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...Extending Homomorphism Theorem to Multi-Systems Abstract: The multi-laterality of WORLD implies multi-systems to be its best candidate model for ones cognition on nature, which is also included in an ancient book of China, TAO TEH KIN...

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...Contents Extending Homomorphism Theorem to Multi-Systems BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01 A Double Cryptography Using the Smarandache Keedwell Cross Inverse Quasigroup BY T. G. JA´IY´EOL´A . ...

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.... .48 2.1.3 Group. . . . . . . . . . . . . . . . . . . . . . . . . . . .50 2.1.4 Isomorphism of systems . . . . . . . . . . . . . . . . . 50 2.1.5 Homomorphism theorem . . . . . . . . . . . . . . . . 51 2.2 Multi-operation systems . . . . . . . . . . . . . . . . . . . . . 55 2.2.1 Multi-operation system. . . . . . . . . . . . . . . . . . 55 2.2.2 Isomorphism of multi...

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....3.8: Let L and M be lattices. A mapping f : L → M is called a i) Join homomorphism if x ∪ y = z ⇒ f(x) ∪ f(y) = f(z). ii) Meet homomorphism if ... ...(z). ii) Meet homomorphism if x ∩ y = z ⇒ f(x) ∩ f(y) = f(z). iii) Order homomorphism if x ≤ y ⇒ f(x) ≤ f(y). 13 f is a homomorphism (or lat... ...orphism if x ≤ y ⇒ f(x) ≤ f(y). 13 f is a homomorphism (or lattice homomorphism) if it is both a join and a meet homomorphism. Injective, sur... ... join and a meet homomorphism. Injective, surjective or bijective lattice homomorphism are called lattice monomorphism, epimorphism, isomorphism res... ...ormal subloop of a loop L is any subloop of L which is the kernel of some homomorphism from L to a loop. Further Pflugfelder [50] has proved the c... ...ITION 2.2.11: A map φ from a loop L to another loop L 1 is called a loop homomorphism if φ(ab) = φ (a) φ (b) for all a, b ∈ L. 21 DEFINITION ... ...o L? 9. For the loops given in examples 2.1.2 and 2.1.3 construct a loop homomorphism. 10. Give an example of a strict non-commutative loop. 11. P... ...is yes; but for this we have to introduce the concept of Smarandache loop homomorphism between loops. DEFINITION 3.3.10: Let L and L' be two Smar... ... subgroups respectively. A map φ from L to L' is called Smarandache loop homomorphism (S-loop homomorphism) if φ restricted to A is mapped onto a s...

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... Special elements in rings 24 2.3 Substructures of a ring 26 2.4 Homomorphism and quotient rings 28 2.5 Special rings 30 2.6 Mod... ...n radical are introduced in section three. Recollection of the concept of homomorphisms and quotient rings are carried out in section four. Special r... ...re not ideals in M 3×3 = {(a ij )/ a ij ∈ Z 6 ={0, 1, … , 5}}. 2.4 Homomorphism and Quotient Rings In this section we recall the basic con... ...ism and Quotient Rings In this section we recall the basic concepts of homomorphism and quotient rings and give some examples. DEFINITION 2.4... ...DEFINITION 2.4.1: Let R and S be two rings. A mapping f: R→ S is called a homomorphism of rings if for all a, b ∈ R we have (1) f(a+b) = f(a) + f(b) ... ...∈ R we have (1) f(a+b) = f(a) + f(b) and (2) f(ab) = f(a) f(b). If f is a homomorphism, it is easy to verify f(0)=0, f(-x) = -f(x), and f(1 R )= 1 S... ... we say the map f is unitary. DEFINITION 2.4.2: Let f: R→ S be a ring homomorphism, the kernel of the homomorphism f is defined to be the set = ... ... {x ∈ R / f(x) = 0} and is denoted by Ker f = {x ∈ R / f(x)=0}. A ring homomorphism f: R→ S is called 1) a monomorphism if f is injective 2)... ...tomorphism if R = S and f is an isomorphism or equivalently; we can say a homomorphism f: R→ S is a monomorphism if and only if Ker (f) = {0}. C...

