i issue- 4 2010 2010 s - scholar.oauife.edu.ng · muktibodh p. s.: problems of standard model in...
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2010 2010I
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ISSUE- 4 2010
EDITORIAL BOARD
(i) Dr. R. V. SARAYKAR, Editor - in - Chief,
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NUMTA BULLETIN, 2008 ISSN 0974-1623
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Contents
1. Temıto.pe. Gbo. lahan Jaıyeo. la : On the Factor Set of Code Loops 1-16
2. Kumbhojkar H. V. : Mathematics: Brother of Humor 17-32
3. Muktibodh P. S.: Problems of standard Model in Cosmology 33-51
4. Shobhane A. R. and Vyawahare A.W. : Some Pythagorian Triples
and their Properties 52-57
5. Surendran K. K. : The Realm of Energy and Elementary Particles 58-79
6. Vyawahare A.W. : Madhava’s Contribution to Mathematics 80-97
D.T.P. Work Using LATEX by :
A.S.Muktibodh,
179, Sahakarnagar, (Khmala)
Nagpur-440025
Ph.(0712)2292290
Unity in Mathematics 1-23
Beyond Farey Fibonacci Sequence24-30
Mathematics: Brother of Humor (Contd.) 31-39
Magical Geometrical Figures40-44
Relationship between code loops and some other loops5 45-525.
6. Conformal Killing Vectors in Spherically Symmetric,Conformal Killing Vectors in Spherically Symmetric,
Inhomogeneous, Shear-free, Separable Metric
Spacetimes
Conformal Killing Vectors in Spherically Symmetric,
Inhomogeneous, Shear-free, Separable Metric
Spacetimes
Conformal Killing Vectors in Spherically Symmetric,
Inhomogeneous, Shear-free, Separable Metric
Spacetimes
Conformal Killing Vectors in Spherically Symmetric,
Inhomogeneous, Shear-free, Separable Metric
Spacetimes
53-67
77. Integrated approach and Trace of Ancient
Tradition in SREDHIVYAVHARASredhivyavhara
68-91
8. Ganit- Sar-Sangraha of Mahaviracharya 92-95
: Sanjay Wagh
Kiran SisodiaKiran Sisodia
H.V.Kumbhojkar:
:
: A.W.Vyawahare
: Jayeola Temitope
: S.M.Wagh, R.V.Saraykar, P.S.Muktibodh, K.S.Govinder
: V.D.Heroor
: A.W.Vyawahare
Kiran Sisodia
Jaiyeola Temitope
Relationship between code loops and some other loops ∗†
Temıto.pe. Gbo. lahan Jaıyeo. la‡
Department of Mathematics,Obafemi Awolowo University, Ile Ife, Nigeria.
[email protected], [email protected]
Abstract
Necessary and sufficient conditions for a code loop L(φ) on a binary linear code Cthat is doubly even with a factor set φ to be an homogenoeus loop, a Kikkawa loop, a K-loop and a cross inverse property loop are established. If c(u, v, w) represent the numberof corresponding co-ordinates of code words u, v and w that are non-zero and u·v denotethe number of the corresponding co-ordinates of code words u and v that are non-zero,then it is shown that; c(u, v, x + y) ≡ [c(u, v, x) + c(u, v, y)] mod 2, c(x + y, u, v) ≡[c(x, u, v) + c(y, u, v)] mod 2 and u·x+u·y
2 ≡ u·(x+y)2 mod 2 whenever L(φ) is an A-loop.
