Codes of groupoids with one-sided quasigroup conditions
We propose a method of description of quasigroup conditions defining one-sided quasigroup classes and onesided quasigroup varieties by number codes. This method can be used for all one-sided quasigroups. In order to construct the code words we use the technique of Steinitz numbers.
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nasplib_isofts_kiev_ua-123456789-1545622025-02-23T17:04:18Z Codes of groupoids with one-sided quasigroup conditions Ga̷luszka, J. We propose a method of description of quasigroup conditions defining one-sided quasigroup classes and onesided quasigroup varieties by number codes. This method can be used for all one-sided quasigroups. In order to construct the code words we use the technique of Steinitz numbers. 2009 Article Codes of groupoids with one-sided quasigroup conditions / J. Ga̷luszka // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 27–44. — Бібліогр.: 10 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:05A05, 05B15, 20N02, 20N05. https://nasplib.isofts.kiev.ua/handle/123456789/154562 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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We propose a method of description of quasigroup conditions defining one-sided quasigroup classes and onesided quasigroup varieties by number codes. This method can be used for all one-sided quasigroups. In order to construct the code words we use the technique of Steinitz numbers. |
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Ga̷luszka, J. Codes of groupoids with one-sided quasigroup conditions Algebra and Discrete Mathematics |
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Codes of groupoids with one-sided quasigroup conditions |
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Codes of groupoids with one-sided quasigroup conditions |
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Codes of groupoids with one-sided quasigroup conditions |
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Codes of groupoids with one-sided quasigroup conditions |
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Codes of groupoids with one-sided quasigroup conditions |
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codes of groupoids with one-sided quasigroup conditions |
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Інститут прикладної математики і механіки НАН України |
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Codes of groupoids with one-sided quasigroup conditions / J. Ga̷luszka // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 27–44. — Бібліогр.: 10 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT galuszkaj codesofgroupoidswithonesidedquasigroupconditions |
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2025-11-24T02:42:16Z |
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2025-11-24T02:42:16Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2009). pp. 27 – 44
c⃝ Journal “Algebra and Discrete Mathematics”
Codes of groupoids with one-sided quasigroup
conditions
Jan Gal̷uszka
Communicated by V. I. Sushchansky
Abstract. We propose a method of description of quasi-
group conditions defining one-sided quasigroup classes and one-
sided quasigroup varieties by number codes. This method can be
used for all one-sided quasigroups. In order to construct the code
words we use the technique of Steinitz numbers.
1. Introduction
We propose a technique of describing quasigroup conditions defining one-
sided quasigroup classes and one-sided quasigroup varieties by number
codes. To obtain these descriptions we apply some combinatorial meth-
ods involving, among other things, the concept of the cycle type of a per-
mutation. We use a technique which arises from an idea of E. Steinitz
(see [10, p. 250]). Today this construction is known as Steinitz numbers
([3, 9]) (or supernatural or surnatural numbers ([8])). These numbers
have found applications in various parts of algebra, especially in group
theory, field theory and some related areas. Examples of such applications
can be found in [5, 6, 7]. Construction of the Steinitz numbers — slightly
reformulated — is recalled briefly to make the paper self-contained.
Our method has common roots with one introduced in [4], but is
slightly more general and applies to all one-sided quasigroups. In the
main theorem (Theorem 9.2) we show that the quasigroup conditions
defining one-sided quasigroup classes and one-sided quasigroup varieties
2000 Mathematics Subject Classification: 05A05, 05B15, 20N02, 20N05.
Key words and phrases: groupoid, quasigroup, right quasigroup, left quasi-
group, one-sided quasigroup, cycle type description.
28 Codes of groupoids
as well as conditions establishing relations between these classes can be
precisely expressed by sequences of extended natural numbers interpreted
as Steinitz numbers. This result was announced without proof at the
AAA 78 – 78th Conference on General Algebra (Bern 2009). Some ex-
amples and applications are also given.
2. Basic notation
To avoid misunderstandings, we recall some basic notation and terminol-
ogy:
∙ ℕ denotes the set of natural numbers (nonnegative integers).
∙ ℕ1 := ℕ− {0} denotes the set of positive integers.
∙ n ∣ m where n,m ∈ ℕ1 means that n divides m.
∙ ℙ = {p1, p2, . . .} where p1 < p2 < ⋅ ⋅ ⋅ denotes the set of prime
numbers.
∙ ℕ̄ := ℕ ∪ {ℵ0} denotes the extension of ℕ by adding a new largest
element ℵ0.
∙ ℕ̄1 := ℕ̄− {0}.
∙ N̄ = (ℕ̄,≤) and N̄1 = (ℕ̄1,≤) are both bounded chains.
