k ∈ N,
l ∈ N∪{0}, l is the first place in which ∗ appears, x ∈ A, x0 = x∗, x1 = x,
i.e., if l = 1, then i1 = 0, i2r = 0, i2r+1 = 1 for r ∈ N , and if l > 1, then
ir = 1 for r < l; il+2s = 0, il+2s+1 = 1 for r, s ∈ N ∪ {0}.
Then we have (see [3]):
Theorem 1. Every unary term operation in a commutative ∗-associative
groupoid (A; +,∗ ) is of the form g(k,l)(x).
A. Chwastyk, K. G lazek 131
Define h(p,r)(y, x) = (. . . (y+xi1)+xi2)+ . . .)+xir)+ . . .)+xip , where
p, r ∈ N ∪ {0}, h(0,0)(y, x) = y, is = 0 for s < r; ir+2s = 0, ir+2s+1 = 1
for s ∈ N ∪ {0}.
For simplicity of notation, we write g(k,l)(x) ⊕ h(p,r)(x) instead of
h(p,r)(g(k,l)(x), x). We obtain (see [3]):
Proposition 2. Every n-ary term operation in a commutative ∗-associ-
ative groupoid (A,+,∗ ) is of the following form:
f(x1, x2, . . . , xn) = (. . . (g(k1,l1)(x1)⊕h(k2,l2)(x2))⊕ . . .)⊕h(kn,ln)(xn).
2. ∗-Associative quasigroups
We have the following characterization of ∗-associative quasigroups
(see [2]):
Theorem 2. Let (A; +,∗ ) be a ∗-associative groupoid. Then (A; +) is a
quasigroup if and only if the following two conditions hold:
(i) (∃ε ∈ A) (∀a ∈ A) ε+ a = a∗,
(ii) (∀a ∈ A) (∃b ∈ A) b+ a = ε.
Let A = (A; +,∗ , ε) be a ∗-associative quasigroup. Define −a as an
element such that a + (−a) = ε. To shorten notation, we write a − b
instead of a + (−b). The unary operation a 7→ −a can be added to
the set of fundamental operations of A. Now, we prove several simple
properties of ∗-associative quasigroups.
Proposition 3. Let (A; +,−,∗ , ε) be a ∗-associative quasigroup. Then
(a) (−a)∗ = −(a∗);
(b) −(a+ b) = (−b) + (−a);
(c) a = b⇔ a− b = ε; and
(d) [a = b+ c] ⇔ a− b∗ = c, for all a, b, c ∈ A.
Proof. The first equality is obvious. For the second we have:
(a+ b) + ((−b) + (−a)) = (a+ b) + ((−a∗) + (−b∗))∗ =
= ((a+ b) + (−a∗))∗ + (−b∗) = ((b∗ + a∗)∗ + (−a∗))∗ + (−b∗) =
= (b∗ + (a∗ + (−a∗))∗)∗ + (−b∗) = (b∗ + ε)∗ + (−b∗) = b∗ + (−b∗) = ε.
The proofs of the last two equivalences are straightforward.
132 Pseudo-nearrings and quasi-modules over them
If there exists an idempotent e (i.e., e+e = e, e∗ = e) in a ∗-associative
groupoid (A; +,∗ ), then we can define a set Qe as follows:
Qe = {a ∈ A : e+ a = a+ e = a∗, a+ b = b+ a = e for some b ∈ A}.
Proposition 4. Let A = (A; +,∗ ) be a ∗-associative groupoid. Then Qe
is a ∗-associative quasigroup and
Qe = {a ∈ A : a∗ ∈ (e+A) ∩ (A+ e), e ∈ (a+A) ∩ (A+ a)}.
Proof. We first prove that Qe is a subalgebra of A. Let a ∈ Qe. Then
a + e = e + a = a∗ and a + b = b + a = e for some b ∈ A. This yields
e+ a∗ =(a+ e)∗ = (a∗)∗ = a = a∗ + e, a∗ + b∗ = (b+ a)∗ = e = b∗ + a∗.
This clearly forces a∗ ∈ Qe.