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...ο) be two semigroups. We say a map φ from (S, ∗) → (S 1 , ο) is a semigroup homomorphism if φ (s 1 ∗ s 2 ) = φ(s 1 ) ο φ(s 2 ) for all s 1 , s 2 ∈ ... ...G, ο) and (G 1 , ∗) be two groups. A map φ: G to G 1 is said to be a group homomorphism if φ (a • b) = φ(a) ∗ φ(b) for all a, b ∈ G. DEFINITION 1... ... 2, …, 27}, the semigroup under multiplication modulo 28. 4. Construct a homomorphism between the semigroups. S 3 ×3 given in problem 1 and Z 28... ...3 given in problem 1 and Z 28 given in problem 3. Find the kernel of this homomorphism. ( φ: S 3 ×3 → Z 28 ) where ker φ = {x ∈ S 3 ×3 / φ(x) = ... ...tween the groups G = 〈g/g 6 = 1 〉 and S 3 ? 9. Can you construct a group homomorphism between g = 〈g/g 6 = 1 〉 and S 4 ? Prove or disprove. 1... ... 〈g/g 6 = 1 〉 and S 4 ? Prove or disprove. 10. Does there exist a group homomorphism between G = 〈g/g 11 = 1 〉 and the symmetric group S 5 ? ... ...G = 〈g/g 11 = 1 〉 and the symmetric group S 5 ? 11. Can we have a group homomorphism between G = 〈g/g p = 1〉 and the symmetric group S q (wher... ...tric group S q (where p and q are two distinct primes)? 12. Find a group homomorphism between D 2n and S n . (D 2n is called the dihedral group... ... Let L and M be any two lattices. A mapping f: L → M is called a 1. Join homomorphism if x ∪ y = z ⇒ f(x) ∪ f(y) = f(z) 2. Meet homomorphism if x...

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... R, then the map f : R → R µ defined by f(x) = µ x ∗ for all x ∈ R is a homomorphism with kernel µ t , where t = µ (0). THEOREM [109]: If µ is ... ...ssume the subring contains the identity of R. We let g denote the natural homomorphism of R onto R/M. If S is a subset of R, we let 〈S 〉 denote the ... ...he extension of fuzzy prime ideals. Let R and S be rings and let f be a homomorphism of R into S. Let T denote f (R). If I is an ideal in R, then ... ... be obtained as a matter of routine. THEOREM 1.4.72: If f : R → R' is a homomorphism of rings and J : R → L and J' : R' → L are fuzzy ideals, then... ...s no properly fuzzy prime ideal. The following results are relation on homomorphism and epimorphism of rings. THEOREM 1.4.78: i. Let f : R ... ...ker f then f (P) : R' → L is a prime fuzzy ideal. ii. If f : R → R' is a homomorphism of rings. If P' : R' → L is a fuzzy prime ideal then f –1 (... ...prove. As in case of prime ideals and its relation with epimorphism and homomorphism, we give here without proof the relation of primary ideals an... ...here without proof the relation of primary ideals and the epimorphism and homomorphism. THEOREM 1.4.80: i. Let f : R → R' be an epimorphism of... ...ker f, then f(Q) is a primary fuzzy ideal of R'. ii. Let f : R → R' be a homomorphism of rings. If Q' : R' → L is a primary fuzzy ideal of R' then ...

