American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Updating the Fundamental Theorem of Homomorphism of General Universal Algebras Gezahagne Mulat Addis*, Department of Mathematics, Dilla University, Dilla, Ethiopia Email: buttu412@yahoo.com Abstract From the fundamental theorem of homomorphisms, it is well known that any homomorphism of groups (or rings, or modules, or vector spaces and of general universal algebras) can be decomposed as a composition of a monomorphism and an epimorphism. This paper provides the uniqueness of such decomposition up to the level of associates in the case of general universal algebras Keywords: Algebra of a given type; Homomorphisms; Kernel of a homomorphism; Congruence Relations and an associate of a homomorphism. 1. Introduction It is well known that, if 𝑓:𝐺 β†’ 𝐺 β€² is a homomorphism of groups (or rings or modules) the quotient group 𝐺 ker𝑓� is isomorphic to the image of 𝑓 which is a subgroup of the codomain group 𝐺 β€². This isomorphism 𝑔 is simply induced by 𝑓, in the sense that, 𝑔 can be defined by 𝑔(π‘Ž + ker 𝑓) = 𝑓(π‘Ž) for any coset π‘Ž + ker 𝑓 , π‘Ž ∈ 𝐺 Also, we have the natural epimorphism β„Ž:𝐺 β†’ 𝐺 ker 𝑓� , defined by β„Ž(π‘Ž) = π‘Ž + ker𝑓 for all π‘Ž ∈ 𝐺. In other words, we have a decomposition𝑓 = 𝑔 ∘ β„Ž;where g is a monomorphism and h is an epimorphism. This is known as the Fundamental Theorem of Homomorphisms [3, 11].This result can be extended to any homomorphism 𝑓:𝐴 β†’ 𝐴′ of universal algebras of the same type by considering the binary relation πœƒ on the domain algebra A defined by, πœƒ = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ 𝑓(π‘Ž) = 𝑓(𝑏)} ------------------------------------------------------------------------ * Corresponding author. E-mail address: buttu412@yahoo.com.. 61 http://asrjetsjournal.org/ mailto:buttu412@yahoo.com American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 Figure1: Decomposition of homomorphism of groups and the quotient algebra 𝐴 πœƒοΏ½ . In the general case of universal algebras, this πœƒ is defined as the kernel of 𝑓 and is actually a congruence relation on the domain algebra A and 𝐴 πœƒοΏ½ is the set of all congruence classes of elements of 𝐴 corresponding to πœƒ [8]. In the familiar cases of homomorphisms of groups (or rings or modules) the kernel of f is a normal subgroup (or an ideal or a submodule respectively) of the domain and the congruence classes are precisely the cosets of the conventional kernels [5, 9, 10]. In all these cases there is an order isomorphism of the set of normal subgroups(or ideals or submodules) onto the set of congruence relations on the domain algebra. In this scenario, we can