[ Pobierz całość w formacie PDF ]

THEOREM 3.1.3: Every diassociative loop is a S-loop.
Proof: Since a loop L is diassociative if every pair of elements of L generate a
subgroup in L. Thus every diassociative loop is a S-loop.
THEOREM 3.1.4: Let L be a Moufang loop which is centrally nilpotent of class 2.
Then L is a S-loop.
Proof: We know if L is a Moufang loop which is centrally nilpotent of class 2, that is, a
Moufang loop L such that the quotient of L by its centre Z(L) is an abelian group; and
let L denote the set of all elements of L whose order is a power of p. Recall that the
p
nuclearly derived subloop, or normal associator subloop of L, which we denote by L"
is the smallest normal subloop of L such that L/L" is associative (i.e. a group). Recall
48
also that the torsion subloop (subloop of finite order elements) of L is isomorphic to
the (restricted) direct product of the subloops L where p runs over all primes.
p
(Theorem 6.2. of R. H. Bruck [6] and [7] some theorems on Moufang loops and
Theorem 3.9 of Tim Hsu [63] or in the finite case, Corollary 1 of Glauberman and
Wright [22]). Using the main theorem A of Tim Hsu [63] we see that for p > 3, L is a
p
group. So L is S-loop when L is a Moufang loop which is centrally nilpotent of class 2.
In view of the Theorem 2.11 of Tim Hsu [63] we have the following:
THEOREM 3.1.5: Let L be a Moufang loop. Then L is a S-loop.
Proof: Now according to the theorem 2.11 of Tim Hsu [63] we see if L is a Moufang
loop then L is diassociative that is for (x,y) " L, )#x, y*# is associative. By theorem
3.1.3. of ours a Moufang loop is a S-loop.
Now we know Bol loops are power-associative (H.O. Pflugfelder [49, 50] and D. A.
Robinson [55]) leading to the following theorem.
THEOREM 3.1.6: Every Bol loop is a S-loop.
Proof: Bol loops are power-associative from the above references and by our theorem
3.1.2. All Bol loops are S-loops.
Now Michael Kinyon [42] (2000) has proved every diassociative A-loop is a Moufang
loop in view of this we have.
THEOREM 3.1.7: Let L be a A-loop. If L is diassociative or L is a Moufang loop
then L is a S-loop.
Proof: In view of Corollary 1 of Kinyon [42] we see for an A-loop L. The two concepts;
L is diassociative is equivalent to L is a Moufang loop so we have L to be S-loop.
In view of this we propose an open problem in chapter 5.
From the results of Kinyon [41] 2001 we see that every ARIF loop is a diassociative
loop in this regard we have the following.
THEOREM 3.1.8: Every ARIF loop is a S-loop.
Proof: Kinyon [41] has proved in Theorem 1.5 that every ARIF loop is diassociative
from our theorem 3.1.3 every diassociative loop is a S-loop. So all ARIF loops are S-
loops.
Further using the Lemma 2.4 of Kinyon [41] we have another interesting result.
49
THEOREM 3.1.9: Every RIF-loop is a S-loop.
Proof: According to Kinyon's [41] Lemma 2.4. every RIF loop is an ARIF loop. But by
our Theorem 3.1.8 every ARIF loop is a S-loop. Hence every RIF loop is a S-loop.
Fenyves [20](1969) has introduced the concept of C-loops.
By Corollary 2.6 of Kinyon [41], every flexible C-loop is a ARIF-loop, we have a
theorem.
THEOREM 3.1.10: Every flexible C-loop is a S-loop.
Proof: We see every flexible C-loop is a ARIF loop by Corollary 2.6. of Kinyon [41],
Hence by Theorem 3.1.8 every flexible C-loop is a S-loop.
This theorem immediately does not imply that all non-flexible C-loops are not S-loops;
for we have the example 4.2 given in Kinyon [41] is a C-loop which is not flexible
given by the following table:
0 1 2 3 4 5 6 7 8 9 10 11
"
0 0 1 2 3 4 5 6 7 8 9 10 11
1 1 2 0 4 5 3 7 8 6 10 11 9
2 2 0 1 5 3 4 8 6 7 11 9 10
3 3 4 5 0 1 2 10 11 3 8 6 7
4 4 5 3 1 2 0 11 9 10 6 7 8
5 5 3 4 2 0 1 9 10 11 7 8 8
6 6 7 8 11 9 10 0 1 2 4 5 3
7 7 8 6 9 10 11 1 2 0 5 3 4
8 8 6 7 10 11 9 2 0 1 3 4 5
9 9 10 11 7 8 6 5 3 4 0 1 2
10 10 11 9 8 6 7 3 4 5 1 2 0
11 11 9 10 6 7 8 4 5 3 2 0 1
This C-loop is a S-loop as the set {0, 1, 2, " } is a group given by the following table:
0 1 2
"
0 0 1 2
1 1 2 0
2 2 0 1
Thus we have a non-flexible C-loop which is a S-loop leading to the following open
problem proposed in Chapter 5. For more about C-loops please refer (F. Fenyves)
[20].
50
Thus we have seen the definition of S-loops and the classes of loops which are S-
loops. Now we proceed on to define substructures in S-loops.
PROBLEMS:
1. Give an example of a loop of order 8 which is not a S-loop.
2. Can a Bruck loop be a S-loop?
3. Find a S-loop of order 9 which is not a Moufang loop.
4. What is the order of the smallest S-loop?
5. Does there exist a S-loop of order p, p a prime?
3.2 Smarandache substructures in Loops
In this section we introduce the concept of Smarandache subloops (S-subloops) and
Smarandache normal subloops (S-normal subloops). The absence of S-normal
subloops in a loop L leads us to the definition of Smarandache simple loops (S-simple
loops). We prove that the class of loops L is a S-simple loop of order n + 1. We
n
obtain some results about these definitions.
DEFINITION 3.2.1: Let L be a loop. A proper subset A of L is said to be a
Smarandache subloop (S-subloop) of L if A is a subloop of L and A is itself a S-
loop; that is A contains a proper subset B contained in A such that B is a group
under the operations of L. We demand A to be a S-subloop which is not a
subgroup.
In view of this we have the following:
THEOREM 3.2.1: Let L be a loop. If L has a S-subloop then L is a S-loop.
Proof: If a loop L has S-subloop then we have a subset A ‚" L such that A is a subloop
and contains a proper subset B such that B is a group. Hence B ‚" A ‚" L so L is a S- [ Pobierz caÅ‚ość w formacie PDF ]