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... and A' its subgroups respectively. A map φ from L to L' is called S-loop homomorphism if φ restricted to A is mapped to a subgroup A' of L', i.e. φ... ...stricted to A is mapped to a subgroup A' of L', i.e. φ: A → A' is a group homomorphism. It is not essential that φ be defined on whole of A. The... ...ny two S-groupoids. A map φ from G 1 to G 2 is said to be a Smarandache homomorphism (S-homomorphism) if φ: A 1 → A 2 where A 1 ⊂ G 1 and A 2 ... ...et S and S' be two semirings. A mapping φ : S → S' is called the semiring homomorphism if φ (a + b) = φ (a) + φ(b) and φ (a b) = φ (a) φ (b) f... ...X is any non empty set) for all N ∈ V and for all g: X → N there exists a homomorphism h ∈ Hom (F x , N): h o f = g X F X f g ... ...rings are defined as in case of near-rings. The concept of loop near-ring homomorphism is also defined as in case of near-rings. Now we proceed on t... ...the near field contained in N. Now we proceed on to define Smarandache homomorphism. DEFINITION 1.6.27: Let N and N 1 be two S-near-rings P an... ... 1 be S-N –subgroups. i. h: N → N 1 is called a Smarandache near-ring homomorphism (S-near-ring homomorphism) if for all, m, n ∈ M, (M ⊂ N, M 1 ... ...d on whole of N. 51 ii. h: P → P 1 is called Smarandache N subgroup-homomorphism (S-N subgroup- homomorphism) if for all p, q in S (S ⊂ P; S 1...

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...defined on sets X and Y, respectively then a function h : X → Y is said to be a homomorphism from (X, R) to (Y, Q) if (x 1 , x 2 ) ∈ R implies (h(x ... ... ) ∈ R implies (h(x 1 ), h(x 2 )) ∈ Q for all x 1 , x 2 ∈ X. In other words a homomorphism implies that for every two elements of the set X which ... ...never the case under a homomorphic function h, the function is called a strong homomorphism. It satisfies the two implications 〈x 1 , x 2 〉 ∈ R ... ...eria that a many-to- 28 one function h must satisfy in order to be a strong homomorphism are somewhat modified. The function h imposes a partiti... ...ments of B to some element y 2 ∈ Y. Then the function h is said to be a strong homomorphism from 〈X, R〉 to 〈Y, Q〉 if and only if the degree of the... ... equality must be satisfied for each pair of blocks of the partition π h . If a homomorphism exists from (X, R) to (Y, Q) then the relation Q (X, Y)... ...that are not the counterparts of any number of R. This is not the case when the homomorphism is strong. Here more properties of the relation R are p... ...desirable properties in sets such as ordering or similarity. If h : X → Y is a homomorphism from (X, R) to (Y, Q) and if h is completely specified,... ...n R and Q are equal. Each of these terms applies without modification to fuzzy homomorphisms. Now we just recall the notions of Sup-i-compositions...

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...defined on sets X and Y, respectively then a function h : X → Y is said to be a homomorphism from (X, R) to (Y, Q) if (x 1 , x 2 ) ∈ R implies (h(x ... ... ) ∈ R implies (h(x 1 ), h(x 2 )) ∈ Q for all x 1 , x 2 ∈ X. In other words a homomorphism implies that for every two elements of the set X which ... ...never the case under a homomorphic function h, the function is called a strong homomorphism. It satisfies the two implications 〈x 1 , x 2 〉 ∈ R ... ...eria that a many-to- 28 one function h must satisfy in order to be a strong homomorphism are somewhat modified. The function h imposes a partiti... ...ments of B to some element y 2 ∈ Y. Then the function h is said to be a strong homomorphism from 〈X, R〉 to 〈Y, Q〉 if and only if the degree of the... ... equality must be satisfied for each pair of blocks of the partition π h . If a homomorphism exists from (X, R) to (Y, Q) then the relation Q (X, Y)... ...that are not the counterparts of any number of R. This is not the case when the homomorphism is strong. Here more properties of the relation R are p... ...desirable properties in sets such as ordering or similarity. If h : X → Y is a homomorphism from (X, R) to (Y, Q) and if h is completely specified,... ...n R and Q are equal. Each of these terms applies without modification to fuzzy homomorphisms. Now we just recall the notions of Sup-i-compositions...