say that any homomorphism f of algebras of any type can be decomposed as a composition of a monomorphism and an epimorphism. It is natural to question ourselves that, is this decomposition done uniquely in one way? Of course not in one and only one way and this paper provides the uniqueness of such a decomposition of homomorphismsupto the level of associate in the case of homomorphisms of general universal algebras. 2. Method/Approach Let A and B be algebras of a given type β„± and let 𝑓:𝐴 β†’ 𝐡, be any homomorphism. Let us consider the binary relation 𝐾𝑓 defined by: 𝐾𝑓 = {(π‘₯,𝑦) ∈ 𝐴 Γ— 𝐴 ∢ 𝑓(π‘₯) = 𝑓(𝑦)} Then 𝐾𝑓 becomes a congruence relation on 𝐴 [6, 11]. Also, the natural map β„Ž:𝐴 β†’ 𝐴 𝐾𝑓� , defined by β„Ž(π‘Ž) = 𝐾𝑓(π‘Ž), the congruence class of π‘Ž corresponding to 𝐾𝑓, is an epimorphism and the function 𝑔: 𝐴 𝐾𝑓� β†’ 𝐡, defined by 𝑔 �𝐾𝑓(π‘Ž)οΏ½ = 𝑓(π‘Ž) for all π‘Ž ∈ 𝐴, is a monomorphism. Now, we have the decomposition 𝑓 = 𝑔 ∘ β„Ž. To discuss about the uniqueness of the monomorphism𝑔 and the epimorphismβ„Ž in this decomposition,we first define the notion of an associate of a homomorphism of algebras and then we prove the uniqueness of such a decomposition upto the level of associates; that is, , if𝑓 can be decomposed in two ways as 𝑓 = π‘”π‘œβ„Ž and 𝑓 = π‘”β€²π‘œβ„Žβ€², where 𝑔 and 𝑔′ are monomorphisms and β„Ž and β„Žβ€² are epimorphisms then we will prove that 62 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 𝑔~𝑔′and β„Ž~β„Žβ€². This says that, the decomposition of a homomorphism𝑓 as a composition of amonomorphism and anepimorphism is unique upto associate. In this vein we generalize and unify all the fundamental theorems of homomorphisms and isomorphisms. 3. Preliminaries 3.1. Definition of Algebras Definition 3.1.1. Fora nonempty set A and a nonnegative integer n, we define 𝐴0 = {βˆ…}, and for 𝑛 > 0, 𝐴𝑛is the set of 𝑛-tuples of elements from A. An 𝑛 βˆ’ π‘Žπ‘Ÿπ‘¦ operation (or function) on A is any function f from An to A; n is the arity(or rank) of f. A finitaryoperation is an 𝑛- ary operation, for some n. The image of (π‘Ž1, π‘Ž2, . . . , π‘Žπ‘›) under an 𝑛 βˆ’ary operation 𝑓is denoted by 𝑓(π‘Ž1, π‘Ž2, . . . , π‘Žπ‘›).An operation 𝑓on 𝐴is called a nullaryoperation (or constant) if its arity is zero; it is completely determined by the image 𝑓(βˆ…) in A of the only element βˆ… in 𝐴0, and as such it is convenient to identify it with the element 𝑓(βˆ…). Thus a nullary operation is thought of as an element of A. An operation 𝑓on A is unary, binary, or ternary if its arity is 1, 2, or 3, respectively. [1] Definition 3.1.2. A type (or language) of algebras is a set β„± of function symbols such that a nonnegative integer 𝑛is assigned to each member 𝑓of β„±. This integer is called the arity (or rank) of 𝑓, and 𝑓is said to be an 𝑛 βˆ’ary operation symbol. The subset of 𝑛 βˆ’ary function symbols in F is denoted by ℱ𝑛.