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can also occur then we denote this class by G""(n). If we have one of t or u to be zero we
denote it by G"""(n).
Thus, we have groupoids with identity as G (n) ‚" G"(n) ‚" G""(n) ‚" G"""(n). Now our
aim is to study which of these groupoids are SGs, Smarandache P-groupoids and so on.
Example 5.6.1: Let G = (Z4 *" {e}) define " on G by a " b = 2a + 3b (mod 4) for a, b " Z4 is
given by the following table:
e 0 1 2 3
"
e e 0 1 2 3
0 0 e 3 2 1
1 1 2 e 0 3
2 2 0 3 e 1
3 3 2 1 0 e
This is a SG with identity of order 5 as {e, m} is a semigroup for m = 1, 2, 3.
THEOREM 5.6.1: All groupoids in G"""(n) are SGs.
Proof: Clearly by the definition of G"""(n) we have {e, m} for m = 1, 2, 3, ... , n 1 are
semigroups. So every groupoid in G"""(n) are SGs.
Remark: Since G (n) ‚" G"(n) ‚" G""(n) ‚" G"""(n) for every n we see all groupoids in G (n),
G"(n), G""(n) are SGs as we have proved in Theorem 5.6.1 that all groupoids in G"""(n) is a
SG.
THEOREM 5.6.2: No groupoid in G"""(n) is a Smarandache idempotent groupoid.
Proof: Since by the very definition of G we see x " x = e, so no element in G is an
idempotent element, so no groupoid in G"""(n) is a Smarandache idempotent groupoid.
Example 5.6.2: Let G = Z6 *" {e} for a, b " Z6 define a " b = 5a + 3b. The groupoid is given
by the following table:
e 0 1 2 3 4 5
"
e e 0 1 2 3 4 5
0 0 e 3 0 3 0 3
1 1 5 e 5 2 5 2
2 2 4 1 e 1 4 1
90
3 3 3 0 3 e 3 0
4 4 2 5 2 5 e 5
5 5 1 4 1 4 1 e
This groupoid is Smarandache strong right alternative, but it is not even Smarandache
left alternative. Now we give an example of a left alternative groupoid in G (6), which is not
right alternative.
Example 5.6.3: The groupoid in G (6) is given by the following table:
e 0 1 2 3 4 5
"
e e 0 1 2 3 4 5
0 0 e 5 4 3 2 1
1 1 4 e 2 1 0 5
2 2 2 1 e 5 4 3
3 3 0 5 3 e 2 1
4 4 4 3 2 1 e 5
5 5 2 1 0 5 4 e
This is a Smarandache strong left alternative groupoid for (x " x) " y = x " (x " y).
Consider (x " x) " y = y. Now x " (x " y) = x " [4x + 5y] = 4x + 20x + 25y = y. Hence the
claim.
This is not Smarandache right alternative, it is left for the reader to verify. In view of
this, we have the following theorem.
THEOREM 5.6.3: Groupoids in G"""(n) are Smarandache strong right alternative groupoid if
and only if t2 a" 1 (mod n) and tu + u a" 0 (mod n).
Proof: A SG G in G"""(n) is Smarandache strong right alternative if for x, y " G we must
have (x " y) " y = x " (y " y). (x " y) " y = (tx + uy) " y = t2x + tuy + uy (mod n). Now x " (y
" y) = x. (x " y) " y a" x " (y " y) (mod n) if and only if t2 a" 1 (mod n) and tu + u a" 0 (mod n).
Similarly we get the following characterization theorem for a groupoid in G"""(n) to be
Smarandache strong left alternative.
THEOREM 5.6.4: Let G be a groupoid in G"""(n). G is a Smarandache strong left alternative
if and only if u2 a" 1 (mod n) and (t + tu) a" 0 (mod n).
Proof: On similar lines, the proof can be given as in Theorem 5.6.3.
THEOREM 5.6.5: Let G (n) be a groupoid in G"""(n), n not a prime. G (n) is a Smarandache
strong Bol groupoid, Smarandache strong Moufang groupoid, Smarandache strong P-
groupoid only when t2 a" t (mod n) and u2 a" u (mod n).
Proof: The verification is left for the reader.