Now, suppose that a, b ∈ Qe. Therefore a + c = c + a = e and
b+ d = d+ b = e for some c, d ∈ A. Then
e+ (a+ b)∗ = (e+ a)∗ + b = a∗∗ + b = a+ b,
(a+ b)∗ + e = a+ (b+ e)∗ = a+ b, (a+ b)∗ + (d+ c)∗ =
= a+ (b+ (d+ c)∗)∗ = a+ ((b+ d)∗ + c)∗ = a+ (e+ c)∗ =
= (a+ e)∗ + c = a+ c = e = (d+ c)∗ + (a+ b)∗.
This implies (a+ b)∗ ∈ Qe, and consequently a+ b ∈ Qe.
To prove that Qe ⊇ {a ∈ A : a∗ ∈ (e + A) ∩ (A + e), e ∈ (a + A) ∩
(A + a)}, let a∗ ∈ (e + A) ∩ (A + e) and e ∈ (a + A) ∩ (A + e). Hence
a∗ = e + p, a∗ = q + e, e = a + r and e = t + a for some p, q, r, t ∈ A.
So e + a = e + (e + p)∗ = (e + e)∗ + p = e + p = a∗, and analogously
a+ e = a∗.
Now, put b = (t + e)∗. Then t + e = t + e∗ = t + (a + r)∗ =
(t+a)∗+r = e+r = (t+a)+r. Thus (t+a)+r = b∗ = t+(a+r) and so
a+b = a+((t+a)+r)∗ = a+(e+r)∗ = (a+e)∗+r = a∗∗+r = a+r = e.
In the same manner, we can see that b+a = e. The result is a ∈ Qe.
Here and subsequently, we denote i = 0, 1, j = i+ 1(mod 2).
Lemma 1. The following properties hold in every commutative ∗-associ-
ative quasigroup (A; +,−,∗ , ε):
(a) −(a− a∗) = (a− a∗)∗,
(b) ai + (a− a∗)i = ai,
(c) (a+ a∗) + (a− a∗)i = aj + aj,
A. Chwastyk, K. G lazek 133
Proof. It is immediate.
Let g(k,l)(x) be as in Section 1, and let T1 and T2 denote the sets of
all terms of the form g(k,l)(x) fulfilling the following conditions:
1) l = 0 for k = 2, l = 1 for k odd, l = 3 for k > 2, k even;
2) l = 0 for k = 1, l = 1 for k even, l = 3 for k > 1, k odd, respectively.
Denote
f(k,l,m,n)(x) = g(k,l)(x) + g(m,n)(x− x∗),
where k ∈ N, l,m,n ∈ N ∪ {0}, and g(0,0)(x) = ε.
If m 6= 0, then [g(k,l)(x) ∈ T1 and g(m,n)(x) ∈ T2] or [g(k,l)(x) ∈ T2
and g(m,n)(x) ∈ T1].
Theorem 3. Every unary term operation in a commutative ∗-associative
quasigroup (A; +,−,∗ , ε) is of the form ±f(k,l,m,n)(x).
Proof. Obviously, every ∗-associative quasigroup is also a ∗-associative
groupoid. So, by Theorem 1, the term operations of the form g(k,l)(x)
and also −g(k,l)(x) belong to the set of term operations of a commutative
∗-associative quasigroup.
From Propositions 1(b), 3(a) and 3(b) we deduce that the term opera-
tions are also of the form
g(k,l)(x) ± g(m,n)(x− x∗) or − g(k,l)(x) ± g(m,n)(x− x∗).
By Lemma 1(a), these forms are equivalent to
±[g(k,l)(x) + g(m,n)(x− x∗)] = ±f(k,l,m,n)(x).
We consider only the term operations of the form f(k,l,m,n)(x), because
for −f(k,l,m,n)(x) the verification is similar.
We first observe that, for m 6= 0, it is enough to consider the terms
g(k,l)(x) which belong to T1 or T2, because the other forms of term op-
erations can be rewritten in a suitable form, i.e. f(k′,l′,m′,n′)(x).
For k = 2, by Lemma 1(c) we have
f(2,2,m,n)(x) = (x+ x∗) + g(m,n)(x− x∗) =
= (x+ x∗) + [g(m−1,n)(x− x∗) + (x− x∗)i] =
= [(x+ x∗) + (x− x∗)i] + g∗(m−1,n)(x− x∗) =
= (xj + xj) + g∗(m−1,n)(x− x∗),
where xj + xj = g(2,i)(x) ∈ T1 or T2.
