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An approach to the isotheory by means of extended pseudoisotopisms on 1 , Juan N nez 2 Ra ul Falc u rafalgan@us.es 1 Department of Applied Mathematics I. University of Seville. 2 Department of Geometry and Topology. University of
Ra´ ul Falc´
u˜ nez2 rafalgan@us.es
1Department of Applied Mathematics I. University of Seville. 2Department of Geometry and Topology. University of Seville.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
1 Preliminaries. 2 Extended isotopisms. 3 Extended pseudoisotopisms. Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
1 Preliminaries. 2 Extended isotopisms. 3 Extended pseudoisotopisms. Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
1942: Algebra isotopism (Albert). 1944: Quasigroup isotopism (Bruck). 1948: Lie-admissible algebras (Albert). 1967: Lie-Santilli admissibility (Santilli). 1978: Lie-Santilli isotheory (Santilli). 1983: Metric isospace (Santilli). 1992: Isoanalisis (Kadeisvili). 1993: Isotopology, isomanifold. (Tsagas-Sourlas). 1996: Isocalculus (Santilli). 2002: Isotopology (Tsagas-Sourlas-Santilli-Falc´
u˜ nez). 2005: Isomanifolds based on generalized isotopisms (Falc´
Non-injective isoalgebras (Falc´
u˜ nez). 2006: Santilli isotopisms of partial Latin squares (Falc´
u˜ nez). 2007: Extended isotopisms of partial Latin squares (Falc´
2015: Santilli autotopisms of partial groups (Falc´
u˜ nez).
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Abraham Adrian Albert 1905-1972
Two algebras (A1, ·) and (A2, ◦) are isotopic if there exist three regular linear transformations α, β and γ from A1 to A2 such that α(x) ◦ β(y) = γ(x · y), for all x, y ∈ A1. The algebra A2 is said to be isotopic to A1. The triple (α, β, γ) is said to be an isotopism between both algebras A1 and A2. If α = β = γ, then this is an isomorphism.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
A quasigroup [Haussmann and Ore, 1937] is a nonempty set Q endowed with a product ·, such that if any two of the three symbols u, v and w in the equation u · v = w are given as elements of Q, then the third is uniquely determined as an element of Q.
Richard Hubert Bruck 1914-1991
Two quasigroups (Q1, ·) and (Q2, ◦) are isotopic if there exist three bijections α, β and γ from Q1 to Q2 such that α(x) ◦ β(y) = γ(x · y), for all x, y ∈ Q1. A quasigroup with identity element is a loop. A principal loop isotopism between (Q1, ·) and (Q2, ◦) is an isotopism of the form x ◦ y = (x · u) · (v · y), where u, v ∈ Q1. (Q2, ◦) is a loop with unit element I = u · v. If (Q1, ·) is a group, then x ◦ y = x · T · y, where T = I −1 ∈ Q1.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Marius Sophus Lie 1842-1899
A Lie algebra is an anticommutative algebra A that satisfies the Jacobi identity J(x, y, z) = (xy)z + (yz)x + (zx)y = 0, ∀ x, y, z ∈ A. Commutator product: [x, y] = xy − yx. Minus algebra: A → (A−, [., .]).