[1] Definition 3.1.3. If β„± is a type of algebras then an algebra A of type β„± is an ordered pair 〈𝐴,𝐹βŒͺ where A is a nonempty set and 𝐹is a family of finitary operations on A indexed by the language β„± such that corresponding to each 𝑛 βˆ’ary function symbol 𝑓in β„± there is an 𝑛 βˆ’ary operation 𝑓𝐴 on A. The set A is called the universe (or underlying set) of 〈𝐴,𝐹βŒͺ, and the 𝑓𝐴’s are called the fundamental operations of A.[1] If 𝐹 is finite, say β„± = {𝑓1, 𝑓2, . . . , π‘“π‘˜}, we often write 〈𝐴, 𝑓1, 𝑓2, . . . ,π‘“π‘˜βŒͺfor 〈𝐴,𝐹βŒͺ, usually adopting the convention: π‘Žπ‘Ÿπ‘–π‘‘π‘¦ 𝑓1 β‰₯ π‘Žπ‘Ÿπ‘–π‘‘π‘¦ 𝑓2 β‰₯ Β· Β· Β· β‰₯ π‘Žπ‘Ÿπ‘–π‘‘π‘¦ π‘“π‘˜. 3.2. Homomorphisms of Algebras Definition 3.2.1: Let 𝐴 and 𝐡 be algebras of a given type β„±. A mapping 𝛼: 𝐴 β†’ 𝐡 is called a homomorphism, if the following are satisfied. [1, 4 &7] i. If 𝑓 ∈ β„± is a nullary operation symbol, then 𝛼(𝑓𝐴) = 𝑓𝐡 ii. If 𝑓 ∈ β„± is an n-ary operation symbol,𝑛 > 0 and π‘Ž1, π‘Ž2, … . . π‘Žπ‘› ∈ 𝐴, then 𝛼(𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›) = 𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2), … … ,𝛼(π‘Žπ‘›)) Theorem 3.2.1: Let A, B and C be algebras of a given type β„±. Let 𝛼:𝐴 β†’ 𝐡 and 𝛽:𝐡 β†’ 𝐢 be homomorphisms. Then π›½π‘œπ›Ό:𝐴 β†’ 𝐢 is also a homomorphism [7]. 63 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 Proof: Let 𝑓 ∈ β„± be a nullary operation symbol. Then 𝛽 ∘ 𝛼(𝑓𝐴) = 𝛽(𝛼(𝑓𝐴)) = 𝛽(𝑓𝐡) (∡ 𝛼is a homomorphism) = 𝑓𝐢 (∡ 𝛽is a homomorphism) Let𝑓 ∈ β„± be an n-ary operation symbol, 𝑛 > 0 and π‘Ž1, π‘Ž2, … . . π‘Žπ‘› ∈ 𝐴. Then, 𝛽 ∘ 𝛼(𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›) = 𝛽(𝛼(𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›)) = 𝛽(𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2), … ,𝛼(π‘Žπ‘›))) (∡ 𝛼is a homomorphism) = 𝑓𝐢(𝛽(𝛼(π‘Ž1)) , 𝛽�𝛼(π‘Ž2)οΏ½, … …𝛽�𝛼(π‘Žπ‘›)οΏ½) (∡ 𝛽is a homomorphism) = 𝑓𝐢(𝛽 ∘ 𝛼(π‘Ž1) , 𝛽 ∘ 𝛼(π‘Ž2), … … … …𝛽 ∘ 𝛼(π‘Žπ‘›)) Therefore, π›½π‘œπ›Ό:𝐴 β†’ 𝐢is also a homomorphism. Definition 3.2.2: Let A and B be algebras of a given type β„± and 𝛼: 𝐴 β†’ 𝐡 be any function. Then i. 𝛼 is called a monomorphism, if it is an injective homomorphism. ii. 𝛼 is called an epimorphism, if it is a surjective homomorphism. iii. 𝛼 is called an isomorphism, if it is a bijective homomorphism.[1, 6& 7] 3.3. Congruence Relations Definition 3.3.1: Let A be an algebra of type β„± and πœƒ be an equivalence relation on 𝐴. Then ΞΈ is said to be a congruence relation on A, if the following is satisfied for any π‘Ž1, π‘Ž2, … . . π‘Žπ‘›, 𝑏1, 𝑏2, … . . 