∙ ∣A∣ denotes the cardinality of the set A.
∙ If A ⊆ CnCnCn is a set of cardinal numbers, then lcm(A) denotes the
least common multiple of A.
∙ Let A and B be sets. Then AB := {f ∣ f : B −→ A} denotes the
set of all maps from B to A, often identified with their graphs, so
viewed as subsets of B ×A.
3. Steinitz numbers
3.1. Construction
The Steinitz numbers ([10]) are the relational system S = (S,≤) where
S := ℕ̄ℙ is the set of sequences (sp)p∈ℙ, where each sp is a natural number
or ℵ0, and ≤ is defined coordinatewise by the relation from N̄ = (ℕ̄,≤).
More precisely, if s1 = (s1,p)p∈ℙ, s2 = (s2,p)p∈ℙ ∈ S then
s1 ≤ s2 :⇔ ∀ p ∈ ℙ s1,p ≤ s2,p.
J. Gal̷uszka 29
For reasons that will become clear shortly, we sometimes write s1 ∣ s2
instead of s1 ≤ s2 and call this relation ‘divisibility’.
3.2. Closed Steinitz numbers
In the next sections we show that one can assign Steinitz numbers to
quasigroups. To enable this assignment for every right (and left) quasi-
group we need to extend the Steinitz numbers, just as ℕ was extended
to ℕ̄.
Let ℙ̄ = ℙ∪{ℵ0} and S̄ := ℕ̄ℙ̄. Members of S̄ are called closed Steinitz
numbers. Just as in S, the relation ≤ on S̄ is defined coordinatewise, and
sometimes called ‘divisibility’. We write S̄ = (S̄,≤) = (S̄, ∣).
3.3. Selected properties
We simplifiy the notations as follows:
∙ (n1, . . . , nk, ∞. . .) denotes the Steinitz number (sp)p∈ℙ such that:
sp1 = n1,. . . , spk = nk and spi = nk for i ∈ ℕ and i > k.
∙ (n1, . . . , nk, ∞. . .)(nℵ0) denotes the closed Steinitz number (sp)p∈ℙ̄
such that:
sp1 = n1,. . . , spk = nk, spi = nk for i ∈ ℕ, i > k and sℵ0 = nℵ0 .
The easy proof of the following fact is omitted.
Fact 3.1.
(i) The relational systems S and S̄ are both complete distributive bounded
lattices. (ℵ0, ∞. . .) and (0, ∞. . .) are the unit and null respectively in S,
whereas (ℵ0, ∞. . .)(ℵ0) and (0, ∞. . .)(0) are the unit and null respectively
in S̄.
(ii) There are two natural monomorphic embeddings of relational sys-
tems:
( ) : N1
mon
−→ S (n 7→ (n)), (1)
⟨ ⟩ : N̄1
mon
−→ S̄ (n 7→ ⟨n⟩), (2)
defined as follows:
∙ (1) = (0, ∞. . .), ⟨1⟩ = (0, ∞. . .)(0).
∙ If n ∈ ℕ, n > 1 and n = 2n13n2 . . . pnk is the prime factoriza-
tion of n, then
(n) = (n1, . . . , nk, ∞. . .), ⟨n⟩ = (n1, . . . , nk, ∞. . .)(0).
30 Codes of groupoids
∙ ⟨ℵ0⟩ = (0, ∞. . .)(1).
(iii) S is a proper sublattice of S̄ but because these lattices have different
units, S is not a bounded sublattice of S̄.
It is now clear that if n,m ∈ ℕ and n,m ≥ 1, then (n) ≤ (m) means
exactly that n ∣ m, which explains the name ‘divisibility’ for the order
relation in N.
3.4. Examples
(i) Consider the number 360 in the decimal system. Then:
(360) = (3, 2, 1, 0, ∞. . .) or less formally but more intuitively
(360) = 23325170 . . . p0i . . .;
⟨360⟩ = (3, 2, 1, 0, ∞. . .)(0) or less formally but more intuitively
⟨360⟩ = 23325170 . . . p0i . . .ℵ
0
0.
(ii) The sequence (3, 2, 1, 0, ∞. . .)(1) is a ‘proper’ closed Steinitz number,
i.e.
∀n ∈ ℕ1 ⟨n⟩ ∕= (3, 2, 1, 0, ∞. . .)(1).
(iii) Evidently ⟨360⟩ = (3, 2, 1, 0, ∞. . .)(0) ∣ (3, 2, 1, 0, ∞. . .)(1).
4. Groupoids with right (left) quasigroup properties
4.1. Right (left) quasigroups and quasigroups
Definitions
By a groupoid we mean a pair G = (G, ⋅ ) with universe G and binary
operation
⋅ : G×G −→ G ((x, y) 7→ xy).
