INTRODUCTION
In a draft paper (Manes, 1989), the equational theory of disjoint alternatives, around 1989, Manes introduced the concept of Ada (Algebra of disjoint alternatives) (A, ∧, V, ()^{I}, ()_{π}, 0, 1, 2) (Where ∧,V are binary operations on A, ()^{I}, ()_{π }are unary operations and 0, 1, 2 are distinguished elements on A) which is however differ from the definition of the Ada of his later paper (Manes, 1993) Adas and the equational theory of ifthenelse in 1993. While the Ada of the earlier draft seems to be based on extending the IfThenElse concept more on the basis of Boolean algebras and the later concept is based on Calgebras (A, ∧,V, ()^{ ~ })) (where ∧,V are binary operations on A, ()^{ ~} is a unary operation) introduced by Guzman and Squier (1990). Rao (1994) first introduced the concept of A*algebra (A, ∧, V, _{*}, ()^{ ~}, ()_{π}, 0, 1, 2)) (where ∧,V,_{*} are binary operations on A, ()^{ ~ },()_{π} are unary opePrarooparations and 0, 1, 2 are distinguished elements on A) not only studied the equivalence with Ada, Calgebra, Ada’s connection with 3ring, stone type representation but also introduced the concept of A*clone, the IfThenElse structure over A*algebra and Ideal of A*algebra. Rao (2000) introduced the concept pre A*algebra (A, ∨, ∧, ()^{~}) (where ∧, V are binary operations on A, ()^{~} is a unary operation on A analogous to Calgebra as a reduct of A*algebra, studied their subdirect representations, obtained the results that 2 = {0, 1} and 3 = {0, 1, 2} are the subdirectly irreducible preA*algebras and every preA*algebra can be imbedded in 3^{X}( where 3^{X} is the set of all mappings from a nonempty set X into 3 = {0, 1, 2}). Praroopa (2012) introduced the specific concepts on pre A*algebra and of the papers Praroopa and Rao (2011a), Praroopa and Rao (2011b) studied pre A*algebra as a semilattice, lattice in pre A*algebra.
PRELIMINARIES
Definition: Pre A*Algebra (Rao, 2000): An algebra (A, ∨, ∧, ()^{~}) satisfying:
•  (x^{~})^{~} = x, ∀x∈A 
• 
x∧x = x, ∀x∈A 
• 
x∧y = y∧x, ∀ x, y∈A 
• 
(x∧y)^{~} = x^{~}∨y^{~}, ∀x, y∈A 
• 
x∧(y∧z) = (x∧y)∧z , ∀ x, y, z∈A 
• 
x∧(y∨z) = (x∧y)∨(x∧z), ∀x, y, z∈A 
• 
x∧y = x∧(x^{~}∨y), ∀x, y∈A 
is called a pre A*algebra.
Definition: (Praroopa, 2012) pre A* Homomorphism: Let (A_{1}, ∧, ∨, ()^{~}) and (A_{2}, ∧, ∨, ()^{~}) be two pre A*algebras. A mapping f: A_{1}→A_{2} is called an pre A*–homomorphism, if:
•  The homomorphism f: A_{1}→A_{2} is onto, then f is called epimorphism 
• 
The homorphism f: A_{1}→A_{2} is oneone, then f is called monomorphism 
• 
The homomorphism f : A_{1}→A_{2} is oneone and onto then f is called an isomorphism and A_{1}, A_{2} are isomorphic, denoted by A_{1} ≅ A_{2} 
Definition: (Praroopa, 2012) kernel of pre A*homomorphism: By the definition of pre A*homomorphism, define Kernel of pre A*homomorphism Let A_{1}, A_{2} be two pre A*algebras and f: A_{1}→A_{2}be a pre A*homomorphism then the set {x∈A/f(x) = 0} is called the kernel of f and it is denoted by kerf.
Example: Let A be a pre A*algebra with 1, 0. Suppose that for every x∈A{0, 1}, x∨x~≠1. Define f:A→{0, 1, 2} by f(1) = 1, f(0) = 0 and f(x) = 2 if x≠0, 1. Then f is a pre A*homomorphism.
Theorems on Pre A*homomorphism (Praroopa, 2012): Theorem: Let f:A→B be a pre A*homomorphism from a pre A*algebra A into a pre A*algebra B ad Kerf = {x∈A/f(x) = 0} is the kernel then kerf = {0} if and only if f is one one.
Proof:  Suppose Kerf = {0}
To show that f is oneone:
For any x, y∈A, consider f(x) = f(y)
⇒ 
f(x)f(y) = 0 