𝑏𝑛 ∈ 𝐴 , If οΏ½π‘Ž1,𝑏1οΏ½, οΏ½π‘Ž2,𝑏2οΏ½, … … … … … … . . , οΏ½π‘Žπ‘›,𝑏𝑛� ∈ πœƒ,𝑓 ∈ β„±is an 𝑛-ary operation symbol and 𝑛 > 0,then (𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›) , 𝑓𝐴(𝑏1, 𝑏2, … . . 𝑏𝑛)) ∈ πœƒ. [1, 2, 6& 7] In other words, a congruence relation is an equivalence relation on an algebraic structure (such as groups, rings or vector spaces and of general universal algebras) that is compatible with the structure. In the next two theorems we will observe that kernels of homomorphisms and congruence relations are same. Theorem 3.3.1: Let A and B be algebras of a given type β„±. Let 𝛼:𝐴 β†’ 𝐡 be a homomorphism. Then kernel of 𝛼 defined by;ker𝛼 = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ 𝛼(π‘Ž) = 𝛼(𝑏)} is a congruence relation on A. [1, 2, 6] 64 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 Proof:Put πœƒ = π‘˜π‘’π‘Ÿπ›Ό. Then it is clear that, ΞΈ is an equivalence relation on A. Now, for any π‘Ž1, π‘Ž2, … . . π‘Žπ‘› ,𝑏1, 𝑏2, … . . 𝑏𝑛 ∈ 𝐴, let οΏ½π‘Ž1,𝑏1οΏ½, οΏ½π‘Ž2,𝑏2οΏ½, … … . . , οΏ½π‘Žπ‘›,𝑏𝑛� ∈ πœƒ and let 𝑓 ∈ β„± be an 𝑛- ary operation symbol and 𝑛 > 0, then 𝛼(π‘Žπ‘–) = 𝛼(𝑏𝑖) for all 1 ≀ 𝑖 ≀ 𝑛. Therefore, 𝛼( 𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›) ) = 𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2), … … ,𝛼(π‘Žπ‘›)) (∡ 𝛼is a homomorphism) = 𝑓𝐡(𝛼(𝑏1),𝛼(𝑏2), … … ,𝛼(𝑏𝑛)) = 𝛼(𝑓𝐴(𝑏1, 𝑏2, … . . 𝑏𝑛)) (∡ 𝛼is a homomorphism) Thus, �𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›), 𝑓𝐴(𝑏1, 𝑏2, … . . 𝑏𝑛)οΏ½ ∈ ker𝛼 = πœƒ and hence, ker𝛼 is a congruence relation on A. Theorem 3.3.2: Let A be an algebra of typeβ„± and πœƒ be a congruence relation on A. Then there exists an algebra 𝐡 of type β„± and a homomorphism 𝛼:𝐴 β†’ 𝐡 such that πœƒ = ker𝛼. [2] Proof: Put πœƒ = π‘˜π‘’π‘Ÿπ›Ό = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ 𝛼(π‘Ž) = 𝛼(𝑏)} and let 𝐴 πœƒοΏ½ = {πœƒ(π‘Ž): π‘Ž ∈ 𝐴} be the quotient of A over πœƒ, where πœƒ (π‘Ž) = {𝑏 ∈ 𝐴: (π‘Ž, 𝑏) ∈ πœƒ }. Then 𝐴 πœƒοΏ½ is an algebra of the given type β„±. Now consider a natural epimorphism𝛼:𝐴 β†’ 𝐴 πœƒοΏ½ defined by 𝛼(π‘Ž) = πœƒ(π‘Ž) for all π‘Ž ∈ 𝐴. Then, π‘˜π‘’π‘Ÿ 𝛼 = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ 𝛼(π‘Ž) = 𝛼(𝑏)} = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ πœƒ(π‘Ž) = πœƒ(𝑏)} = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ (π‘Ž, 𝑏) ∈ πœƒ} = πœƒ Therefore, the required algebra and a homomorphism are 𝐴 πœƒοΏ½ and 𝛼 such that ker𝛼 = πœƒ. ∎ Now, from the above two theorems we observe that, kernels of homomorphisms from an algebra 𝐴 and congruence relations on 𝐴 are the same. 4. The Updated