In the following, the Gothic letters (without or with subscripts), like G,
ℌ, Gi and ℌj stand for groupoids only.
By a right (resp. left) quasigroup we mean a groupoid G such that
for all a, b ∈ G the equation xa = b (resp. ax = b) has a unique solution.
By a quasigroup we mean a groupoid which is a right and left quasigroup
simultaneously.
Duality
For a groupoid G = (G, ⋅ ) we have the dual groupoid G← = (G, ∘ ) where
x ∘ y := yx. Clearly (G←)← = G. Let t be a term over a language appro-
priate for groupoid theory. Let G be a groupoid. Then the interpretation
tG
←
is named the dual sentence to the interpretation tG. If a groupoid G
J. Gal̷uszka 31
is a right quasigroup then its dual groupoid is a left quasigroup and vice
versa. This duality establishes a symmetrical correspondence between
‘right’ and ‘left’ versions of statements. For conciseness we formulate
almost all statements below in one (right) version only.
Combinatorial approach
In combinatorial terminology, a groupoid G is a right quasigroup if and
only if for every a ∈ G the right translation
rGa : G −→ G (x 7→ xa)
is a bijection. Thus, we have the function
rG : G −→ SG (x 7→ rGx )
where SG denotes the set of bijections of G. Denote by QG∗ the class
of right quasigroups and by QG∗[G] the set of right quasigroups with
universe G. In this way we obtain the bijection
%∗G : QG∗[G] −→ (SG)
G (x 7→ rx). (3)
Informally but intuitively, a right quasigroup structure on G can be
seen as a bundle of bijections (rGa )a∈G. More formally, one can define a
map ' : G×G → G by
'(u, v) = a :⇔ ua = v
Then for each a ∈ G the fibre '−1(a) = {(u, v) ∣ '(u, v) = a} = {(u, ua) ∣
a ∈ G}, so it is precisely (the graph of) rSa (cf. [1]).
This interpretation is illustrated in Figure 1. The fibres are presented
as vertical lines. Horizontal coordinates determine the values of the right
translations. The universe of the quasigroup is G = {a� ∣ � ∈ I}.
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
rGa�
a�
aϰ
�
��+
Q
Q
Q
Qs
rGa�
(aϰ)
Figure 1: Combinatorial interpretation of a right quasigroup G
32 Codes of groupoids
Cycle types
For any set G, every permutation of G can be (uniquely) decomposed
into disjoint cycles. This yields the map
� : SG −→ CnCnCnℕ̄1 (x 7→ �(x) = x⃗ = (x⃗i)i∈ℕ̄1
) (4)
where fi ∈ Cn denotes the number of cycles of length i appearing in
the disjoint cycle decomposition of f (fℵ0 denotes the number of infinite
cycles). This map is called the cycle type description of the permutation
f (see [2]).
If a right quasigroup G is fixed then we write simply ra instead of rGa
and r instead of rG. The map giving the cycle type description of all the
permutations ra for a ∈ G,
r⃗ := �r : G −→ CnCnCnℕ̄1 (x 7→ �(rx) = r⃗x = (r⃗xi)i∈ℕ̄1
)
is called the cycle type of G. Recall that %∗G(G) = rG = r ∈ (SG)
G. Thus
we have the following map (cf. (3)):
�G%∗G : QG∗[G] −→ (CnCnCnℕ̄1)
G
(x 7→ r⃗x) (5)
where �G is the product map defined on (SG)
G as follows:
�G((fa)a∈G) := (�(fa))a∈G.
5. Cycle codes. Code numbers
5.1. Definitions and main concepts
Full cycle codes
Let r⃗ be the cycle type of a right quasigroup G. Let i ∈ ℕ̄1. The i-th
support of G is the set
suppir⃗ := {a ∣ r⃗ai ∕= 0}. (6)
By the full cycle code of G we mean the sequence
&(r⃗) = (&i)i∈ℕ̄1
:=
∑
a∈G
r⃗a =
∑
a∈G
(r⃗ai)i∈ℕ̄1
= (
∑
a∈G
r⃗ai)i∈ℕ̄1
(7)
where
&i =
∑
a∈G
ra,i :=
⎧
⎨
⎩
∑
a∈ suppir⃗
r⃗ai if
⎧
⎨
⎩
∣ suppir⃗ ∣ < ℵ0
and
∀ a ∈ suppir⃗ r⃗ai < ℵ0
;
∣ suppir⃗ ∣(
∪
{ r⃗ai ∣ a ∈ suppir⃗ }) otherwise.
The full cycle code has a connection with the cardinality of the groupoid.
This connection can be easily seen for finite groupoids (cf. (10) below).