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Abraham Adrian Albert 1905-1972
An algebra A over a field F is said to be Lie-admissible if the related minus algebra A− is a Lie algebra. Associative algebras. Lie algebras with product xy = x · y − y · x. Quasi-associative algebras (λ-mutations): ch(F) = 2 and λ ∈ F ⇒ A(λ) with product (x, y)λ = λ · xy + (1 − λ) · yx. [x, y]λ = (2λ − 1) · [x, y]. A is Lie-admissible ⇔ A(λ) is Lie-admissible for all λ.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
(λ, µ)-mutation of an algebra A: (x, y)λ,µ = λ · xy + µ · yx, where λ, µ ∈ F \ {0} and λ = µ. Hence, [x, y]λ,µ = (λ − µ) · [x, y] Lie-admissible algebras → Physics. Classical level for non-conservative systems: In pseudo-Hamiltonian Mechanics [Duffin, 1962] [x, y]λ,µ =
n
∂qi ∂y ∂pi + µ ∂x ∂pi ∂y ∂qi
Quantum-mechanical level for elementary-particle interacting or decaying regions:
Fixed parameters: (Non) observability of quarks. Variable parameters: Indetermination principle.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
Operator-deformations of an algebra A: (x, y)P,Q = xPy − yQx, where P, Q ∈ A. P = Q ⇒ The algebra with product x ×Q y = xQy is Lie-admissible. Classical mechanics (modified Hamilton’s equations): (x, H) =
2n
∂x ∂αi Sij(t, α) ∂H ∂αj = dx(α) dt , where α is a local chart in a manifold, H is a Hamiltonian and Sij is a non-singular C ∞-tensor in a region.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
Operator-deformations of an algebra A: (x, y)P,Q = xPy − yQx, where P, Q ∈ A. P = ±Q ⇒ The algebra with product (x, y)P,Q is also Lie-admissible. [x, y]P,Q = x(P + Q)y − y(P + Q)x. Hadronic mechanics (generalization of Heisenberg equation): (x, H) = xPH − HQx, where P and Q are non-Hermitian and non-singular operators in a physical system that represent non-self-adjoint forces. It makes possible to consider extended particles that admits additional contact, non-potential and non-Hamiltonian interactions.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
Operator-deformations of an algebra A: (x, y)P,Q = xPy − yQx, where P, Q ∈ A. P = ±Q ⇒ The algebra with product (x, y)P,Q is also Lie-admissible. [x, y]P,Q = x(P + Q)y − y(P + Q)x. Question: How the product in an associative algebra can be modified to yield as general as possible a Lie-admissible algebra? Study of isotopisms that preserve axioms of Lie-admissibility.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
Operator-deformations of an algebra A: (x, y)P,Q = xPy − yQx, where P, Q ∈ A. P = ±Q ⇒ The algebra with product (x, y)P,Q is also Lie-admissible. [x, y]P,Q = xTy − yTx. Question: How the product in an associative algebra can be modified to yield as general as possible a Lie-admissible algebra? Study of isotopisms that preserve axioms of Lie-admissibility.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
A ≡ Associative algebra. T ≡ Isotopic element. In the more general version, T can be taken outside the algebra A. In quantum mechanics, T depends on the state space determined by a set S of parameters: x, v, t, δ, . . .. T : S → A \ {0}. Isoproduct:
(x, y, s) → x ×sy = xT(s)y. Lie-isotopic product: ˆ [., .ˆ ] : A × A × S → A (x, y, s) → ˆ [x, yˆ ]s = xT(s)y − yT(s)x = x ×sy − y ×sx.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
A ≡ Associative algebra. S ≡ State space. T(s) ≡ Isotopic element. x ×sy = xT(s)y. ˆ [x, yˆ ]s = x ×sy − y ×sx. Lie-Santilli isotheory: Step-by-step axiom-preserving construction
isoproduct ×s: Numbers, spaces, algebras, groups, symmetries, ... In Quantum mechanics, deformation of associative algebras are based on non-Hamiltonian effects, characteristics and interactions. Question: How to represent non-Hamiltonian terms in an invariant way?
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
A ≡ Associative algebra. S ≡ State space. T(s) ≡ Isotopic element. x ×sy = xT(s)y. ˆ [x, yˆ ]s = x ×sy − y ×sx. Santilli proposes to represent non-Hamiltonian terms by means of a generalization of the basic unit I of the structure in which is embedded T. I → I(s) = T −1(s). Hence, x ×s I(s) = I(s) ×sx = x, ∀x ∈ A, s ∈ S. All the underlying structures must admit I as their unit (isounit) → Isomathematics.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
A ≡ Associative algebra. S ≡ State space. T(s) ≡ Isotopic element.
x ×sy = xT(s)y. ˆ [x, yˆ ]s = x ×sy − y ×sx. Isoelements of the system: (x, s) ∈ A × S → xs = x I(s). Hence,
×s ys = (x I(s))T(s)(y I(s)) = x( I(s)T(s))(y I(s)) = x(y I(s)) = (xy) I(s) = xy s.
