[ Pobierz całość w formacie PDF ]
72
2. Similarly when n is non prime that is a composite number and if p / n then Zn (1.
p) is a SG.
Thus the class of groupoids Z (n) when n > 3 and n `" 5 has SGs.
Note: When n = 3 it is left for the reader to verify Z (3) has no SGs.
THEOREM 5.1.7: The SG Zn (t, u) " Z (n) with t + u a" 1 (mod n) is a Smarandache
idempotent groupoid.
Proof: Clearly x " x = x for all x " Zn as x " x = tx + ux a" (t + u) x (mod n) as t + u a" 1 (mod
n) we have x * x = x in Zn. Hence the claim.
THEOREM 5.1.8: The SG Zn (t, u) " Z (n) with t + u a" 1(mod n) is a Smarandache P-
groupoid if and only if t2 a" t (mod n) and u2 a" u (mod n).
Proof: Clearly we have proved if t + u a" 1 (mod n), then Zn (t, u) is a SG in Z(n). To show Zn
(t, u) is a Smarandache P-groupoid we have to prove (x " y) " x = x " (y " x) for all x, y "
Zn(t,u). Now to show Zn (t, u) to be SG we have to prove (x " y) " x = x " (y " x).
Consider (x " y) " x = (tx + uy) " x = t2x + tuy + ux. Now x " (y " x) = x " (ty + ux) =
tx + tuy + u2x. Now t2x + tuy + ux = tx + tuy + u2x if and only if t2 a" t (mod n) and u2 a" u
(mod n).
Example 5.1.8: Let Z12 (4, 9) " Z (12). Clearly, Z12 (4, 9) is a SG, which is also a
Smarandache P-groupoid by theorem 5.1.8. Z12 (4, 9) is defined by x " y = 4x + 9y (mod 12).
It is left for the reader to prove Z12 (4, 9) is a Smarandache P-groupoid.
THEOREM 5.1.9: The groupoid Zn (t, u) " Z (n) with t + u a" 1 (mod n) is a Smarandache
alternative groupoid if and only if t2 a" t (mod n) and u2 a" u (mod n).
Proof: Zn (t, u) when t + u a" 1 (mod n) is a SG. Clearly, (x " y) " y = x " (y " y) and (x " x) "
y = x " (x " y) if and only if t2 a" t (mod n) and u2 a" u (mod n). Thus, we see from this theorem
example 5.1.8 the groupoid Z12 (4, 9) is also a Smarandache alternative groupoid.
THEOREM 5.1.10: The groupoid Zn (t, u) " Z (n) with t + u a"1 (mod n) is a SG. This
groupoid is a Smarandache strong Bol groupoid if and only if t3 a" t (mod n) and u2 a" u (mod
n).
Proof: We know Zn (t, u) " Z (n) is a SG when t + u a" 1 (mod n). This groupoid is a
Smarandache strong Bol groupoid if and only if t3 a" t (mod n) and u2 a" u (mod n). It essential
to prove ((x * y) * z) * x = x * ((y * z) * x) for all x, y, z " Zm (t, u).
((x * y) * z) * x = [(tx + uy) * z] * x
= (t2x + tuy + uz) * x
= t3x + t2uy + tuz + ux.
73
Now, x * ((y * z) * x) = x * [(ty + uz) * x]
= x * [t2y + tuz + ux]
= tx + t2uy + tu2z + u2x.
((x * y) * z) * x a" x * ((y * z) * x) (mod n) if and only if t3 a" t (mod n) and u2 a" u
(mod n). Hence the claim.
Example 5.1.9: Let Z6 (3, 4) " Z (6) be the SG given by the following table:
0 1 2 3 4 5
"
0 0 4 2 0 4 2
1 3 1 5 3 1 5
2 0 4 2 0 4 2
3 3 1 5 3 1 5
4 0 4 2 0 4 2
5 3 1 5 3 1 5
This groupoid is a Smarandache strong Bol groupoid as 43 a" 4(mod 6) and 32 a" 3(mod 6),
which is the condition for a groupoid in Z (n) to be a Smarandache strong Bol groupoid.