Fundamental Theorem of Homomomorphisms of General Universal Algebras 4.1. An associate of a Homomorphism of General Universal Algebras Definition 4.1.1: Let 𝐴,𝐡,𝐢 and 𝐷 be algebras of a given type β„±. Let𝑔: 𝐴 β†’ 𝐡 and β„Ž:𝐢 β†’ 𝐷 be 65 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 homomorphisms. Then 𝑔is said to be an associate of β„Ž, if there exist two isomorphisms 𝛼: 𝐢 β†’ 𝐴 and 𝛽: 𝐷 β†’ 𝐡such that, the diagram Figure 2: An associate of a homomorphism is commutative; in the sense that, 𝑔 ∘ 𝛼 = π›½π‘œβ„Ž. Remark: Following the above definition we can observe that the binary relation β€œ~” forms an equivalence relation on the class of all algebras of a given type. Furthermore, if any two homomorphisms are given to be an associate to each other, then we can draw the properties of a homomorphism like injectivity, surjectivity and bijectivity which is held in one of the two associate homomorphisms to the other. 4.2. The Updated Fundamental Theorem of Homomorphisms of Algebras Theorem 4.2.1(The Updated Fundamental Theorem of Homomorphisms of Algebras): Let 𝐴 and 𝐡 be algebras of a given type β„± and πœ‹: 𝐴 β†’ 𝐡 be a homomorphism. Then, 1. There exists an algebra 𝑋 of the given type β„±, an epimorphism β„Ž:𝐴 β†’ 𝑋 and a monomorphism 𝑔:𝑋 β†’ 𝐡 such that; πœ‹ = 𝑔 ∘ β„Ž (𝑖. 𝑒. πœ‹ (π‘Ž) = 𝑔 ∘ β„Ž(π‘Ž)for all π‘Ž ∈ 𝐴) This property is known as decomposition of homomorphisms and 2. This decomposition is unique upto associate; in the sense that, if πœ‹ = 𝑔 ∘ β„Ž and πœ‹ = 𝑔’ ∘ β„Žβ€™ are two decompositions of πœ‹ where 𝑋 and π‘Œ are algebras of the given type β„±, β„Ž:𝐴 β†’ 𝑋and β„Žβ€²:𝐴 β†’ π‘Œ are epimorphisms and 𝑔:𝑋 β†’ 𝐡 and 𝑔’:π‘Œ β†’ 𝐡 are monomorphisms, then 𝑔~𝑔′and β„Ž~β„Žβ€². 66 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 Proof: 1. Put πœƒ = kerπœ‹ = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ πœ‹(π‘Ž) = πœ‹(𝑏)}. Then it is clear that 𝐴 πœƒοΏ½ = {πœƒ(π‘Ž): π‘Ž ∈ 𝐴}, the quotient of A over πœƒ, is an algebra of the given type β„±. Now consider a natural epimorphism 𝛼:𝐴 β†’ 𝐴 πœƒοΏ½ defined by 𝛼(π‘Ž) = πœƒ(π‘Ž) for all π‘Ž ∈ 𝐴 and a monomorphism 𝑔:𝐴 πœƒοΏ½ β†’ 𝐡 defined by π‘”οΏ½πœƒ(π‘Ž)οΏ½ = πœ‹(π‘Ž) for all πœƒ ∈ 𝐴 πœƒοΏ½ , π‘Ž ∈ 𝐴. Then, for any π‘Ž ∈ 𝐴 consider, 𝑔 ∘ β„Ž(π‘Ž) = π‘”οΏ½β„Ž(π‘Ž)οΏ½ = π‘”οΏ½πœƒ(π‘Ž)οΏ½ (by the definition of β„Ž) = πœ‹(π‘Ž) (by the definition of 𝑔) Thus,𝑔 ∘ β„Ž = πœ‹ is the required decomposition of πœ‹. 