[ xs ,s ys] = ˆ [x, y]s. If x = y, then xs = ys, for all s ∈ S.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
A ≡ Associative algebra. S ≡ State space. T(s) ≡ Isotopic element.
ˆ xx ×s ˆ ys = ˆ xy s. ˆ [ˆ xs, ˆ ysˆ ]s = ˆ [x, y]s. Let s ∈ S. We define ˆ As = {ˆ xs | x ∈ A} αs : A → ˆ As such that αs(x) = ˆ xs, ∀x ∈ A. Hence, [ xs ,s ys] = ˆ [x, y]s ⇒ [αs(x) ,sαs(y)] = αs([x, y]). The family FS = {(αs, αs, αs): s ∈ S} constitutes a local state isomorphism of algebras on the underlying dynamical system. Similar families are defined for any other isostructure based on isoelements.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
A ≡ Associative algebra. S ≡ State space. T(s) ≡ Isotopic element.
ˆ xx ×s ˆ ys = ˆ xy s. ˆ [ˆ xs, ˆ ysˆ ]s = ˆ [x, y]s. This opens two possibilities for further research in isomathematics:
1 To develop a comprehensive study of isotheory and Hadronic
mechanics by means of the theory of local state isomorphisms [Sussmann, 1977].
2 To generalize the isotheory to a non-isomorphic frame closer to the
classical theory of isotopisms.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ruggero Maria Santilli Born in 1935
A ≡ Associative algebra. S ≡ State space. T(s) ≡ Isotopic element.
ˆ xx ×s ˆ ys = ˆ xy s. ˆ [ˆ xs, ˆ ysˆ ]s = ˆ [x, y]s. This opens two possibilities for further research in isomathematics:
1 To develop a comprehensive study of isotheory and Hadronic
mechanics by means of the theory of local state isomorphisms [Sussmann, 1977].
2 To generalize the isotheory to a non-isomorphic frame and
bring it closer to the classical theory of isotopisms.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
1 Preliminaries. 2 Extended isotopisms. 3 Extended pseudoisotopisms. Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
A Latin square L of order n (L ∈ LS(n)) is a n × n array with elements chosen from [n] = {1, 2, ..., n}, such that each symbol
Sn: Symmetric group on [n]. An isotopism of L is a triple Θ = (α, β, γ) ∈ S3
n, where α, β and γ
are respectively, permutations of rows, columns and symbols of L.
L = 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 Θ = ((1 2)(3 4), (2 3), Id) ⇒ LΘ = 2 4 3 1 1 3 2 4 4 2 1 3 3 1 4 2
An isotopism mapping L to itself is an autotopism. The stabilizer subgroup of L by the action of S3
n is its autotopism group:
A(L) = {Θ ∈ In : LΘ = L}.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
A partial Latin square P of order n (P ∈ PLS(n)) is a n × n array with elements chosen from a set of n symbols, such that each symbol occurs at most once in each row and in each column. It is the multiplication table of a partial quasigroup.
P = 1 − − 4 − 1 4 − 3 − 2 1 4 3 1 − ∈ PLS(4)
Isotopisms and autotopisms of partial Latin squares are defined similarly to those of Latin squares, although now π(∅) = ∅, for all π ∈ Sn.