134 Pseudo-nearrings and quasi-modules over them
For k > 2 and k even, by Lemma 1(b), we obtain
f(k,2,m,n)(x) = g(k,2)(x) + g(m,n)(x− x∗) =
[(x+ x∗) ⊕ g(k−2,2)(x)] + [g(m−1,n)(x− x∗) + (x− x∗)i] =
[(xj ⊕ g(k−2,2)(x)) + xi] + [g(m−1,n)(x− x∗) + (x− x∗)i] =
[xi + (x− x∗)i] + [(xj ⊕ g(k−2,2)(x)) + g(m−1,n)(x− x∗)] =
xi + [(xj ⊕ g(k−2,2)(x)) + g(m−1,n)(x− x∗)] =
[xj + (xj ⊕ g(k−2,2)(x)] + g∗(m−1,n)(x− x∗) =
[(xj + xj) ⊕ g(k−2,2)(x)] + g∗(m−1,n)(x− x∗).
After a finite number of similar steps, we get the term operation of the
form f(k′,l′,m′,n′)(x), where m′ = 0 or g(k′,l′)(x) belongs to T1 or T2. The
same conclusion can be drawn for k odd. Therefore we can assume l 6= 2.
Applying the equality (x+x)+x = (x+x∗)+x∗, by the same method
as before, we can see that l ≤ 3 for k ≥ 3. Consequently, g(k,l)(x) ∈ T1
or g(k,l)(x) ∈ T2.
Since
g(2,2)(x− x∗) = (x− x∗) + (x− x∗)∗ = ε
and
g(3,0)(x− x∗) = ((x− x∗) + (x− x∗)) + (x− x∗) = (x− x∗),
we conclude that the term operations of the form g(m,n)(x − x∗) belong
to T1 or T2.
Now, let g(k,l)(x), g(m,n)(x) ∈ T1. Then for k = 1, by Lemma 1(b),
we get
f(1,1,m,n)(x) =
g(1,1)(x) + g(m,n)(x− x∗) = x∗ + [g(m−1,n)(x− x∗) + (x− x∗)] =
[x+ (x− x∗)] + g∗(m−1,n)(x− x∗) = x+ g∗(m−1,n)(x− x∗) =
g(1,0)(x) + g∗(m−1,n)(x− x∗),
where g(1,0)(x) ∈ T2 and g∗(m−1,n)(x− x∗) ∈ T1.
For k > 1,m = 1, we see that
f(k,l,1,1)(x) = g(k,l)(x) + g(1,1)(x− x∗) =
[g(k−1,l)(x) + x] + (x− x∗)∗ = [x+ (x− x∗)] + g∗(k−1,l)(x) =
g∗(k−1,l)(x) + x.
A. Chwastyk, K. G lazek 135
So, we can rewrite this term operation in the form f(k′,l′,m′,n′)(x), where
m′ = 0.
For k = 2 and m = 2, by Proposition 1(b) and Lemma 1(b), we have
f(2,0,2,0)(x) = g(2,0)(x).
For k = 2 and m > 2, we obtain
f(2,0,m,n)(x) = [x+ x] + [g(m−1,n)(x− x∗) + (x− x∗)] =
[x+ (x− x∗)] + [x+ g(m−1,n)(x− x∗)] = x+ [x+ g(m−1,n)(x− x∗)] =
[x+ x∗] + g∗(m−1,n)(x− x∗) = [x+ x∗] + [g∗(m−2,n)(x− x∗) + (x− x∗)] =
[(x+ x∗)∗ + (x− x∗)] + g(m−2,n)(x− x∗) = [(x+ x∗) + (x− x∗)]+
g(m−2,n)(x− x∗) = [x∗ + x∗] + g(m−2,n)(x− x∗),
where g(2,1)(x) ∈ T2 and g(m−2,n)(x− x∗) ∈ T1.