J. Gal̷uszka 33
Cycle codes
By the cycle code of G we mean the sequence of closed Steinitz numbers
G⃗ := ("i)i∈ℕ̄1
(8)
associated with the full cycle code &(r⃗) (cf. (7)) as follows:
"i :=
{
⟨i⟩ if &i ∕= 0,
⟨1⟩ = (0, ∞. . .)(0) otherwise.
The cycle code carries information about the ‘torsion’ structure of the
groupoid (cf. (∗t) below). From the technical point of view G⃗ ∈ S̄ℕ̄1 and
can be interpreted as an infinite matrix.
Code numbers
By the code number of G we mean the closed Steinitz number
⟨G⟩ := max{"i ∣ i ∈ ℕ̄1}. (9)
The code number carries summary information about the ‘torsion’ and
‘equational’ structure of the groupoid (see Theorem 9.2 below).
5.2. Examples
Let G1 be the right quasigroup with universe ℕ1 determined by the family
of permutations (fn)n∈ℕ1
defined by
fn = (1 2) ∪ . . . ∪ (2n− 1 2n) ∪ (2n+ 1 . . . 2n+ pn) ∪ id (n ∈ ℕ1).
Thus
f1 = (1 2) ∪ (3 4) ∪ id,
f2 = (1 2) ∪ (3 4) ∪ (5 6 7) ∪ id,
f3 = (1 2) ∪ (3 4) ∪ (5 6) ∪ (7 8 9 10 11) ∪ id,
. . .
34 Codes of groupoids
G1 has the infinite Cayley table:
f1 f2 f3 . . .
G1 1 2 3 4 5 6 7 8 9 10 11 . . .
1 2 2 2 2 2 2 2 2 2 2 2 . . .
2 1 1 1 1 1 1 1 1 1 1 1 . . .
3 4 4 4 4 4 4 4 4 4 4 4 . . .
4 3 3 3 3 3 3 3 3 3 3 3 . . .
5 5 6 6 6 6 6 6 6 6 6 6 . . .
6 6 7 5 5 5 5 5 5 5 5 5 . . .
7 7 5 8 8 8 8 8 8 8 8 8 . . .
8 8 8 9 7 7 7 7 7 7 7 7 . . .
9 9 9 10 10 10 10 10 10 10 10 10 . . .
10 10 10 11 11 9 9 9 9 9 9 9 . . .
11 11 11 7 12 12 12 12 12 12 12 12 . . .
...
...
...
...
...
...
...
...
...
...
...
...
. . . .
Thus:
∙ r⃗ = ((r⃗j i)i∈ℕ̄1
)j∈ℕ1
where
r⃗j i =
⎧
⎨
⎩
ℵ0 if j ∈ ℕ1, i = 1;
2 if j = 1, i = 2;
j if j ≥ 2, i = 2;
1 if j ≥ 2, i = pj ;
0 otherwise.
∙ &(r⃗) = (&i)i∈ℕ̄1
where
&i =
⎧
⎨
⎩
ℵ0 if i ∈ {1, 2};
1 if i ∈ ℙ, i ∕= 2;
0 otherwise.
∙ G⃗1 = ("i)i∈ℕ̄1
where
"i =
{
⟨pi⟩ if i ∈ ℕ1;
⟨1⟩ if i = ℵ0.
∙ ⟨G1⟩ = (1, ∞. . .)(0).
J. Gal̷uszka 35
Let G2 be the right quasigroup with universe ℕ1 determined by the
family of permutations (fn)n∈ℕ1
defined by
fn :=
{
g if n = 1,
(1 . . . n− 1) ∪ idℕ1−{1,...,n−1} if n > 1,
where g is the permutation of ℕ1 defined as follows:
g(k) =
⎧
⎨
⎩
k − 2 if k is odd, k ≥ 3,
k + 2 if k is even, k ≥ 2,
2 if k = 1.
Thus:
∙ r⃗ = ((r⃗j i)i∈ℕ̄1
)j∈ℕ1
where
r⃗j i =
⎧
⎨
⎩
0 if j = 1, i ∕= ℵ0;
1 if j = 1, i = ℵ0;
ℵ0 if j > 1, i = 1;
1 if j > 1, i = j;
0 otherwise.
∙ &(r⃗) = (&i)i∈ℕ̄1
where
&i =
{
ℵ0 if i = 1;
1 otherwise.
∙ G⃗2 = (⟨i⟩)i∈ℕ̄1
.
∙ ⟨G2⟩ = (ℵ0, ∞. . .)(1).
There is no positive integer n such that ⟨n⟩ = ⟨G2⟩.
6. Cycle codes and code numbers of finite groupoids
For all finite and some other one-sided quasigroups one can construct so
called code tables. These tables — presented as matrices — allow us to
clearly present the meaning of the notions introduced above.