⇒f(xy) = 0 

⇒xy∈Kerf = {0} 

⇒xy = 0 

⇒f(x)f(y) = 0 

⇒x = y, ∀x, y∈A 
Therefore f is oneone
Converse: Suppose that f is oneone
⇒f(x) = f(y) ⇒ x = y, ∀ x, y∈A
To show that Kert f = {0}
Let x∈ Kerf
⇒f(x) = 0
⇒ x = 0 (since f is oneone)
Therefore Kerf = {0}
Lemma (Rao, 2000): Let f: A_{1}→A_{2} be pre A*homomorphism where A_{1}, A_{2} are pre A*algebras with 1_{1} and 1_{2}Then:
(i)  If A_{1} has the element 2, then f(2) is the element of A_{2} 
(ii) 
If a∈B(A_{1}), then f(a)∈B(A_{2}) 
Where:
B(A_{1}) = {x/x∨ x ^{~ }=1}
B(A_{2}) = {x/x∨ x ^{~ }=1}
Note: If f:A→B and g:B→C are pre A*homomorphisms.
So their composition or product gof: A→C which is defined by gof (a) = g(f(a) is also pre A*homomorphism.
Proposition (Praroopa, 2012): If f:A→B and g:B→C are pre A*homomorphisms. Then:
(i)  If f and g are mono, so is gof 
(ii) 
If f and g are epi, so is gof 
(iii) 
If gof is mono, so is f 
(iv) 
If gof is epi, so is g 
Proof: If f:A→B and g:B→C are pre A*homomorphisms. Then gof: A→C is also a pre A*homomorphisms.
(i)  Suppose f, g are oneone 
Now suppose (gof)(a_{1}) = (gof)(a_{2})
⇒g(f(a_{1})) = g(f(a_{2}))
⇒f(a_{1}) = f(a_{2}) ( Since g is one–one)
⇒a_{1} = a_{2} (Since f is oneone)
(ii)  Suppose f, g are onto 

Let c∈C, since g is onto, there exists b∈B such that g(b) = c. 

Since b∈B and f is onto there exists a∈A such that f(a) = b. 

Therefore, g(b) = g(f(a)) = c, = (gof)(a) = c 

Hence, for c∈C, there exists a∈A such that (gof)(a) = c 

This is true for every c∈C 

Therefore gof is onto 

Therefore gof is epimorphism. 
(iii)  Suppose gof is mono i.e., gof is oneone 



We have to show f is oneone 

Suppose f(a_{1}) = f(a_{2}) 

⇒g(f(a_{1})) = g(f(a_{2})) 

⇒(gof) (a_{1})) = (gof)(a_{2}) 

⇒a_{1} = a_{2} (Since gof is oneone ) 

Therefore f is oneone. 

Hence, f is mono. 
(iv)  Suppose gof is epi⇒ gof is onto 



We have to show g is onto 

Since gof: A→C is onto, for any c∈C, there exist a∈A such that 

(gof)(a) = c 

⇒g(f(a)) = c where f(a)∈B 

⇒f(a) = b, where b∈B 

Therefore g(b) = c, for some b∈B 

Therefore for c∈C, ∃b∈B∋g(b) = c , 

this is true for all c∈C 

Hence g is onto 

Thus g is epimorphism. 
Corollary (Praroopa, 2012): The pre A*homomorphisms f:A→B is an isomorphism, if and only if, there exists a pre A*homomorphism g:B→C such that fog is an automorphism of B and gof is an automorphism of A.
Proof: Suppose that the pre A*homomorphism f:A→B is an isomorphism.

i.e., f is a bijection 

Then f^{1}:B→A is a bijection such that fof^{1} = I_{B} 

and f^{1}of = I_{A.} 

Hence, f^{1}:B→A is a mapping such that fof^{1} = I_{B} which is an automoptism of B. 

And f^{1}of = I_{A} which is an automorphism of A 

Now we show that f^{1} is a Pre A*homomorphism : 

Since f is Pre A*homomorphism, we have f(1) = 1 

⇒f^{1}(1) = 1 

Let b_{1}, b_{2}∈B. 