1 Introduction
Let L be a loop and Z ≤ Z(L). If L/Z ∼= C where C is an abelian group then, L ∼= L(φ)where φ : C × C → F , F been a field and L(φ) = F × C with binary operation ∗ suchthat for all (a, u), (b, v) ∈ L(φ) ;
(a, u) ∗ (b, v) = (a + b + φ(u, v), u + v) (1)
and (L(φ), ∗) is a loop. The fact that (L(φ), ∗) is a loop is shown in [14].Let V be a finite dimensional vector space over the field F and C ≤ V . With |F | = 2
such that F = ZZ2 = {0, 1}, C is called a binary linear code and its elements (vectors) arecalled code words. For all u, v ∈ C, let ||u|| denote the number of non-zero co-ordinates in u(this is simply equal to the ’norm squared’ of u if V = IRn), let u ·v denote the number of thecorresponding co-ordinates of u and v that are non-zero (this is simply equal to the ’innerproduct’ of u and v if V = IRn) and let c(u, v, w) represent the number of correspondingco-ordinates of u, v and w that are non-zero.
C is called a double even code if :
∗2000 Mathematics Subject Classification. Primary 20NO5.†Keywords : code loop, binary linear code, A-loop, homogeneous loop, Kikkawa loop, K-loop‡All correspondence to be addressed to this author.
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NUMTA BULLETIN,2010, 45-52NUMTA BULLETIN, 2010, 45-52
45
1. u · v is even
2. ||u|| is divisible by 4.
φ is called a factor set on C if it satisfy the conditions :
φ(u, u) ≡ ||u||4
mod 2 (2)
φ(u, v) + φ(v, u) ≡ u · v2
mod 2 (3)
φ(u, v) + φ(u + v, w) + φ(v, w) + φ(u, v + w) ≡ c(u, v, w) mod 2 (4)
As mentioned in [4], for all u ∈ C, φ(u, 0) = φ(0, u) = 0.For more on the varieties of loops, definitions and their general properties, readers can
check [1],[2], [3], [7] and [13].Code loops were first shown to be Moufang loops by Griess [8] and was later shown to
be Moufang loops with unique non-identity square in [5]. They have also been of interest toChein and Goodaire [6] and P. Vojtechovsky [10], [11], [12] and [14]. Here we shall approachthem in the style of [5].
Necessary and sufficient conditions for a code loop L(φ) on a binary linear code C thatis doubly even with a factor set φ to be an homogenoeus loop, a Kikkawa loop, a K-loopand a cross inverse property loop are established. It is shown that
c(u, v, x+y) ≡ [c(u, v, x)+ c(u, v, y)] mod 2, c(x+y, u, v) ≡ [c(x, u, v)+ c(y, u, v)] mod 2
andu · x + u · y
2≡ u · (x + y)
2mod 2 ∀ u, v, x, y ∈ C
whenever L(φ) is an A-loop.The following definitions are needed. In all, LIPL, RIPL, IPL and AIPL mean a left
inverse property loop, right inverse property loop, inverse property loop and automorphicinverse property loop respectively.
Definition 1.1 Let (G, ◦) be a loop.
1. G is called an Aλ(Aρ,Aµ)-loop if and only if all left(right,middle) inner mappings in Gare automorphisms. It is called an A-loop if and only if it is an Aλ-loop, an Aρ-loopand an Aµ-loop
2. G is called a left(right) homogeneous loop if and only if it is an Aλ(Aρ)-loop and aLIPL(RIPL). It is called an homogeneous loop if and only if it is an A-loop and anIPL.
3. G is called a left(right) K-loop loop if and only if it is a left(right) Bol loop and anAIPL It is called a K-loop if and only if it is a Moufang loop and an AIPL.
4. G is called a left(right) Kikkawa loop if and only if it is a left(right) homogeneous loopand an AIPL. It is called a Kikkawa loop if and only if it is an homogeneous loop andan AI PL.
The automorphism group of a loop (G, ◦) is represented by A(G, ◦) = A(G).
2
T. Jaiyeola 46
Main Results
2 A-Loops And Code Loops
Theorem 2.1 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. If
A(u, v, w) = φ(u, v) + φ(u, v + w) + φ(u + v, w) + φ(v, w) and F (u, v) = φ(u, v) + φ(v, u),then
1. L(φ) is an Aλ-loop ⇔ A(u, v, x + y) = A(u, v, x) + A(u, v, y).
2. L(φ) is an Aρ-loop ⇔ A(x + y, v, u) = A(x, v, u) + A(y, v, u).
3. L(φ) is an Aµ-loop ⇔ F (u, x + y) = F (u, x) + F (u, y).
ProofLet α = (a, u), β = (b, v), γ = (c, x), δ = (d, y) ∈ L(φ).