6.1. Code tables
Let G = (G, ⋅) where G = {g1, g2, . . . , gn} is a finite right quasigroup with
Cayley table
36 Codes of groupoids
G g1 g2 . . . gn
g1 g11 g12 . . . g1n
g2 g21 g22 . . . g2n
...
...
...
...
...
gn gn1 gn2 . . . gnn .
For finite groupoids the code parameters defined in (7), (8) and (9)
can be clearly presented in the following code table (the formally infinite
sequences are written in shorter forms, with null elements omitted):
(rg)g∈G r⃗g1 r⃗g2 . . . r⃗gn &(r⃗) G⃗
1 rg1,1 rg2,1 . . . rgn,1 &1 "1
2 rg1,2 rg2,2 . . . rgn,2 &2 "2
...
...
...
...
...
...
...
n rg1,n rg2,n . . . rgn,n &n "n
⟨G⟩ .
On can check that the following formula holds:
∑
i∈ℕ1
i&i = ∣G∣. (10)
6.2. Example
Let G3 be the right quasigroup with Cayley table
G3 0 1 2 3 4
0 0 0 0 0 1
1 1 1 1 2 2
2 2 2 3 3 3
3 3 4 4 4 4
4 4 3 2 1 0 .
The code table of G3 is:
(ra)a∈G3
r⃗0 r⃗1 r⃗2 r⃗3 r⃗4 &(r⃗) G⃗3
1 5 5 2 1 0 13 ⟨1⟩ = (0, ∞. . .)(0)
2 0 0 0 0 0 0 ⟨1⟩ = (0, ∞. . .)(0)
3 0 0 1 0 0 1 ⟨3⟩ = (0, 1, 0, ∞. . .)(0)
4 0 0 0 1 0 1 ⟨4⟩ = (2, 0, ∞. . .)(0)
5 0 0 0 0 1 1 ⟨5⟩ = (0, 0, 1, 0, ∞. . .)(0)
⟨G3⟩ = (2, 1, 1, 0, ∞. . .)(0) .
Thus ⟨G3⟩ = (2, 1, 1, 0, ∞. . .)(0) = ⟨60⟩.
J. Gal̷uszka 37
7. Connections of codes of groupoids with other notions
For convenience of the reader we repeat some definitions from [4] to make
our presentation self-contained.
Consider the function
� : CnCnCnℕ̄1 −→ P(ℕ̄1)
where P(ℕ̄1) denotes the power set of ℕ̄1 and �(g) := {i ∈ ℕ̄1 ∣ g(i) ∕= 0}
for g ∈ CnCnCnℕ̄1 . We call �(g) the support of g. The composition �� (cf.
(4)) assigns to every permutation f ∈ SG its cycle support �(f⃗ ). For its
product function we have (��)G((fa)a∈G) = �G(f⃗ ) = (�(f⃗a))a∈G. Let
%∗�G be the function defined on QG∗[G] as follows:
%∗�G (G) := (��)G(%∗G(G)).
We call %∗�G (G) the cycle support of the right quasigroup G. Consider
im %∗�G (G), the image of the function %∗�G (G). Define
Σ∗G :=
∪
im %∗�G (G).
It is clear that the set Σ∗
G
is nonempty. Moreover, n ∈ Σ∗
G
if and only if
there exists a ∈ G such that there exists a cycle of length n ∈ ℕ̄1 in the
disjoint cycle decomposition of rGa . By the cycle weight of G we mean
(cf. [4]) the number
!∗G := lcm(Σ∗G).
It is clear that this is a positive integer and it does not exist for all right
quasigroups. As in [4] we say that a right quasigroup G has finite cycle
type if
0 < ∣Σ∗G∣ < ℵ0 ∧ Σ∗G ⊆ ℕ1. (11)
A right quasigroup G has countable cycle type if Σ∗
G
⊆ ℕ1. Some appli-
cations of the above notions apear in Theorems 9.1 and 10.1 of Section 9.
It is clear that G has finite cycle type if and only if !∗
G
is ‘well defined’,
i.e. !∗
G
∈ ℕ1. Evidently there exist right quasigroups not having finite
cycle type (for example G1 and G2 from Section 5.2). For them, the
cycle weight does not exist. The code number introduced in this paper is
a more general notion and applies for all right quasigroups. The following
theorem gives connections between these notions:
Theorem 7.1. Let G be a right quasigroup. Then
∀n ∈ ℕ1 (!∗G = n ⇐⇒ ⟨G⟩ = ⟨n⟩ ). (12)
38 Codes of groupoids
Proof. ⇒. Because (11) is satisfied in G, the following formula holds:
∃ p ∈ ℙ ∀m ∈ Σ∗G ∃ (�m,2, . . . , �m,p) ∈ ℕ
p m = 2�m,23�m,3 . . . p�m,p .