Since f:A→B is isomorphism, we have f is onto 

Then ∃ a_{1}, a_{2}∈A ∋ f(a_{1}) = b_{1}, f(a_{2}) = b_{2}. 

Therefore f(a_{1}va_{2) =} f(a_{1}) v f(a_{2}) 

= b_{1}∨b_{2} (Since f is pre A*homomorphism) 

⇒a_{1}∨a_{2} = f^{1}(b_{1}∨b_{2}) 

⇒ f^{1}(b_{1}∨b_{2}) = f^{1}(b_{1})∨f^{1}(b_{1})∨ f^{1}(b_{2}) 

and f(a_{1}∧a_{2}) = f(a_{1})∧f(a_{2}) 

= b_{1}∨b_{2} 

⇒a_{1}∧a_{2} = f^{1}(b_{1}∧v_{2}) 

⇒f^{1}(b_{1}∧b_{2}) = f^{1}(b_{1}) ∧f^{1}(b_{2}) 

Since f^{1}(b_{1}^{~})f = (f^{1}(b_{1}))^{~} 

Therefore f^{1} is a Pre A*homomorphism 
By taking g = f^{1} we have g:B→A is a Pre A*homo, such that fog and gof are automorphisms of B and A, respectively.
Converse: Conversely, assume that g:B→A is a pre A*homomorphism such that fog and gof are auto of B and A, respectively, where f:A→B is a pre A*homomorphism
Since fog: B→B is an auto we have fog is an epimorphism.
⇒ f is epi (by proposition1.8)
⇒ f is onto
Since gof is auto, we have gof is mono
⇒ f is mono (Since by proposition1.8)
⇒f is one–one
Therefore f:A→B is an isomorphism
Theorem (Praroopa, 2012): Under any pre A*homomorphism f of a pre A*algebra A onto a pre A* algebra A_{1} with 0, the set kerf (kernel of f) is an ideal in A.
Proof: Let f : A→A1 be a pre A*homomorphism
Then Kerf = {x∈A/f(x) = 0}
(i)  If f(a) = 0 ad f(b) = 0 
Then f(a∨b) = f(a)∨f(b) = 0 V 0 = 0
⇒a∨b∈Kerf
i.e., if a∈Kerf, b∈Kerf⇒a∨b∈kerf
_{(ii) a∈}Kerf ⇒f(a) = 0
for b∈B(A) f(a∧b)= f(a) ∧ f(b) = 0 ∧ f(b)
= 0 Since f(b)∈B(A1), . .
⇒a∧b∈Kerf
Therefore, from (i) and (ii), Kerf is an ideal in A
Proposition (Praroopa, 2012): If f is a pre A*homomorphism of a pre A*algebra A into another pre A*algebra, then f(A)≅A/f^{1}(0). Where f(A) is called the image,
f^{1}(0) = {a∈A/f(a) = 0} the Kernel of f.
Proof : f: A→B is a pre A*homomorphism.
Then by Proposition 3.6 there exists a congruence relation θ_{x} on A, an epimorphism α_{x}: A→A_{x} and a monomorphism α: A_{x}→B such that αoα_{x} = f
⇒αoα_{x}(a) = f(a), ∀a∈A
⇒f(a) = αoα_{x}(a)
= α(α_{x}(a))
≅α_{x}(a) (Since αis mono)
= A_{x} (Since α_{x} is onto)
∴f(a)≅A_{x}, ∀a∈A
Hence, f(a)≅A_{x}→(a)
Since α_{x}: A→A_{x} is onto then by fundamental theorem of homomorphism we have
Ker 
α_{x}={(s,t)∈AxA/α_{x} (s) = α_{x} (t)} 

={(s,t)∈AxA/x∧s = x∧t} 

=θ_{x} 

∴A/θ_{x}≅A_{x} 

Since Kerα_{x} = Ker f; 

(Verification: Let s ∈Kerα_{x} 

⇒α_{x}(s) = 0 

⇒x∧s = 0 

⇒α(x∧s) = 0 (Since α is oneone) 

⇒f(s) = 0 (Since α: A_{x}→B by α(x∧s) = f(s), ∀s∈A) 

⇒s∈Kerf) 
Hence A/ Kerf ≅A_{x}≅f(A) (by (1))
Therefore f(A)≅A/Kerf.
CONCLUSION
Established the concept of kernel of pre A*homomorphism and proved some theorems on these pre A*homomorphisms. Establish its useful theorems and related with these concepts of pre A*homomorphisms.