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... . DEFINITION: A mapping φ from a group G to a group G is said to be a group homomorphism if for all a, b ∈ G, φ(ab) = φ(a) φ(b). Remarks: If φ... ...G, φ(ab) = φ(a) φ(b). Remarks: If φ(a) = e for all a ∈ G. We call φ a trivial homomorphism. Likewise if φ is a map from G to G such that φ(x) = x ... ... from G to G such that φ(x) = x for every x ∈ G then also we say φ is a trivial homomorphism or the identity homomorphism from G to G . The kernel of... ...by K φ = {x ∈ G / φ(x) = e , e is the identity element of G }. DEFINITION: A homomorphism G into G is said to be an isomorphism if φ is one to one... ... is defined by ψ(g) = τ g , the relation τ gh = τ g τ h tells us that ψ is a homomorphism. What is the kernel K of ψ? If g 0 ∈ K, then ψ(g 0 ) =... ...ressions for 25 0 g eτ we conclude that g 0 = e, whence K = (e). We know a homomorphism ψ of G into A(S) with kernel K is an isomorphism of G i... ...l Cayley's theorem of group for S-semigroups we need the concept of S-semigroup homomorphism using which we will prove our result. DEFINITION: Let... ...nd S' be any two S-semigroups. A map φ from S to S' is said to be a S-semigroup homomorphism if φ restricted to a subgroup A ⊂ S → A' ⊂ S' is a grou... ...stricted to a subgroup A ⊂ S → A' ⊂ S' is a group homomorphism. The S-semigroup homomorphism is an isomorphism if φ: A → A' is one to one and onto. ...

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...L). DEFINITION 2.1.14: A map θ from a loop L to a loop L’ is called a homomorphism if θ (ab) = θ (a) θ (b) for all a, b ∈ L. DEFINITION 2.1.... ...subgroups respectively. A map φ from L to L' is called a Smarandache loop homomorphism (S- loop homomorphism) if φ restricted to A is mapped onto the... ...d to A is mapped onto the subgroup A' in L', that is φ: A → A' is a group homomorphism. The concept of S-loop isomorphism and automorphism are de... ... +, .) be any two SNA-rings. A map φ : R to R 1 is a Smarandache NA-ring homomorphism (SNA-ring homomorphism) if φ : A → A 1 is a ring homomorphis... ...for loop ring Z 7 L 13 (5) and Z 9 L 15 (8) maps φ such that φ is a SNA-homomorphism? Find Ker φ 1 if φ 1 : Z 7 A 1 → Z 9 A 2 where A 1 = {e... ...apping X → X 1 from a Lie algebra L to Lie algebra L 1 is said to be a homomorphism if it satisfies; (1) if X → X 1 , Y → Y 1 then for any λ... ..., Y] → [X 1 , Y 1 ]. If this mapping is onto then L 1 is said to be a homomorphism image of L. Example 4.1.4: All trace zero matrices of gl(n,... ...s, which are S-Lie algebras. We say φ: R → S is a Smarandache Lie algebra homomorphism (S-Lie algebra homomorphism) if φ restricted to the subsets W... ...SNA-subrings which satisfy the Lie identity and φ: W → V is a Lie algebra homomorphism. Note: It is important to note that the mapping need not ev...