THEOREM 5.1.11: Let Zn (t, u) " Z (n) with t + u a" 1 (mod n) be a SG. Zn (t, u) is a
Smarandache strong Moufang groupoid if and only if t2 a" t (mod n) and u2 a" u (mod n).
Proof: The Moufang identity is (x " y) " (z " x) = (x " (y " z)) " x for all x, y, z " Zn.
Now (x " y) " (z " x) = (tx + uy) " (tz + ux) = t2x + tuy + tux + u2x.
Further (x " (y " z)) " x = (x " (ty + uz)) " x
= (tx + tuy +u2z) " x
= t2x + t2uy + tu2z + ux.
Now t2x + tuy + tuz + u2x a" t2x + t2uy + tu2z + ux (mod n) if and only if u2 a" u (mod
n) and t2 a" t (mod n). Hence the claim.
Thus we see the class of groupoids Z (n) contain SGs and also Smarandache strong
Bol groupoids, Smarandache strong Moufang groupoids, Smarandache idempotent groupoids
and Smarandache strong P-groupoids.
PROBLEM 1: Is Z8 (3, 5) a SG?
PROBLEM 2: Show Z11 (5, 7) " Z (11) is a SG. Does Z11 (5, 7) have Smarandache
subgroupoids?
PROBLEM 3: Does Z (19) have SGs, which are Smarandache strong Bol groupoids?
74
PROBLEM 4: Give an example of a SG, which is not a Smarandache strong Moufang
groupoid in Z (16).
PROBLEM 5: Find all the SGs in the class of groupoids Z (24).
PROBLEM 6: Find all SGs, which are Smarandache strong P-groupoids in the class of
groupoids Z (18).
PROBLEM 7: Find all the Smarandache subgroupoids of Z16 (9, 8).
PROBLEM 8: Does Z27 (11, 17) have non trivial Smarandache subgroupoids?
PROBLEM 9: Find all SGs in Z (22).
PROBLEM 10: Is Z22 (10, 13) a Smarandache Moufang groupoid? Justify!
PROBLEM 11: Prove Z17 (11, 7) is a SG.
PROBLEM 12: Can Z (5) have SGs?
PROBLEM 13: Which groupoids in Z (14) are SGs?
PROBLEM 14: Prove Z25 (11, 15) is a SG.
PROBLEM 15: Find all SGs in Z (15).
PROBLEM 16: Find all Smarandache P-groupoids in Z (9).
PROBLEM 17: Find all Smarandache strong Bol groupoids in Z (27).
PROBLEM 18: Can Z (8) have Smarandache strong Moufang groupoid?
PROBLEM 19: Find a Smarandache P- groupoid in Z (121).
"
"
"
5.2 Smarandache Groupoids in Z"(n)
In this section, we study the conditions for the class of groupoids in Z"(n) to have
SGs. Throughout this section we will be interested in studying those groupoids in Z"(n)
which are not in Z (n) that is only groupoids in Z"(n) \ Z (n).
Example 5.2.1: Let Z5 (2, 4) " Z" (5). Z5 (2, 4) is a SG given by the following table:
0 1 2 3 4
"
0 0 4 3 2 1
1 2 1 0 4 3
2 4 3 2 1 0
75
3 1 0 4 3 2
4 3 2 1 0 4
Clearly, this is a SG as {1}, {2}, {3} and {4} are subsets which are semigroups.
It is important and interesting to note that the class of groupoids Z"(3) and Z"(4) are
such that Z"(3) = Z (3) and Z"(4) = Z(4). But Z"(5) has only two SGs given by Z5(2,4) and Z5
(4, 2).
As in the case of groupoids in Z (n), the groupoids in Z"(n) are SGs when t + u a" 1
(mod n).
Example 5.2.2: Let Z6 (2, 4) " Z"(6) be the groupoid given by the following table:
0 1 2 3 4 5
"
0 0 4 2 0 4 2
1 2 0 4 2 0 4
2 4 2 0 4 2 0
3 0 4 2 0 4 2
4 2 0 4 2 0 4
5 4 2 0 4 2 0
This is a SG as the set {0, 3} is a semigroup of Z6 (2, 4).