1. Consider L(α, β) = LαLβL−1βα. Lβα = L(a+b+φ(v,u),u+v). γLβα = (a + b + c + φ(v, u) +
φ(u + v, x), u + v + x) ⇔ (a + b + c + φ(v, u) + φ(u + v, x), u + v + x)L−1βα = γ = (c, x).
L(φ) is an Aλ-loop ⇔ L(α, β) ∈ A(L(φ)) ∀ α, β ∈ L(φ) ⇔ (γδ)L(α, β) =γL(α, β) ∗ δL(α, β) ⇔ (γδ)LαLβL−1
βα = γLαLβL−1βα ∗ δLαLβL−1
βα.
L. H. S.= (γδ)LαLβL−1βα = (a+ b+ c+d+φ(x, y)+φ(u, x+ y)+φ(v, u+x+ y), u+ v +
x + y)L−1βα = (a + b + c + d + φ(v, u) + φ(v, x + y + u) + φ(u + v, x + y) + φ(u, x + y) +
φ(x, y)+φ(v, u)+φ(u+v, x+y), u+v+x+y)L−1βα = (a+b+c+d+φ(x, y)+A(v, u, x+
y)+φ(v, u)+φ(u+v, x+y), u+v +x+y)L−1βα = (c+d+A(v, u, x+y)+φ(x, y), x+y).
R. H. S.= γLαLβL−1βα ∗δLαLβL−1
βα = (a+c+b+φ(u, x)+φ(v, u+x), u+x+v)L−1βα ∗(a+
d+b+φ(u, y)+φ(v, u+y), u+y+v)L−1βα = (a+b+c+φ(v, u)+φ(v, u+x)+φ(v+u, x)+
φ(u, x) + φ(v, u) + φ(u + v, x), u + x + v)L−1βα ∗ (a + d + b + φ(v, u) + φ(v, u + y) + φ(v +
u, y) +φ(u, y) +φ(v, u) +φ(u + v, y), u+ y + v)L−1βα = (a+ b+ c+ A(v, u, x) +φ(v, u) +
φ(u+v, x), u+v +x)L−1βα ∗ (a+ b+d+A(v, u, y)+φ(v, u)+φ(u+v, y), u+v +y)L−1
βα =(c + A(v, u, x), x) ∗ (d + A(v, u, y), y) = (c + d + A(v, u, x) + A(v, u, y) + φ(x, y), x + y).
L(φ) is an Aλ-loop ⇔ L. H. S.=R. H. S. ⇔ A(v, u, x + y) = A(v, u, x) + A(v, u, y).
2. L(φ) is an Aρ-loop ⇔ R(α, β) ∈ A(L(φ)) ∀ α, β ∈ L(φ) ⇔ (γδ)R(α, β) =γR(α, β) ∗ δR(α, β) ⇔ (γδ)RαRβR−1
αβ = γRαRβR−1αβ ∗ δRαRβR−1
αβ .
3
Code Loops and Some Other Loops 47
L. H. S.= (γδ)RαRβR−1αβ = (c+d+a+ b+φ(x, y)+φ(x+y, u)+φ(x+y +u, v), x+y +
u + v)R−1αβ = (a + b + c + d + φ(x + y, u) + φ(x + y, u + v) + φ(x + y + u, v) + φ(u, v) +
φ(u, v) + φ(x + y, u + v) + φ(x, y), x + y + u + v)R−1αβ = (a + b + c + d + A(x + y, u, v) +
φ(x, y)+φ(u, v)+φ(x+y, u+v), x+y+u+v)R−1αβ = (c+d+A(x+y, u, v)+φ(x, y), x+y).