Thus !∗
G
= 2�23�3 . . . p�p where �i = max{�m,i ∣ m ∈ Σ∗
G
}. Hence
⟨!∗
G
⟩ = (�2, �3, . . . , �p, 0, ∞. . .)(0) where �i ∕= 0 if and only if for some a ∈
G there exists a cycle of length k ∈ ℕ̄1 in the disjoint cycle decomposition
of rGa and i ∣ k.
Let us consider the cycle code of G, i.e. the sequence G⃗ = ("i)i∈ℕ̄1
(see (8)). Thus there exists l ∈ ℕ̄1 such that "i = ⟨1⟩ for all i ≥ l.
Moreover, for i < l we know that "i ∕= ⟨1⟩ if and only if for some a ∈ G
there exists a cycle of length i ∈ ℕ̄1 and i ∕= 1 in the disjoint cycle
decomposition of rGa . Thus "i ∕= ⟨1⟩ if and only if i ∈ Σ∗
G
and i ∕= 1.
Hence ⟨G⟩ = max{"i ∣ i ∈ Σ∗
G
}. Thus ⟨G⟩ = ⟨!∗
G
⟩.
⇐. Let ⟨G⟩ = ⟨n⟩ for n ∈ ℕ1. Then Σ∗
G
satisfies (11). Thus !∗
G
exists
and !∗
G
= m for some m ∈ ℕ1. Applying the part (⇒) proved previously
we see that ⟨G⟩ = ⟨n⟩ = ⟨m⟩. Thus n = m and !∗
G
= n.
8. Power conditions. Power classes
In this section we focus our attention on conditions defining quasigroup
properties.
8.1. Main notions
The family of ‘power’ terms {xyn ∣ n ∈ ℕ1} is inductively defined as
follows:
xy1 := xy, xyn := (xyn−1)y,
and similarly for {nyx ∣ n ∈ ℕ1}. We recall some ‘power’ conditions
associated with these terms (see [4]). For n ∈ ℕ1 we have the identity
xyn = x. (∗n)
The variety of groupoids defined by the identity (∗n) is denoted by Qn.
The elements of Qn are called groupoids of right exponent n. We set
Q := {Qn ∣ n ∈ ℕ1}, Q∗ :=
∪
Q. (13)
The groupoids in Q∗ are said to be of finite right exponent. As a gener-
alization of (∗n) we propose the following conditions:
∀x, y ∃n ∈ ℕ1 xyn = x. (∗t)
J. Gal̷uszka 39
We call G right-torsion if the formula (∗t) is satisfied in G. The class of
right-torsion groupoids is denoted by T ∗ (in [4] this class is denoted by
QG∗t ).
Let s ∈ S. The following formulas can be seen as generalizations of
the identities (∗n) for Steinitz numbers:
∃n ∈ ℕ1 ∀x, y n ∣ s, xyn = x. (∗s)
Thus G satisfies (∗s) if and only if there exists n ∈ ℕ1 such that n ∣ s
and G satisfies (∗n). Qs̄ denotes the class of groupoids satisfying (∗s).
The members of Qs̄ are called s-bounded right exponent groupoids. If
m ∈ ℕ1 then Qm = Qm̄. Thus the bar over s in Qs̄ can be omitted.
Clearly if s ∈ S then
Qs =
∪
n∈ℕ1
n∣s
Qn.
Thus
∪
s∈S
Qs =
∪
s∈S
(
∪
n∈ℕ1
n∣s
Qn) = Q∗.
Let s ∈ S. As a generalization of the condition (∗t) for Steinitz
numbers we have:
∀x, y ∃n ∈ ℕ1 n ∣ s, xyn = x. (∗ts)
The class of all groupoids satisfying (∗ts) is denoted by Ts. The elements
of Ts are called s-bounded right torsion groupoids. We have the following
equality:
∪
s∈S
Ts = T ∗.
To establish connections between the above classes we record the fol-
lowing proposition:
Proposition 8.1 ([4], Proposition 14).
Q∗ ⊈ T ∗ ⊈ QG∗. (14)
40 Codes of groupoids
9. Main results
We start by recalling a useful theorem from [4].
Theorem 9.1 ([4], Theorem 5).
(i) G ∈ Qn if and only if G is a right quasigroup of finite cycle type
and its cycle weight !∗
G
divides n.
(ii) Let G be a right quasigroup of finite cycle type. Then
G ∈ Q!∗
G
−
∪
i<!∗
G
Qi.