2. Now we prove the uniqueness (upto associate) of the above decomposition.Suppose that πœ‹ = 𝑔 ∘ β„Ž and πœ‹ = 𝑔’ ∘ β„Žβ€™ are two decompositions of πœ‹ where 𝑋 and π‘Œ are algebras of the given type β„±, β„Ž:𝐴 β†’ 𝑋and β„Žβ€²:𝐴 β†’ π‘Œ are epimorphisms and 𝑔:𝑋 β†’ 𝐡 and𝑔’:π‘Œ β†’ 𝐡 are monomorphisms. Claim 1:- β„Ž~β„Žβ€² (h and h’ are associate to each other) and 𝑔~𝑔′ Figure 3: Two decompositions of πœ‹ Now consider the identity map 𝐼𝐴 on an algebra 𝐴, which is an isomorphism. On the other hand since β„Žβ€²:𝐴 ⟢ π‘Œ is an epimorphism, every element of Y can be expressed as β„Žβ€™(π‘Ž) for some π‘Ž ∈ 𝐴. Define 𝛼:π‘Œ β†’ 𝑋 by: π›ΌοΏ½β„Žβ€²(π‘Ž)οΏ½ = β„Ž(π‘Ž)for all β„Žβ€²(π‘Ž) ∈ π‘Œ where π‘Ž ∈ 𝐴. Then we prove that: a) 𝛼 is well defined; for, 67 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 For any 𝑦1, 𝑦2 ∈ π‘Œ there exists π‘Ž1 andπ‘Ž2∈ 𝐴 such that𝑦1 = β„Žβ€²(π‘Ž1) and 𝑦2 = β„Žβ€²(π‘Ž2) Therefore, 𝑦1 = 𝑦2 ⟹ β„Žβ€²(π‘Ž1) = β„Žβ€²(π‘Ž2) inπ‘Œ ⟹ 𝑔′(β„Žβ€²(π‘Ž1)) = π‘”β€²οΏ½β„Žβ€²(π‘Ž2)οΏ½ in𝐡 ⟹ 𝑔′ ∘ β„Žβ€²(π‘Ž1) = 𝑔′ ∘ β„Žβ€²(π‘Ž2) in𝐡 ⟹ 𝑓(π‘Ž1) = 𝑓(π‘Ž2) in𝐡 (∡ 𝑓 = 𝑔′ ∘ β„Žβ€²) ⟹ 𝑔 ∘ β„Ž(π‘Ž1) = 𝑔 ∘ β„Ž(π‘Ž2) in𝐡 (∡ 𝑓 = 𝑔 ∘ β„Ž) ⟹ 𝑔(β„Ž(π‘Ž1)) = 𝑔(β„Ž(π‘Ž2)) in𝐡 ⟹ β„Ž(π‘Ž1) = β„Ž(π‘Ž2) in𝑋 (∡ 𝑔 is an injection) Therefore, 𝛼 is well defined. b) 𝛼 is an injection; for, Let 𝑦1 = β„Žβ€²(π‘Ž1) and 𝑦2 = β„Žβ€²(π‘Ž2) ∈ π‘Œ for some π‘Ž1 andπ‘Ž2 ∈ 𝐴. Then 𝛼(𝑦1) = 𝛼(𝑦2)𝑖𝑛𝑋 ⟹ π›ΌοΏ½β„Žβ€²(π‘Ž1)οΏ½ = π›ΌοΏ½β„Žβ€²(π‘Ž2)οΏ½ in𝑋 ⟹ β„Ž(π‘Ž1) = β„Ž(π‘Ž2) 𝑖𝑛𝑋 (by the definition of 𝛼) ⟹ π‘”οΏ½β„Ž(π‘Ž1)οΏ½ = π‘”οΏ½β„Ž(π‘Ž2)οΏ½ in𝐡 ⟹ 𝑔 ∘ β„Ž(π‘Ž1) = 𝑔 ∘ β„Ž(π‘Ž2) in𝐡 ⟹ 𝑓(π‘Ž1) = 𝑓(π‘Ž2) in𝐡 (∡ 𝑓 = 𝑔 ∘ β„Ž) ⟹ 𝑔′ ∘ β„Žβ€²(π‘Ž1) = 𝑔′ ∘ β„Žβ€²(π‘Ž2)in𝐡 (∡ 𝑓 = 𝑔′ ∘ β„Žβ€²) ⟹ π‘”β€²οΏ½β„Žβ€²(π‘Ž1)οΏ½ = 𝑔′(β„Žβ€²(π‘Ž2) in𝐡 ⟹ β„Žβ€²(π‘Ž1) = β„Žβ€²(π‘Ž2) inπ‘Œ (∡ 𝑔′ is an injection) ⟹ 𝑦1 = 𝑦2 inπ‘Œ Therefore, 𝛼 is an injection. 68 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 c) 𝛼 is a surjection; for, π‘₯ ∈ 𝑋 β‡’ π‘₯ = β„Ž(π‘Ž)for some π‘Ž ∈ 𝐴 (∡ β„Žis a surjection) β‡’ π‘₯ = 𝛼(β„Žβ€²(π‘Ž))for some π‘Ž ∈ 𝐴, where β„Žβ€²(π‘Ž) ∈ π‘Œ β‡’ π‘₯ = 𝛼(𝑦)for some 𝑦 ∈ π‘Œ, where 𝑦 = β„Žβ€²(π‘Ž) Therefore, 𝛼 is a surjection. d) 𝛼 is a homomorphism; for, Let 𝑓 ∈ β„± be a nullary operation symbol. Then π‘“π‘Œ ∈ π‘Œ and 𝑓𝐴 ∈ 𝐴. Since β„Žβ€² is a homomorphism, we have β„Žβ€²(𝑓𝐴) = π‘“π‘Œ . Therefore, 𝛼(π‘“π‘Œ) = 𝛼(β„Žβ€²(𝑓𝐴)) = β„Ž(𝑓𝐴) (by the definition of 𝛼) = 𝑓𝑋 (∡ β„Žis a homomorphism) Also, Let 𝑓 ∈ β„± be an 𝑛-ary operation symbol,𝑛 > 0and 𝑦1,𝑦2 , … … . . ,𝑦𝑛 ∈ π‘Œ. Then since β„Žβ€² is an epimorphism, there existsπ‘Ž1, π‘Ž2, … . . π‘Žπ‘› ∈ 𝐴, such that 𝑦𝑖 = β„Žβ€²(π‘Žπ‘–) for all 1 ≀ 𝑖 ≀ 𝑛 and hence, π›ΌοΏ½π‘“π‘Œ(𝑦1 ,𝑦2, … … . . ,𝑦𝑛)οΏ½ = π›ΌοΏ½π‘“π‘Œ(β„Žβ€²(π‘Ž1), β„Žβ€²(π‘Ž2), … … . . , β„Žβ€²(π‘Žπ‘›))οΏ½ = 𝛼(β„Žβ€²(𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›))) (∡ β„Žβ€²is a homomorphism) = β„Ž(𝑓𝐴(π‘Ž1, π‘Ž2, … . . π‘Žπ‘›)) = 𝑓𝑋(β„Ž(π‘Ž1), β„Ž(π‘Ž2), … . . β„Ž(π‘Žπ‘›)) (∡ β„Žis a homomorphism) = 𝑓𝑋(𝛼(β„Žβ€²(π‘Ž1)),𝛼(β„Žβ€²(π‘Ž2)), … . .