R. H. S.= γRαRβR−1αβ ∗ δRαRβR−1
αβ = (c + a + b + φ(x, u) + φ(x + u, v), x + u + v)R−1αβ ∗
(d+a+b+φ(y, u)+φ(y+u, v), y+u+v)R−1αβ = (a+b+c+φ(x, u)+φ(x, u+v)+φ(x+
u, v)+φ(u, v)+φ(u, v)+φ(x, u+v), u+v+x)R−1αβ ∗(a+b+d+φ(y, u)+φ(y, u+v)+φ(y+
u, v)+φ(u, v)+φ(u, v)+φ(y, u+ v), u+ v + y)R−1αβ = (a+ b+ c+A(x, u, v)+φ(u, v)+
φ(x, u+v), u+v +x)R−1αβ ∗ (a+ b+d+A(y, u, v)+φ(u, v)+φ(y, u+v), u+v +y)R−1
αβ =(c + A(x, u, v), x) ∗ (d + A(y, u, v), y) = (c + d + A(x, u, v) + A(y, u, v) + φ(x, y), x + y).
L(φ) is an Aρ-loop ⇔ L. H. S. =R. H. S. ⇔ A(x + y, v, u) = A(x, v, u) + A(y, v, u).
3. Consider Tα = RαL−1α . γLα = (a + c + φ(u, x), u + x) ⇔ (a + c + φ(u, x), u + x)L−1
α =(c, x).
L(φ) is an Aµ-loop ⇔ (γδ)Tα = γTα ∗ δTα.
L. H. S. = (γδ)Tα = (c + d + a + φ(x, y) + φ(x + y, u), x + y + u)L−1α =
(c + d + a + φ(u, x + y) + φ(x + y, u) + φ(u, x + y) + φ(x, y), x + y + u)L−1α = (a + c +
d+F (u, x+y)+φ(x, y)+φ(u, x+y), u+x+y)L−1α = (c+d+F (u, x+y)+φ(x, y), x+y).
R. H. S.= γTα ∗ δTα = (c + a + φ(x, u), x + u)L−1α ∗ (d + a + φ(y, u), y + u)L−1
α =(a+c+φ(u, x)+φ(x, u)+φ(u, x), x+u)L−1
α ∗(a+d+φ(u, y)+φ(y, u)+φ(u, y), y+u)L−1α =
(a + c + F (u, x) + φ(u, x), x + u)L−1α ∗ (a + d + F (u, y) + φ(u, y), y + u)L−1
α =(c + F (u, x), x) ∗ (d + F (u, y), y)L−1
α = (c + d + F (u, x) + F (u, y) + φ(x, y), x + y).
L(φ) is an Aµ-loop ⇔ L. H. S.=R. H. S. ⇔ F (u, x + y) = F (u, x) + F (u, y).
Corollary 2.1 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. L(φ) is an A-loop if and only if the following identities are true for all codewords u, v, x, y in C.
1. A(u, v, x + y) = A(u, v, x) + A(u, v, y).
2. A(x + y, u, v) = A(x, u, v) + A(y, u, v).
3. F (u, x + y) = F (u, x) + F (u, y).
ProofThe loop L(φ) is an A-loop if and only if it is an Aλ-loop, an Aρ-loop and an Aµ-loop. Thus,by Theorem 2.1, the claim follows.
4
T. Jaiyeola 48
Lemma 2.1 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. If L(φ) is an A-loop, then the following are true in L(φ).
(a) A(u, v, x + y) = A(u, v, x) + A(u, v, y).
(b) A(x + y, u, v) = A(x, u, v) + A(y, u, v).
(c) F (u, x + y) = F (u, x) + F (u, y).
(d) c(u, v, x + y) ≡ [c(u, v, x) + c(u, v, y)] mod 2.
(e) c(x + y, u, v) ≡ [c(x, u, v) + c(y, u, v)] mod 2.
(f) u·x+u·y2
≡ u·(x+y)2
mod 2.