In the next theorem, for uniformity of notation, we apply among
other things the map described in (1) and we identify ℕ1 with its image
under ( ) in S as well as S with its image under inclusion in S̄. We
consequently identify N1 with its isomorphic image in S as well as S
with its embedding in S̄. Thus
ℕ1 ⊆ S ⊆ S̄.
Also in the proof of this theorem for n ∈ ℕ1 we write briefly n instead of
⟨n⟩.
Theorem 9.2. Let G be a right quasigroup. Then
G ∈ Q∗ ⇐⇒ ⟨G⟩ ∈ ℕ1. (15)
G ∈ T ∗ −Q∗ ⇐⇒ ⟨G⟩ ∈ S− ℕ1. (16)
G ∈ QG∗ − T ∗ ⇐⇒ ⟨G⟩ ∈ S̄− S. (17)
Proof. (15),⇒. Let G ∈ Q∗. Then using (13) and next Theorem 9.1
we find that G ∈ Q!∗
G
(where !∗
G
∈ ℕ1). Hence by (12) the equality
⟨G⟩ = !∗
G
is satisfied.
(15),⇐. Let ⟨G⟩ = n for some n ∈ ℕ1. Then using Theorem 9.1 and
next (13) we infer that G ∈ Q∗.
(16),⇒. Let G ∈ T ∗ − Q∗. Evidently G ∈ QG∗. By assumption
G is a right torsion quasigroup, i.e. G satisfies (∗t). Thus there is no
element a ∈ G such that ra has an infinite cycle in the disjoint cycle
decomposition. This means that ℵ0 /∈ Σ∗
G
, i.e. Σ∗
G
⊆ ℕ1. Let s ∈ S̄ be
such that ⟨G⟩ = s (recall that for every right quasigroup this number
exists: cf. Section 5.1). Thus sℵ0 = 0 and ⟨G⟩ ∈ S. By assumption,
G /∈ Q∗. Theorem 7.1 implies that (11) is not satisfied in G. Because the
second condition of (11) is satisfied, the first does not hold in G. This
means that ∣Σ∗
G
∣ = ℵ0. Hence ⟨G⟩ ∈ S− ℕ1.
J. Gal̷uszka 41
(16),⇐. Let ⟨G⟩ = s for some s ∈ S − ℕ1. Thus for every i ∈ ℕ1
there exists j ∈ ℕ1 such that i ≤ j, sj ∕= 0. Moreover, sℵ0 = 0. In this
situation:
(a) There is no a ∈ G such that ra has an infinite cycle as factor in
the disjoint cycle decomposition.
(b) For every i ∈ ℕ1 there exists a ∈ G such that in the disjoint cycle
decomposition of ra there exists a factor of length greater than i.
Hence, G ∈ T ∗ (by (a)) and G /∈ Q∗ (by (b)).
(17),⇒. Let G ∈ QG∗−T ∗. This means that G is a right quasigroup
not satisfying (∗t). Thus
∃ y ∃x ∀n ∈ ℕ1 xyn ∕= x.
Hence for some a ∈ G the following formula is satisfied:
∃x ∀n ∈ ℕ1 xan ∕= x.
Thus for some a ∈ G, ra has an infinite cycle in the disjoint cycle decom-
position. Hence ⟨G⟩ ∈ S̄− S.
(17),⇐. Let ⟨G⟩ ∈ S̄− S. Thus ⟨G⟩ℵ0 ∕= 0. Thus for some a ∈ G, ra
has an infinite cycle in the disjoint cycle decomposition. Hence G does
not satisfy (∗t) and G ∈ QG∗ − T ∗.
10. Examples and applications
10.1. Examples
s-bounded right exponent groupoids
Let (ℌn)n∈ℕ1
be a sequence of finite right quasigroups such that ℌn is
a right quasigroup associated with the family of permutations (fi)i∈n
defined by
fi :=
{
(0 . . . 2j − 1) ∪ idn−{0,...,2j−1} if i = 2j − 1 for some j,
idn otherwise.
Using Cayley tables we can illustrate this sequence as follows:
ℌ1 0
0 0
,
ℌ2 0 1
0 0 1
1 1 0
,
ℌ3 0 1 2
0 0 1 0
1 1 0 1
2 2 2 2
,
ℌ4 0 1 2 3
0 0 1 0 3
1 1 0 1 0
2 2 2 2 1
3 3 3 3 2
, . . . ,
42 Codes of groupoids
ℌ2n 0 1 2 3 4 ⋅ ⋅ ⋅ 2n − 1
0 0 1 0 3 0 ⋅ ⋅ ⋅ 2n − 1
1 1 0 1 0 1 ⋅ ⋅ ⋅ 0
2 2 2 2 1 2 ⋅ ⋅ ⋅ 1
3 3 3 3 2 3 ⋅ ⋅ ⋅ 2
4 4 4 4 4 4 ⋅ ⋅ ⋅ 3
...