𝛼(β„Žβ€²(π‘Žπ‘›))) = 𝑓𝑋(𝛼(𝑦1),𝛼(𝑦2), … . .𝛼(𝑦𝑛)) Therefore, 𝛼 is a homomorphism. Thus, By the results in (a), (b), (c) and (d) we get that 𝛼 is an isomorphism. Now, for any π‘Ž ∈ 𝐴, consider, 69 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 π›Όπ‘œβ„Žβ€²(π‘Ž) = 𝛼(β„Žβ€²(π‘Ž)) = β„Ž(π‘Ž) (by the definition of 𝛼) = β„Ž (𝐼𝐴(π‘Ž)) (∡ 𝐼𝐴is an identity map on A) = β„Žπ‘œπΌπ΄ (π‘Ž) Therefore, π›Όπ‘œβ„Žβ€² = β„Žπ‘œπΌπ΄ and hence β„Ž ∼ β„Žβ€² (or β„Ž and β„Žβ€² are associates). Claim 2:- 𝑔~𝑔′ (or 𝑔 and 𝑔′ are associates) Figure 4: Two decompositions of πœ‹ From claim (1) we have, the identity map 𝐼𝐴 on the domain algebra 𝐴and 𝛼:π‘Œ β†’ 𝑋, which is defined by π›ΌοΏ½β„Žβ€²(π‘Ž)οΏ½ = β„Ž(π‘Ž) for all β„Žβ€²(π‘Ž) ∈ π‘Œ where π‘Ž ∈ 𝐴, which are isomorphisms. Also, consider the identity map 𝐼𝐡 on 𝐡 which is also an isomorphism. Since β„Žβ€²:𝐴 β†’ π‘Œ is an epimorphism and hence a surjection, for any 𝑦 ∈ π‘Œ, there exist π‘Ž ∈ 𝐴 such that 𝑦 = β„Žβ€²(π‘Ž). Then consider, 𝐼𝐡 ∘ 𝑔′(𝑦) = πΌπ΅π‘œπ‘”β€²(β„Žβ€²(π‘Ž)) in𝐡 = 𝐼𝐡(π‘”β€²οΏ½β„Žβ€²(π‘Ž)οΏ½) in𝐡 = π‘”β€²οΏ½β„Žβ€²(π‘Ž)οΏ½in𝐡 = 𝑔′ ∘ β„Žβ€²(π‘Ž)in𝐡 = 𝑓(π‘Ž) 𝑖𝑛𝐡 70 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 = 𝑔 ∘ β„Ž(π‘Ž)in𝐡 (∡ 𝑓 = 𝑔 ∘ β„Ž) = 𝑔(β„Ž(π‘Ž))in𝐡 = 𝑔(π›ΌοΏ½β„Žβ€²(π‘Ž)οΏ½)in𝐡 (by the definition of 𝛼) = 𝑔 ∘ π›ΌοΏ½β„Žβ€²(π‘Ž)οΏ½in𝐡 = 𝑔 ∘ 𝛼(𝑦)in𝐡 (∡ 𝑦 = β„Žβ€²(π‘Ž) ) Therefore, 𝐼𝐡 ∘ 𝑔′ = 𝑔 ∘ 𝛼. Thus, we have two isomorphisms, 𝐼𝐡 on 𝐡 and 𝛼:π‘Œ β†’ 𝑋 satisfying that, the diagram; Figure 5: Associate homomorphisms 𝑔 and 𝑔′ is commutative; that is, 𝐼𝐡 ∘ 𝑔′ = 𝑔 ∘ 𝛼. Therefore, 𝑔~𝑔′ (or 𝑔 and 𝑔′ are associate). ∎ Corollary 4.2.1:Let A and B be algebras of a given type β„± and πœ‹: 𝐴 β†’ 𝐡 be a homomorphism. Let πœ‹ = 𝑔 ∘ β„Ž be any decomposition of πœ‹ where β„Ž:𝐴 β†’ 𝑋 is an epimorphism and 𝑔:𝑋 β†’ π΅π‘–π‘ π‘Ž monomorphism for some algebra 𝑋 of the given type β„±. Then there exist an isomorphism of 𝑋 onto the quotient algebra 𝐴 πΎπ‘’π‘Ÿπœ‹οΏ½ . Proof: Put πœƒ = kerπœ‹ = {(π‘Ž, 𝑏) ∈ 𝐴 Γ— 𝐴 ∢ πœ‹(π‘Ž) = πœ‹(𝑏)}. Considering the natural epimorphismβ„Žβ€²:𝐴 β†’ 𝐴 πœƒοΏ½ defined by β„Žβ€²(π‘Ž) = πœƒ(π‘Ž) forall π‘Ž ∈ 𝐴 and a monomorphism 𝑔′:𝐴 πœƒοΏ½ β†’ 𝐡defined by π‘”β€²οΏ½πœƒ(π‘Ž)οΏ½ = 𝑓(π‘Ž) for all πœƒ(π‘Ž) ∈ 𝐴 πœƒοΏ½ , π‘Ž ∈ 𝐴, we get another decomposition πœ‹ = 𝑔′ ∘ β„Žβ€². Then, by the uniqueness of decomposition of homomorphisms of algebras, we get an ismorphism; 𝛼:𝑋 β†’ 𝐴 πœƒοΏ½ 71 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 Figure 6: Two decompositions of πœ‹ such that, 𝛼 ∘ β„Ž = β„Žβ€² and 𝑔′ ∘ 𝛼 = 𝑔. ∎ 5. Conclusion and Recommendation On the basis of the Updated Fundamental Theorem of Homomorphisms