ProofBy (3) and (4), it can be deduced that ; F (u, v) ≡ u·v
2mod 2 and A(u, v, w) ≡ c(u, v, w) mod
2 ∀ u, v, w ∈ C. (a), (b) and (c) follow from Corollary 2.1.
(d)A(u, v, x + y) ≡ c(u, v, x + y) mod 2 (5)
A(u, v, x) ≡ c(u, v, x) mod 2 (6)
A(u, v, y) ≡ c(u, v, y) mod 2 (7)
Adding (5), (6) and (7) and using (a), c(u, v, x + y) ≡ [c(u, v, x) + c(u, v, y)] mod 2.
(e)A(x + y, u, v) ≡ c(x + y, u, v) mod 2 (8)
A(x, u, v) ≡ c(x, u, v) mod 2 (9)
A(y, u, v) ≡ c(y, u, v) mod 2 (10)
Adding (8), (9) and (10) and using (b), c(x + y, u, v) ≡ [c(x, u, v) + c(y, u, v)] mod 2.
(f)
F (u, x + y) ≡ u · (x + y)
2mod 2 (11)
F (u, x) ≡ u · x2
mod 2 (12)
F (u, y) ≡ u · y2
mod 2 (13)
Adding (11), (12) and (13) and using (c), u·x+u·y2
≡ u·(x+y)2
mod 2.
Remark 2.1 As deduced in [14], u · v = ||u||+|v||−||u+v||2
.
5
Code Loops and Some Other Loops 49
3 Homogeneous Loops, Kikkawa Loops, K-Loops,
Cross Inverse Property Loops and Code Loops
Lemma 3.1 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. The following are true.
1. L(φ) is a left homogeneous loop ⇔ A(u, v, x + y) = A(u, v, x) + A(u, v, y).
2. L(φ) is a right homogeneous loop ⇔ A(x + y, u, v) = A(x, u, v) + A(y, u, v).
3. L(φ) is a homogeneous loop ⇔ A(u, v, x + y) = A(u, v, x) + A(u, v, y), A(x + y, u, v) =A(x, u, v) + A(y, u, v) and F (u, x + y) = F (u, x) + F (u, y).
ProofL(φ) is a LIPL and RIPL. The proof of 1. and 2. follow from Theorem 2.1 while 3. followsfrom Corollary 2.1.
Lemma 3.2 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. L(φ) is an AIPL if and only if φ(u + v, u + v) = φ(u, u) + φ(v, v) ∀ u, v ∈ C.
ProofLet x = (a, u), y = (b, v) ∈ L(φ). L(φ) is AIPL if and only if (xy)−1 = x−1y−1 ⇔ (a +b + φ(u, v), u + v)−1 = (a, u)−1 ∗ (b, v)−1 ⇔ (a + b + φ(u, v) + φ(u + v, u + v), u + v) =(a+φ(u, u), u)∗ (b+φ(v, v), v) ⇔ (a+ b+φ(u, v)+φ(u+v, u+v), u+v) = (a+ b+φ(u, u)+φ(v, v) + φ(u, v), u + v) ⇔ φ(u + v, u + v) = φ(u, u) + φ(v, v) ∀ u, v ∈ C.
Lemma 3.3 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. The following are true.
1. L(φ) is a left Kikkawa loop⇔ A(u, v, x+y) = A(u, v, x)+A(u, v, y) and φ(u+v, u+v) =φ(u, u) + φ(v, v).
2. L(φ) is a right Kikkawa loop ⇔ A(x+y, u, v) = A(x, u, v)+A(y, u, v) and φ(u+ v, u+v) = φ(u, u) + φ(v, v).
3. L(φ) is a Kikkawa loop ⇔ A(u, v, x + y) = A(u, v, x) + A(u, v, y), A(x + y, u, v) =A(x, u, v) + A(y, u, v), F (u, x + y) = F (u, x) + F (u, y) and φ(u + v, u + v) = φ(u, u) +φ(v, v).