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...be any two semigroups . We say a map φ: (S, ) → (S', ∗ ∗ ∗ ∗) is a semigroup homomorphism if φ (s 1  s 2 ) = φ(s 1 ) ∗ ∗ ∗ ∗ φ (s 2 ) for all s 1 ... ..., which is not an ideal in S 2×2 given in example 1.3.2. PROBLEM 8: Find a homomorphism between the semigroups S = {Z 30 , ×}, the semigroup unde... ... , ×) be isomorphic with (Z + , +)? Justify your answer. PROBLEM 12: Find a homomorphism between the semigroups Z 7 = {0, 1, 2, ... , 6} under mu... ...roperties about SGs like Smarandache direct product of groupoids, Smarandache homomorphism of groupoids and give some examples of them. DEFINITI... ...and (G', o) be any two SGs. A map φ from (G, ∗) to (G', o) is said to be a SG homomorphism if φ : A → A' is a semigroup homomorphism where A ⊂ G and ... ... respectively. We have φ (a ∗ b) = φ (a) o φ(b) for all a, b ∈ A. We say a SG homomorphism is a SG isomorphism if φ restricted to the semigroups is a... ...: Can Z 13 (3, 2) be SG isomorphic with Z 13 (6, 3)? PROBLEM 4: Give a SG homomorphism between Z 19 (3, 7) and Z 19 (3, 6). PROBLEM 5: Is Z... ...SG isomorphism between Z 8 (4,6) and Z 9 (3,6)? PROBLEM 9: Construct a SG homomorphism between Z 6 (2, 3) and Z 12 (2, 3). 4.5 Smarandach... ..., ) , , B , A s s s s λ δ . A map φ : 1 K to 2 K is a Smarandache automaton homomorphism if φ restricted from K 1 = (Z 1 , A 1 , B 1 , δ 1 , λ 1 )...

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...on the sets X and Y respectively A function h : X → Y is said to be a fuzzy homomorphism from (X, R) to (Y Q) if R (x 1 , x 2 ) ≤ Q (h (x 1 ), h (x ... ...er the case under a homomorphic function h, the function is called a strong homomorphism. It satisfies the two implications (x 1 , x 2 ) ∈ R implies... ... 0 0 6 . 0 5 . 0 0 1 0 9 . 0 0 0 0 5 . 0 The graph of the ordinary fuzzy homomorphism. 63 Now we leave it for the reader to construct a s... ...ism. 63 Now we leave it for the reader to construct a strong fuzzy homomorphism and illustrate it by a graph. Next we just recall the defin... ...Fuzzy Cognitive Maps (FCMs), 16, 63 Fuzzy compatibility relation, 17 Fuzzy homomorphism, 61-3 Fuzzy membership relation, 56 Fuzzy partial ordering... ...64-5 Fuzzy relation, 56-7 G Geodesic, 27 Girth of a graph, 27 Graph homomorphism, 20-1 Graph isomorphism, 20-1 Graph, 18 H Hamilton p... ... Strong digraph, 33 Strong double neutrosophic Eulerian graph, 85-7 Strong homomorphism, 61-2 Strong neutrosophic basis, 11 Strong neutrosophic di...

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... : X α,β →X α,β is a covering mapping, then the induced mapping π ∗ is a homomorphism from Aut M to AutM. Proof First, we prove that for∀g∈Aut M... ...t that πgπ −1 ∈AutM, i.e., π ∗ : Aut M →AutM. Now we prove that π ∗ is a homomorphism from Aut M to AutM. In fact, for ∀g 1 ,g 2 ∈Aut M, we have... ... π −1 )(πg 2 π −1 ) =π ∗ (g 1 )π ∗ (g 2 ). Whence, π ∗ :Aut M →AutM is a homomorphism. ♮ 1.2 Voltage maps For creating a homomorphism between Klein ...

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... = min{x,y} amd C(x \/ y) = C(x) \/ C(y). Therefore, the mapping C is a homomorphism. 57 If (x , 0, 1 – x) = (x 1 , 0, 1 – x 1 ) , then by t... ...0 Highly significant contradiction, 40 Highly significant tautology, 40 Homomorphism, 56 Huntington, E. V., 138 Hypotheses, 94, 135, 136, Hypo...

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...ctor space leading to analogous results as in case of vector spaces. The homomorphism T of a linear bialgebra V = V 1 ∪ V 2 to another linear bi...