L = 1 − − 4 − 3 4 − − − − − − 1 − 3 Θ = ((1 2)(3 4), (2 3), Id) ⇒ LΘ = − 4 3 − 1 − − 4 − − 1 3 − − − −
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
An isotopism Θ = (α, β, γ) between two partial quasigroups (P1, ·) and (P2, ◦) in PLS(n) is said to be a Santilli isotopism [Falc´
N´ u˜ nez, 2006] if there exist three elements iα, iβ, iγ ∈ P1 such that α(x) = x · iα, β(x) = x · iβ, γ(x) = x · iγ, ∀x ∈ P1. Example
P1 ≡ 1 3 2 − 2 4 1 − 3 − 4 1 4 − 3 2 → P2 ≡ 4 2 − 1 3 1 − 2 − 4 2 3 − 3 1 4 . Θ = ((12)(34), (12)(34), Id). α = β = (12)(34) → iα = iβ = 3. γ = Id → iγ = 1.
(x · 3) ◦ (y · 3) = (x · y) · 1, ∀x, y ∈ [4].
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
P ⊆ L ∈ LS(n), F ⊆ A(L). Extended isotopism: [Falc´
PF =
PΘ ∈ PLS(n). Example
P = 1 − − − − − − − − − − − − − − − Θ1 = (Id, (1 2 3 4), (1 4 3 2)) Θ2 = ((1 2 3 4), Id, (1 4 3 2)) ⇒ P{Θ1,Θ2} = − 4 − − 4 − − − − − − − − − − −
Isotopisms of F can be ordered by a set S of indices and then F = {Θs = (αs, βs, γs) | s ∈ S}.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Example
P = 1 − 1 − − − − − ⊆ L = 1 2 1 2 2 1 isotopic to L′ = 2 1 2 1 1 2 F = {Θ1 = ((012), (01), Id), Θ2 = ((012), (12), (021)), Θ3 = (Id, (02), Id)} PΘ1 = − 1 − − − − 1 − , PΘ2 = 2 − − − − − 1 − 2 , PΘ3 = − 1 − − 1 − − − . Hence, PF = 2 1 − − 1 1 2
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Example
P ≡ 1 2 − − 2 1 − − − − − − − − − − ⊆ L ≡ 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 → L′ ≡ 3 4 1 2 4 3 2 1 1 2 3 4 2 1 4 3 Id → iId = 1, π = ((13)(24)) → iπ = 3. F = {Θ1 = (Id, Id, π), Θ2 = (Id, π, Id), Θ3 = (π, Id, Id), Θ4 = (π, π, π)}. PΘ1 ≡ 3 4 − − 4 3 − − − − − − − − − − , PΘ2 ≡ − − 1 2 − − 2 1 − − − − − − − − PΘ3 ≡ − − − − − − − − 1 2 − − 2 1 − − , PΘ4 ≡ − − − − − − − − − − 3 4 − − 4 3 PF = LF = L′.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
The previous concepts can be extended to any other algebraic structure. Example: (Algebras). Let (A1, ·) and (A2, ◦) be two algebras over the same field F. Let S be the state space of an underlying dynamical system. A family F = {Θs = (αs, βs, γs) | s ∈ S} of isotopisms from the algebra (A1, ·) to an algebra (A2, ◦) constitutes an extended isotopism for any subalgebra of A1. αs(x) ◦ βs(y) = γs(x · y), ∀x, y ∈ A1. If there exist an element iπj ∈ A1 such that πj(x) = x · iπj, for all π ∈ {α, β, γ} and j ∈ S, then the family F is a Santilli isotopism of algebras.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Example
For each t ∈ {1, 3}, let αt(x) = xt mod 4, ∀x ∈ F The family {(αt1, αt2, αt1t2
mod 4)} of isotopisms from (Z/4Z, ×) to itself
constitutes a Santilli extended isotopism from the subgroupoid ({0, 1, 2}, ×) to (Z/4Z, ×).