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PROBLEM 1: Find all SGs with identity, which are Smarandache strong Moufang groupoids
in G"""(7) \ G""(7).
PROBLEM 2: Find Smarandache P-groupoids in G""(8) \ G (8).
PROBLEM 3: Find Smarandache Bol groupoids in G"""(9) \ G"(9).
PROBLEM 4: Find the number of SGs in G"(12).
PROBLEM 5: Find the number of Smarandache Bol groupoids in G"(11) \ G (11).
PROBLEM 6: Find the number of Smarandache alternative groupoids in G"(5) \ G (5).
PROBLEM 7: How many Smarandache strong right alternative groupoids exist in G"""(12) \
G""(12)?
Supplementary Reading
1. G. Birkhoff and S. S. Maclane, A Brief Survey of Modern Algebra, New York,
The Macmillan and Co. (1965).
2. R. H. Bruck, A Survey of Binary Systems, Springer Verlag, (1958).
3. W.B.Vasantha Kandasamy, New classes of finite groupoids using Zn,
Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001).
4. W. B. Vasantha Kandasamy, Smarandache Groupoids,
http://www.gallup.unm.edu/~smarandache/Groupoids.pdf.
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CHAPTER SIX
The semi automaton and automaton are built using the fundamental algebraic
structure semigroups. In this chapter we use the generalized concept of semigroups viz.
Smarandache groupoids which always contains a semigroup in them are used to construct
Smarandache semi automaton and Smarandache automaton. Thus, the Smarandache
groupoids find its application in the construction of finite machines. Here we introduce the
concept of Smarandache semi automaton and Smarandache automaton using Smarandache
free groupoids. This chapter starts with the definition of Smarandache free groupoids.
6.1 Basic Results
DEFINITION: Let S be non empty set. Generate a free groupoid using S and denote it by
. Clearly the free semigroup generated by the set S is properly contained in ; as in
we may or may not have the associative law to be true.
Remark: Even (ab) c `" a (bc) in general for all a, b, c " . Thus unlike a free semigroup
where the operation is associative, in case of free groupoid we do not assume the
associativity while placing them in justra position.
THEOREM 6.1.1: Every free groupoid is a Smarandache free groupoid.
Proof: Clearly if S is the set which generates the free groupoid then it will certainly contain
the free semigroup generated by S, so every free groupoid is a Smarandache free groupoid.
We just recall the definition of semi automaton and automaton from the book of
R.Lidl and G.Pilz.
DEFINITION: A Semi Automaton is a triple Y = (Z, A, ´) consisting of two non - empty sets Z
and A and a function ´: Z × A ’! Z, Z is called the set of states, A the input alphabet and ´
the "next state function" of Y.
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Let A = {a1, ..., an} and Z = {z1, ..., zk}. The semi automaton Y = (Z, A, ´). The semi
automaton can also be described by the transition table.
Description by Table:
´ a1 & an
z1 ´(z1,a1) L ´(z1,an )
M M M
zk ´(zk ,a1) L ´(zk,an )
as ´ : Z × A ’! Z, ´(zi, aj) " Z. The semi automaton can also be described by graphs.
Description by graphs:
ai
zr zs
We depict z1, ..., zk as 'discs' in the plane and draw an arrow labeled ai from zr to zs if
´(zr, ai) = zr. This graph is called the state graph.
Example 6.1.1: Z set of states and A input alphabet. Let Z = {0, 1, 2} and A = {0, 1}.
The function ´: Z × A ’! Z defined by ´ (0, 1) = 1 = ´ (2, 1) = ´ (1, 1), ´ (0, 0) = 0, ´ (2, 0) =
1, ´ (1, 0) = 0.
This is a semi automaton, having the following graph.
0 1
2
The description by table is:
0 1
´
0 0 1
1 0 1
2 1 1
DEFINITION: An Automaton is a quintuple K = (Z, A, B, ´, ›) where (Z, A, ´) is a semi
automaton, B is a non-empty set called the output alphabet and »: Z × A ’! B is the output
function.
If z " Z and a " A, then we interpret ´ (z, a) " Z as the next state into which z is
transformed by the input a, » (z, a) " B is the output resulting from the input a.
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Thus if the automaton is in state z and receives input a, then it changes to state ´ (z, a)
with output » (z, a).
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