Now yet another interesting example of a SG is given.
Example 5.2.3: Consider the groupoid Z8 (2, 6) " Z"(8) given by the following table:
0 1 2 3 4 5 6 7
"
0 0 6 4 2 0 6 4 2
1 2 0 6 4 2 0 6 4
2 4 2 0 6 4 2 0 6
3 6 4 2 0 6 4 2 0
4 0 6 4 2 0 6 4 2
5 2 0 6 4 2 0 6 4
6 4 2 0 6 4 2 0 6
7 6 4 2 0 6 4 2 0
This is a SG for the subset {0, 4} is a semigroup. Hence, Z8 (2, 6) is a SG.
Example 5.2.4: The groupoid Z8 (3, 6) is also a SG from the class of groupoids Z"(8).
Further the class Z"(8) has Z8 (2,7), Z8 (4, 5), Z8 (3, 6), Z8 (2, 6), Z8 (4, 6), Z8 (6, 2), Z8 (6, 4)
and Z8 (6, 3) are SGs and the only SGs in Z"(8) \ Z (8) are Z8 (2, 6), Z8 (4, 6), Z8 (6, 2), Z8 (6,
4), Z8 (3, 6) and Z8 (6, 3). Thus there are 6 SGs in Z"(8) \ Z (8).
76
Example 5.2.5: Let Z"(12) be the class of groupoids. We see Z12 (3, 9) is a SG for it contains
P = {0, 6} as a subset which is a semigroup. Z12 (4, 6) is a SG as Q = {4} and P = {0, 6} are
subsets which are semigroups.
Z12 (4, 8) is a SG as the subset {0, 6} is a semigroup.
Z12 (5, 10) is a SG as the subset {6} is a semigroup.
Z12 (8, 10) is a SG as {0, 6} is a semigroup.
Z12 (2, 4) is a SG as {0, 6} is a proper subset which is a semigroup.
Z12 (2, 6) is a SG as {0, 6} is a proper subset which is a semigroup.
Z12 (2, 8) is a SG as {0, 6} is a proper subset which is a semigroup. [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl exclamation.htw.pl
72
2. Similarly when n is non prime that is a composite number and if p / n then Zn (1.
p) is a SG.
Thus the class of groupoids Z (n) when n > 3 and n `" 5 has SGs.
Note: When n = 3 it is left for the reader to verify Z (3) has no SGs.
THEOREM 5.1.7: The SG Zn (t, u) " Z (n) with t + u a" 1 (mod n) is a Smarandache
idempotent groupoid.
Proof: Clearly x " x = x for all x " Zn as x " x = tx + ux a" (t + u) x (mod n) as t + u a" 1 (mod
n) we have x * x = x in Zn. Hence the claim.
THEOREM 5.1.8: The SG Zn (t, u) " Z (n) with t + u a" 1(mod n) is a Smarandache P-
groupoid if and only if t2 a" t (mod n) and u2 a" u (mod n).
Proof: Clearly we have proved if t + u a" 1 (mod n), then Zn (t, u) is a SG in Z(n). To show Zn
(t, u) is a Smarandache P-groupoid we have to prove (x " y) " x = x " (y " x) for all x, y "
Zn(t,u). Now to show Zn (t, u) to be SG we have to prove (x " y) " x = x " (y " x).
Consider (x " y) " x = (tx + uy) " x = t2x + tuy + ux. Now x " (y " x) = x " (ty + ux) =
tx + tuy + u2x. Now t2x + tuy + ux = tx + tuy + u2x if and only if t2 a" t (mod n) and u2 a" u
(mod n).
Example 5.1.8: Let Z12 (4, 9) " Z (12). Clearly, Z12 (4, 9) is a SG, which is also a
Smarandache P-groupoid by theorem 5.1.8. Z12 (4, 9) is defined by x " y = 4x + 9y (mod 12).
It is left for the reader to prove Z12 (4, 9) is a Smarandache P-groupoid.
THEOREM 5.1.9: The groupoid Zn (t, u) " Z (n) with t + u a" 1 (mod n) is a Smarandache
alternative groupoid if and only if t2 a" t (mod n) and u2 a" u (mod n).