...
...
...
...
...
...
...
2n − 1 2n − 1 2n − 1 2n − 1 2n − 1 2n − 1 ⋅ ⋅ ⋅ 2n − 2
, . . . .
Evidently the sequence (ℌn)n∈ℕ1
is strictly increasing. More precisely,
we have the following embeddings:
ℌ1
id
−→ ℌ2
id
−→ ℌ3
id
−→ ℌ4 . . . ℌ2n−1
id
−→ ℌ2n . . . .
One can check that:
⟨ℌ1⟩ = ⟨1⟩ = (0, ∞. . .)(0), ℌ1 ∈ Q1,
⟨ℌ2⟩ = ⟨2⟩ = (1, 0, ∞. . .)(0), ℌ2 ∈ Q2,
⟨ℌ3⟩ = ⟨2⟩ = (1, 0, ∞. . .)(0), ℌ3 ∈ Q2,
⟨ℌ4⟩ = ⟨4⟩ = (2, 0, ∞. . .)(0), ℌ4 ∈ Q4,
. . .
⟨ℌ2n⟩ = ⟨2n⟩ = (n, 0, ∞. . .)(0), ℌ2n ∈ Q2n .
It is clear that
∀n ∈ ℕ ⟨n⟩ ∣ (ℵ0, 0, ∞. . .)(0).
Thus
ℌ1,ℌ2, . . . ,ℌ2n , . . . ∈ Q2ℵ0 .
s-bounded right torsion groupoids
Let us consider the right quasigroup G2ℵ0 associated with the family of
permutations (fi)i∈ℕ defined by
fi :=
{
(0 . . . 2j − 1) ∪ idℕ−{0,...,2j−1} if i = 2j − 1 for some j,
idℕ otherwise.
The upper left corner of the Cayley table of G2ℵ0 looks as follows:
J. Gal̷uszka 43
ℌ2ℵ0 0 1 2 3 4 ⋅ ⋅ ⋅ 2n − 1 2n ⋅ ⋅ ⋅
0 0 1 0 3 0 ⋅ ⋅ ⋅ 2n − 1 0 ⋅ ⋅ ⋅
1 1 0 1 0 1 ⋅ ⋅ ⋅ 0 1 ⋅ ⋅ ⋅
2 2 2 2 1 2 ⋅ ⋅ ⋅ 1 2 ⋅ ⋅ ⋅
3 3 3 3 2 3 ⋅ ⋅ ⋅ 2 3 ⋅ ⋅ ⋅
4 4 4 4 4 4 ⋅ ⋅ ⋅ 3 4 ⋅ ⋅ ⋅
...
...
...
...
...
...
...
...
... ⋅ ⋅ ⋅
2n − 1 2n − 1 2n − 1 2n − 1 2n − 1 2n − 1 ⋅ ⋅ ⋅ 2n − 2 2n − 1 ⋅ ⋅ ⋅
2n 2n 2n 2n 2n 2n ⋅ ⋅ ⋅ 2n 2n ⋅ ⋅ ⋅
...
...
...
...
...
...
...
...
...
. . . .
For this right quasigroup we have
⟨ℌ2ℵ0 ⟩ = (ℵ0, 0, ∞. . .)(0), ℌ2ℵ0 ∈ T2ℵ0 .
10.2. Some applications
Using the notions of number codes, some theorem of [4] can be slightly
enhanced by adding a new fourth item. Below we give two examples of
such enhancements. The proofs are omitted because the equivalence of
the first three items was proved in [4]. The proof of the equivalence of
the first three items with the fourth item is not hard.
Theorem 10.1 (cf. [4], Theorem 9). The following conditions are equiv-
alent:
(i) G is a groupoid of right exponent !∗
G
.
(ii) G is a groupoid of finite right exponent.
(iii) G is a right quasigroup of finite cycle type.
(iv) G is a right quasigroup with ⟨G⟩ = ⟨n⟩ for some positive integer n.
Theorem 10.2 (cf. [4], Theorem 15). The following conditions are equiv-
alent:
(i) G is a right-torsion groupoid.
(ii) G is a right quasigroup satisfying (∗t).
(iii) G is a right quasigroup of countable cycle type.
(iv) G is a right quasigroup with ⟨G⟩ = s for some closed Steinitz num-
ber s.
44 Codes of groupoids
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Contact information
J. Gal̷uszka Institute of Mathematics, Silesian Univer-
sity of Technology, Kaszubska 23, 44-100
Gliwice, Poland
E-Mail: jan.galuszka@polsl.pl
Received by the editors: 15.05.2009
and in final form 12.10.2009.
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