For k > 2,m ≥ 2, we get
f(k,l,m,n)(x) =
[(g(k−2,l)(x) + x∗) + x] + [(g(m−2,n)(x− x∗) + (x− x∗)∗) + (x− x∗)] =
[g∗(k−2,l)(x) + (x∗ + x∗)] + [g∗(m−2,n)(x− x∗) + ((x− x∗)∗ + (x− x∗)∗] =
[g∗(k−2,l)(x) + g∗(m−2,n)(x− x∗)] + [(x∗ + x∗) + ((x− x∗)∗ + (x− x∗)∗)] =
[g∗(k−2,l)(x) + g∗(m−2,n)(x− x∗)] + (x∗ + x∗) =
[(x+ x) + g∗(k−2,l)(x)] + g(m−2,n)(x− x∗),
so m′ = 0 or after a finite number of similar steps, we get the term
operation of the required form.
The same conclusion can be drawn for the case g(k,l)(x), g(m,n)(x) ∈
T2.
3. Pseudo-nearrings
A pseudo-nearrning is an algebra (A; +, ·,∗ , η) of the type (2, 2, 1, 0)
fulfilling the following conditions:
(i) (A; +,∗ , η) is a commutative ∗-associative quasigroup,
(ii) (αβ)γ = α(βγ),
(iii) (αβ)∗ = α∗β,
and
(iv) (α+ β)γ = αγ + βγ,
136 Pseudo-nearrings and quasi-modules over them
for all α, β, γ ∈ A.
Example 4. Let (A; +,∗ , η) be a ∗-associative quasigroup and TA be
the set of all maps from A to itself. Then the algebra (TA; #, ◦,⊗ , fη),
where (f#g)(x) = f(x) + g(x), (f ◦ g)(x) = f(g(x)), f⊗(x) = (f(x))∗
and fη(x) = η, is a pseudo-nearring.
Indeed, it is easy to show that (TA; #,⊗ ) is a commutative ∗-associa-
tive groupoid. We prove that it is a quasigroup. Let h ∈ TA. Then
(fη#h)(x) = fη(x) + h(x) = η + h(x) = (h(x))∗ = h⊗(x), which gives
fη#h = h⊗.
By Lemma 2 of [2], if a ∈ A, then there exists a unique element b,
such that a + b = η = b + a. Define a map gη : A 7→ A as follows
gη(x) = y ⇔ x+ y = η.
Let h ∈ TA. Then ((gη ◦ h)#h)(x) = gη(h(x)) + h(x) = η = fη(x),
and, in consequence, (gη ◦ h)#h = fη.
Example 5. Let (Z;⊕,∗ ) be the ∗-associative quasigroup defined in
Example 3 and a multiplication be given by x◦y = a. Then (Z;⊕, ◦,∗ , a)
is a pseudo-nearring.
Example 6. Let (A; +,∗ , η) be a ∗-associative quasigroup. Define the
operation x ◦ y = x. Then (A; +, ◦,∗ , η) is also a pseudo-nearring.
Proposition 5. Let (A; +, ·,∗ , η) be a pseudo-nearring. Then
ηα = η and − (αβ) = (−α)β.
Proof. Suppose that α ∈ A. By Theorem 2 and (iii), (iv) from the
pseudo-nearring definition, we have:
η = ηα+ (−ηα) = (η∗ + η∗)α+ (−ηα) = (ηα+ ηα)∗ + (−(ηα)) =
= ηα+ (ηα+ (−(ηα))∗ = ηα.
The proof of the second property is immediate.
We use similar notations as in [10]. Define two subsets of a pseudo-
nearring (A; +, ·,∗ , η). A set Aη = {a ∈ A : aη = η} is called the
η-symmetric part of A and Ac = {a ∈ A : aη = a} is called the constant
part of A. It is evident that Aη, Ac are subalgebras of A.
Proposition 6. Let (A; +, ·,∗ , η) be a pseudo-nearring. Then
(∀a ∈ A) (∃aη ∈ Aη) (∃ac ∈ Ac) a = aη + ac.
Proof. Similarly as for nearrings the element a = (a∗ + (−a∗η))∗ + a∗η
will do the decomposition job.
A. Chwastyk, K. G lazek 137
A non-empty subset I of a pseudo-nearring A is said to be a left ideal
of A if:
(∀α, β ∈ I) (∀γ ∈ A) [γα ∈ I, α∗ ∈ I, α− β ∈ I].