of General Universal Algebras which is formulated and proved in this paper, we can conclude that any homomorphism of algebras of a given type can be decomposed as a composition of a monomorphism and an epimorphism and this decomposition is unique upto associate; in the sense that, if a given homomorphism πœ‹ of algebras of a given type can be decomposed in one way as πœ‹ = 𝑔 ∘ β„Ž where 𝑔 is a monomorphism and β„Ž is an epimorphism, and if πœ‹ can also be decomposed in another way as πœ‹ = 𝑔′ ∘ β„Žβ€² where 𝑔′ is a monomorphism and β„Žβ€² is an epimorphism then we get that 𝑔~𝑔′ (or 𝑔 and 𝑔′ are associate to each other) and β„Ž~β„Žβ€² (or β„Ž and β„Žβ€² are associate to each other). We say that two homomorphisms𝑔 and β„Ž of algebras of a given type are associate to each other, if there exists two isomorphisms𝛼 and 𝛽 such that; 𝑔 ∘ 𝛼 = 𝛽 ∘ β„Ž. Since 𝛼 and 𝛽 are isomorphisms and hence invertible it is equivalently saying that one can be expressed as a composition of the other and two isomorphisms; that is, 𝑔 ∘ 𝛼 = 𝛽 ∘ β„Ž ⇔ 𝑔 = 𝛽 ∘ β„Ž ∘ π›Όβˆ’1 ⇔ β„Ž = π›½βˆ’1 ∘ 𝑔 ∘ 𝛼. It follows from this definition that, two associate homomorphisms have different properties in common such as: one of the two associate homomorphisms is an injection (respectively a surjection and a bijection) if and only if the other is an injection (respectively a surjection and a bijection). Moreover, their domain (respectively codomain) are isomorphic to each other. It is finally recommended that it needs additional researches on the class of associate homomorphisms of algebras of a given type to identify and characterize in a more general way. Acknowledgements Above all, I would like to thank the almighty God for his mercy that endowed upon me. Next, I would like to offer my sincerest and deepest gratitude to my advisor Prof. U. M. Swamy, who has supported me throughout this research work with his patency and knowledge whilst allowing me the room to work in my own way. One could not wish for a better or friendly advisor. I am also indebted to my friends helping me throughout my work in different ways. 72 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2014) Volume 10, No 1, pp 61-73 Finally, I would like to express my heartfelt thanks to my parents, mom, dad and my brothers. In my daily work I have been blessed with their unconditional love, kindness, support and encouragement. From the start until the accomplishment of this manuscript, they have been my source of strength and courage. I would not let this opportunity to escape without me saying "mom, I love you and always treasure you. 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