Proof: Zn (t, u) when t + u a" 1 (mod n) is a SG. Clearly, (x " y) " y = x " (y " y) and (x " x) "
y = x " (x " y) if and only if t2 a" t (mod n) and u2 a" u (mod n). Thus, we see from this theorem
example 5.1.8 the groupoid Z12 (4, 9) is also a Smarandache alternative groupoid.
THEOREM 5.1.10: The groupoid Zn (t, u) " Z (n) with t + u a"1 (mod n) is a SG. This
groupoid is a Smarandache strong Bol groupoid if and only if t3 a" t (mod n) and u2 a" u (mod
n).
Proof: We know Zn (t, u) " Z (n) is a SG when t + u a" 1 (mod n). This groupoid is a
Smarandache strong Bol groupoid if and only if t3 a" t (mod n) and u2 a" u (mod n). It essential
to prove ((x * y) * z) * x = x * ((y * z) * x) for all x, y, z " Zm (t, u).
((x * y) * z) * x = [(tx + uy) * z] * x
= (t2x + tuy + uz) * x
= t3x + t2uy + tuz + ux.
73
Now, x * ((y * z) * x) = x * [(ty + uz) * x]
= x * [t2y + tuz + ux]
= tx + t2uy + tu2z + u2x.
((x * y) * z) * x a" x * ((y * z) * x) (mod n) if and only if t3 a" t (mod n) and u2 a" u
(mod n). Hence the claim.
Example 5.1.9: Let Z6 (3, 4) " Z (6) be the SG given by the following table:
0 1 2 3 4 5
"
0 0 4 2 0 4 2
1 3 1 5 3 1 5
2 0 4 2 0 4 2
3 3 1 5 3 1 5
4 0 4 2 0 4 2
5 3 1 5 3 1 5
This groupoid is a Smarandache strong Bol groupoid as 43 a" 4(mod 6) and 32 a" 3(mod 6),
which is the condition for a groupoid in Z (n) to be a Smarandache strong Bol groupoid.
THEOREM 5.1.11: Let Zn (t, u) " Z (n) with t + u a" 1 (mod n) be a SG. Zn (t, u) is a
Smarandache strong Moufang groupoid if and only if t2 a" t (mod n) and u2 a" u (mod n).
Proof: The Moufang identity is (x " y) " (z " x) = (x " (y " z)) " x for all x, y, z " Zn.
Now (x " y) " (z " x) = (tx + uy) " (tz + ux) = t2x + tuy + tux + u2x.
Further (x " (y " z)) " x = (x " (ty + uz)) " x
= (tx + tuy +u2z) " x
= t2x + t2uy + tu2z + ux.
Now t2x + tuy + tuz + u2x a" t2x + t2uy + tu2z + ux (mod n) if and only if u2 a" u (mod
n) and t2 a" t (mod n). Hence the claim.
Thus we see the class of groupoids Z (n) contain SGs and also Smarandache strong
Bol groupoids, Smarandache strong Moufang groupoids, Smarandache idempotent groupoids
and Smarandache strong P-groupoids.
PROBLEM 1: Is Z8 (3, 5) a SG?
PROBLEM 2: Show Z11 (5, 7) " Z (11) is a SG. Does Z11 (5, 7) have Smarandache
subgroupoids?
PROBLEM 3: Does Z (19) have SGs, which are Smarandache strong Bol groupoids?
74
PROBLEM 4: Give an example of a SG, which is not a Smarandache strong Moufang
groupoid in Z (16).
PROBLEM 5: Find all the SGs in the class of groupoids Z (24).
PROBLEM 6: Find all SGs, which are Smarandache strong P-groupoids in the class of
groupoids Z (18).
PROBLEM 7: Find all the Smarandache subgroupoids of Z16 (9, 8).
PROBLEM 8: Does Z27 (11, 17) have non trivial Smarandache subgroupoids?
PROBLEM 9: Find all SGs in Z (22).
PROBLEM 10: Is Z22 (10, 13) a Smarandache Moufang groupoid? Justify!
PROBLEM 11: Prove Z17 (11, 7) is a SG.
PROBLEM 12: Can Z (5) have SGs?