ProofL(φ) is a IPL The proof follows from Theorem 2.1, Corollary 2.1 and Lemma 3.2.
Lemma 3.4 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. The following are true.
1. L(φ) is a left K-loop ⇔ φ(u + v, u + v) = φ(u, u) + φ(v, v).
6
T. Jaiyeola 50
2. L(φ) is a right K-loop ⇔ φ(u + v, u + v) = φ(u, u) + φ(v, v).
3. L(φ) is a K-loop ⇔ φ(u + v, u + v) = φ(u, u) + φ(v, v).
ProofL(φ) is a Moufang loop. The rest of the proof follow from Lemma 3.2.
Theorem 3.1 Let L(φ) be a code loop on a binary linear code C that is doubly even withfactor set φ. L(φ) is a CIPL if and only φ(u, v) = φ(v, u).
ProofL(φ) is a CIPL if and only if x ·yx−1 = y ∀ x, y ∈ L(φ) ⇔ (a, u)∗ [(b, v)∗ (a, u)−1] = (b, v) ⇔(b+φ(u, u)+φ(v, u)+φ(u, u+ v), v) = (b, v) ⇔ φ(u, v +u) = φ(u, u)+φ(v, u) = φ(u+ v, u).
In L(φ), φ(u, v + u) = φ(u + v, u) ⇔ φ(u, v) = φ(v, u). Thence, the proof is complete.
Remark 3.1 In fact, a code loop is a cross inverse property loop if and only if it is com-mutative : L(φ) is commutative if and only if xy = yx ∀ x, y ∈ L(φ) ⇔ (a, u) ∗ (b, v) =(b, v) ∗ (a, u) ⇔ (a + b + φ(u, v), u + v) = (a + b + φ(v, u), u + v) ⇔ φ(u, v) = φ(v, u).
References
[1] R. H. Bruck (1966), A survey of binary systems, Springer-Verlag, Berlin-Gottingen-Heidelberg, 185pp.
[2] O. Chein, H. O. Pflugfelder and J. D. H. Smith (1990), Quasigroups and Loops : Theoryand Applications, Heldermann Verlag, 568pp.
[3] J. Dene and A. D. Keedwell (1974), Latin squares and their applications, the EnglishUniversity press Lts, 549pp.
[4] A. Drapal (2000), A-loops close to code loops are groups, Comment. Math. Carolinae41,2, 245–249.
[5] O. Chein and E.G. Goodaire (1990), Moufang loops with a unique non-identity commu-tator (associator, square), J. Alg. 130, 369–384.
[6] O. Chein and E.G. Goodaire (1990), Code loops are RA2 loops, J. Alg. 130, 385–387.
[7] E. G. Goodaire, E. Jespers and C. P. Milies (1996), Alternative Loop Rings, NHMS(184),Elsevier, 387pp.
[8] R. L. Jr. Griess (1986), Code loops, J. Alg. 100, 224–234.
[9] M. K. Kinyon, K. Kunen, J.D. Phillips (2001), Every Diassociative A-Loop is Moufang,Proc. Amer. Math. Soc. 130,3, 619–624.
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[10] P. Vojtechovsky, Derived forms and binary linear codes, Mathematics Report NumberM99-10, Deparment of mathematics, Iowa State University.
[11] P. Vojtechovsky (2000), Combinatorial aspects of code loops, Comm. Math. Uni. Car-olinae 41,2, 429–435.
[12] P. Vojtechovsky, Combinatorial polarization, code loops and codes of high level, pro-ceedings of CombinaTexas 2003, published in International Journal of Mathematicsand Mathematical Sciences 29 (2004), 1533–1541
[13] H. O. Pflugfelder (1990), Quasigroups and Loops : Introduction, Sigma series in PureMath. 7, Heldermann Verlag, Berlin, 147pp.
[14] P. Vojtechovsky (2003), Connections between codes, groups and loops, Ph.D thesis,Charles University Czech Republic.
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