Remark. The subset Ac is a left ideal of A.
4. Quasi-modules over pseudo-nearrings
Let (V ; +,∗ , ε) be a commutative ∗-associative quasigroup and A =
(A;⊕, ◦,⋆ , η) be a pseudo-nearring. Define a map fα : V → V ; u 7−→ αu
for all α ∈ A. If, for all α, β ∈ A; and for all u, v ∈ V , we have:
(i) (α⊕ β)u = αu+ βu,
(ii) α(u+ v) = αu+ αv,
(iii) (α ◦ β)u = α(βu),
(iv) αε = ε = ηu,
and
(v) αu = βu⇒ α = β,
then V with operations +,∗ , ε, (fα)α∈A is called a quasi-module over A.
Proposition 7. Let (V,+,∗ , ε, (fα)α∈A) be a quasi-module over
a pseudo-nearring (A;⊕, ◦,⋆ , η) and α ∈ A, u ∈ V . Then we have:
(a) α⋆u = (αu)∗ = αu∗;
(b) −(αu) = (−α)u = α(−u);
(c) αu = ε⇒ [α = η or u = ε].
Proof. Let α ∈ A, u ∈ V. Then
α⋆u = (α⊕ η)u = αu+ ηu = αu+ ε = (αu)∗ =
αu+ αε = α(u+ ε) = αu∗.
The rest of the proof is standard.
Denote
F(k,l,m,n,α)(x) = f(k,l,m,n)(x) + αx,
where f(k,l,m,n)(x) is defined as in Section 2, α ∈ A, x ∈ V .
138 Pseudo-nearrings and quasi-modules over them
Theorem 4. Every unary term operation in a quasi-module
(V,+,∗ , ε, (fα)α∈A) over a pseudo-nearring (A,⊕, ◦,⋆ , η) is of the form
±F(k,l,m,n,α)(x).
Proof. The proof is by induction with respect of the complexity of oper-
ations. We first observe that the projection has the required form:
e11(x) = x = (x+ ε) + ηx = F(1,0,0,0,η)(x).
The set of all operations of the form ±F(k,l,m,n,α)(x) is closed under
the quasi-module operations. Indeed, by Propositions 3(b) and 7(b), we
conclude that
−F(k,l,m,n,α)(x) = −f(k,l,m,n)(x) + (−α)x.
Taking into account Proposition 1(b), we get
F(k1,l1,m1,n1,α1)(x) + F(k2,l2,m2,n2,α2)(x) =
[f(k1,l1,m1,n1)(x) + f(k2,l2,m2,n2)(x)] + (α1 + α2)x.
From Proposition 7(a), it follows that
F ∗
(k,l,m,n,α)(x) = f∗(k,l,m,n)(x) + α⋆x.
Now, we verify that
βF(k,l,m,n,α)(x) = β[f(k,l,m,n)(x) + αx] =
β(g(k,l)(x) + g(m,n)(x− x∗)) + (βα)x =
(βg(k,l)(x) + βg(m,n)(x− x∗)) + (βα)x.
By parts (i) and (ii) of the quasi-module definition, and Proposition
7(a), we have
βg(k,l)(x) = β[. . . (xi1 + xi2) + . . .) + xil) + . . .) + xik ] =
(. . . (βxi1 + βxi2) + . . .) + βxil) + . . .) + βxik =
(. . . (βi1x+ βi2x) + . . .) + βilx) + . . .) + βikx =
[(. . . (βi1 + βi2) + . . .) + βil) + . . .) + βik ]x = g(k,l)(β)x.
And also βg(m,n)(x− x∗) = g(m,n)(β)(x− x∗) = [g(m,n)(β) − g∗(m,n)(β)]x.
Here, we use similar notations for term operations g(k,l)(x) in a quasi-
module as for term operations g(k,l)(β) in a pseudo-nearring.
Finally, we deduce that
βF(k,l,m,n,α)(x) = [g(k,l)(β)x+ (g(m,n)(β) − g∗(m,n)(β))x] + (βα)x =
[(g(k,l)(β) + (g(m,n)(β) − g∗(m,n)(β))) + (βα)]x = F(0,0,0,0,γ)(x),
where γ = [(g(k,l)(β) + (g(m,n)(β) − g∗(m,n)(β))) + (βα)]∗.