PROBLEM 13: Which groupoids in Z (14) are SGs?
PROBLEM 14: Prove Z25 (11, 15) is a SG.
PROBLEM 15: Find all SGs in Z (15).
PROBLEM 16: Find all Smarandache P-groupoids in Z (9).
PROBLEM 17: Find all Smarandache strong Bol groupoids in Z (27).
PROBLEM 18: Can Z (8) have Smarandache strong Moufang groupoid?
PROBLEM 19: Find a Smarandache P- groupoid in Z (121).
"
"
"
5.2 Smarandache Groupoids in Z"(n)
In this section, we study the conditions for the class of groupoids in Z"(n) to have
SGs. Throughout this section we will be interested in studying those groupoids in Z"(n)
which are not in Z (n) that is only groupoids in Z"(n) \ Z (n).
Example 5.2.1: Let Z5 (2, 4) " Z" (5). Z5 (2, 4) is a SG given by the following table:
0 1 2 3 4
"
0 0 4 3 2 1
1 2 1 0 4 3
2 4 3 2 1 0
75
3 1 0 4 3 2
4 3 2 1 0 4
Clearly, this is a SG as {1}, {2}, {3} and {4} are subsets which are semigroups.
It is important and interesting to note that the class of groupoids Z"(3) and Z"(4) are
such that Z"(3) = Z (3) and Z"(4) = Z(4). But Z"(5) has only two SGs given by Z5(2,4) and Z5
(4, 2).
As in the case of groupoids in Z (n), the groupoids in Z"(n) are SGs when t + u a" 1
(mod n).
Example 5.2.2: Let Z6 (2, 4) " Z"(6) be the groupoid given by the following table:
0 1 2 3 4 5
"
0 0 4 2 0 4 2
1 2 0 4 2 0 4
2 4 2 0 4 2 0
3 0 4 2 0 4 2
4 2 0 4 2 0 4
5 4 2 0 4 2 0
This is a SG as the set {0, 3} is a semigroup of Z6 (2, 4).
Now yet another interesting example of a SG is given.
Example 5.2.3: Consider the groupoid Z8 (2, 6) " Z"(8) given by the following table:
0 1 2 3 4 5 6 7
"
0 0 6 4 2 0 6 4 2
1 2 0 6 4 2 0 6 4
2 4 2 0 6 4 2 0 6
3 6 4 2 0 6 4 2 0
4 0 6 4 2 0 6 4 2
5 2 0 6 4 2 0 6 4
6 4 2 0 6 4 2 0 6
7 6 4 2 0 6 4 2 0
This is a SG for the subset {0, 4} is a semigroup. Hence, Z8 (2, 6) is a SG.
Example 5.2.4: The groupoid Z8 (3, 6) is also a SG from the class of groupoids Z"(8).
Further the class Z"(8) has Z8 (2,7), Z8 (4, 5), Z8 (3, 6), Z8 (2, 6), Z8 (4, 6), Z8 (6, 2), Z8 (6, 4)
and Z8 (6, 3) are SGs and the only SGs in Z"(8) \ Z (8) are Z8 (2, 6), Z8 (4, 6), Z8 (6, 2), Z8 (6,
4), Z8 (3, 6) and Z8 (6, 3). Thus there are 6 SGs in Z"(8) \ Z (8).
76
Example 5.2.5: Let Z"(12) be the class of groupoids. We see Z12 (3, 9) is a SG for it contains
P = {0, 6} as a subset which is a semigroup. Z12 (4, 6) is a SG as Q = {4} and P = {0, 6} are
subsets which are semigroups.
Z12 (4, 8) is a SG as the subset {0, 6} is a semigroup.
Z12 (5, 10) is a SG as the subset {6} is a semigroup.
Z12 (8, 10) is a SG as {0, 6} is a semigroup.
Z12 (2, 4) is a SG as {0, 6} is a proper subset which is a semigroup.
Z12 (2, 6) is a SG as {0, 6} is a proper subset which is a semigroup.
Z12 (2, 8) is a SG as {0, 6} is a proper subset which is a semigroup. [ Pobierz całość w formacie PDF ]