A. Chwastyk, K. G lazek 139
In the next paper we will use the obtained results, especially the
description of term operations in ∗-associative quasigroups and quasi-
modules over pseudo-nearrings, for the description of independent sets
(in the sense of Marczewski; see [8]) in the above algebras.
References
[1] I. Chajda, K. G lazek, A Basic Course on General Algebra, Technical Univ. Press,
Zielona Góra 2000.
[2] A. Chwastyk, K. G lazek, Remarks on ∗-associative groupoids, Contributions to
General Algebra 13 (2001), 83-89.
[3] A. Chwastyk, K. G lazek, Term operations in commutative ∗-associative groupoids,
Contributions to General Algebra 14 (2003), in print.
[4] K. G lazek, On some non-associative rings (in Polish), Acta Univ. Wratislav. 17
(1961), 15-19.
[5] K. G lazek, ∗-associative and γ-algebras (in Polish), Acta Univ.Wratislav. 58
(1967), 5-19.
[6] J. Hion, Ω-ringoids, Ω-rings and their representations (in Russian), Trudy
Moskov. Mat. Obshch. 14 (1965), 3-47.
[7] A.G. Kurosh, General Algebra Lectures of the 1969-1970 Academic Year (in Rus-
sian), Izdat. “Nauka”, Moscow 1974.
[8] E. Marczewski, Independence and homomorphism in abstract algebras, Fund.
Math. 50 (1961), 45-61.
[9] H.O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann-Verlag,
Berlin 1990.
[10] G. Pilz, Near-Rings, North-Holland Publ. Comp., Amsterdam 1983.
Contact information
Anna Chwastyk Institute of Mathematics, Technical Univer-
sity of Opole, ul. Waryńskiego 4, 45-047,
Opole, Poland
E-Mail: ach@polo.po.opole.pl
Kazimierz G lazek Institute of Mathematics, University of
Zielona Góra, ul. Podgórna 50, 65-246
Zielona Góra, Poland
E-Mail: k.glazek@im.uz.zgora.pl
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| id | nasplib_isofts_kiev_ua-123456789-124613 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-12-07T18:35:10Z |
| publishDate | 2004 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Chwastyk, A. Glazek, K. 2017-09-30T08:19:13Z 2017-09-30T08:19:13Z 2004 Pseudo-nearrings and quasi-modules over them / A. Chwastyk, K. Glazek // Український математичний вісник. — 2004. — Т. 1, № 1. — С. 129-139. — Бібліогр.: 10 назв. — англ. 1810-3200 2000 MSC. 20N02, 20N05, 16Y30, 16W10, 16D99. https://nasplib.isofts.kiev.ua/handle/123456789/124613 In this paper we start to investigate a new notion of pseudo-nearrings and a generalization of linear spaces to quasi-modules over pseudo-nearrings. Pseudo-nearrings can be treated as ringoids in the sense of J. Hion (see [6]). The idea of pseudo-nearings is based on the notion of a ∗-associative quasigroup, i.e. on an involutive groupoid (A;+,* ) in which the following identities hold: (x*)* = x, (x + y)* = y* + x*, (x + y)* + z = x + (y + z)*. We assume also commutativity and quasigroup properties of (A;+). en Інститут прикладної математики і механіки НАН України Український математичний вісник Pseudo-nearrings and quasi-modules over them Article published earlier |
| spellingShingle | Pseudo-nearrings and quasi-modules over them Chwastyk, A. Glazek, K. |
| title | Pseudo-nearrings and quasi-modules over them |
| title_full | Pseudo-nearrings and quasi-modules over them |
| title_fullStr | Pseudo-nearrings and quasi-modules over them |
| title_full_unstemmed | Pseudo-nearrings and quasi-modules over them |
| title_short | Pseudo-nearrings and quasi-modules over them |
| title_sort | pseudo-nearrings and quasi-modules over them |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/124613 |
| work_keys_str_mv | AT chwastyka pseudonearringsandquasimodulesoverthem AT glazekk pseudonearringsandquasimodulesoverthem |