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Example Let t ∈ R and let us define the space vector map αt : R2 → R2 αt − − − → (x, y)
− − − − − → (x, y + t) = − − − → (x, y) + − − − → (0, t) The family F = {(αt1, αt2, αt1+t2) | t1, t2 ∈ R} of isotopisms from R2 to itself is a Santilli extended isotopism from R ∼ = R × {0} to R2. αt1(x)ˆ +αt2(y) = αt1+t2(x + y)
− − → (x, t1) + − − − → (y, t2) = − − − − − − − − − − → (x + y, t1 + t2)
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Example Let us consider the polynomial ring F2[x] over the finite field F2. Let us define the maps α0 and α1 from F2[x] to itself, so that αt(p) = p + t, for all polynomial p ∈ F2[x] and t ∈ {0, 1}. The family F = {(αt1, αt2, α(t1+t2) (mod 2))}t1,t2∈{0,1} is a Santilli extended isotopism from (F2[x], +) to itself. p+q = αt1(p−t1)+αt2(q−t2) = α(t1+t2) (mod 2)(p+q−((t1+t2) (mod 2))), for all t1, t2 ∈ {0, 1}.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
1 Preliminaries. 2 Extended isotopisms. 3 Extended pseudoisotopisms. Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Extended isotopisms make possible to generalize the state isomorphism approach of the isotheory to a state isotopism approach. This approach can be generalized by removing the injectivity in the conditions of the isounit.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Let (G1, ·) and (G2, ◦) be two groupoids. A triple Θ = (α, β, γ) of onto maps from G1 to G2 is a pseudoisotopism if
1 γ(u · v) = γ(u′ · v ′), ∀u, v, u′, v ′ ∈ G1 such that α(u) =
α(u′) and β(v) = β(v ′).
2 α(u) ◦ β(v) = γ(u · v), for all u, v ∈ G1.
α, β and γ injective ⇒ Θ is an isotopism. α = β = γ ⇒ Θ is a pseudoisomorphism. If there exist uα, uβ, uγ ∈ G1 such that π(u) = u · uπ, for all u ∈ G1 and π ∈ {α, β, γ}, then the triple Θ is a Santilli pseudoisotopism.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
If S is a set of indices (state space), then every family F = {(αs, βs, γs)}s∈S of pseudoisotopisms from (G1, ·) to (G2, ◦) constitutes an extended pseudoisotopism for any subgroupoid of G1. It is a Santilli extended pseudoisotopism if each triple of the family F is a Santilli pseudoisotopism.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Example F = C, G =< i >= {1, −1, i, −i}, H = {1, −1} ⊆ G, α : G → H, so that
α(i) = α(−i) = −1 (α, α, α) is a pseudoisomorphism from (G, ·) to (H, ·) that preserves the structure of group.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Example F = (R, +, ×), U ≡ {Differentiable real functions of one real variable}, α : U → R so that α(f ) = f ′(2), ∀f ∈ U. Θ1 = (α, α, α) is a pseudoisomorphism from (U, +) to (R, +) because a + b =
{α(f + g) : f ′(2) = a, g ′(2) = b} . Θ2 = (Id, α, α) is a pseudoisomorphism from (U, ·) to (R, ×) where a × b =
{α(a · f ) : f ′(2) = b} . The pair (Θ1, Θ2) constitutes therefore a pseudoisotopism from (U, +, ·) to (R, +, ×).
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
f : R → R : x → 15x2; g : R → R : x → 5x3 α(f ) = α(g) = 60
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
1 To develop a comprehensive study of isotheory and Hadronic
mechanics by means of the theory of local state isomorphisms [Sussmann, 1977].
2 To study the local state (pseudo)isotopism frame for distinct
algebraic structures in which isotheory can be embedded.
3 To develop examples and applications in Hadronic Mechanics that
incorporate the local state (pseudo)isotopism frame.
4 By focusing on the theory of Latin squares:
How many Santilli autotopisms of a given partial Latin square there exist? Which is the smallest size of a partial Latin square that can be embedded in a Latin square by means of a Santilli extended isotopism? Application in Cryptography.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
19–52, 1944.
Notes Discrete Math. 29: 503–507, 2007.
u˜
Nonlinear Funct. Anal. & Appl. 10(5): 843–863, 2005.
u˜
Hadronic J. 29(3): 285–298, 2006.
u˜
autotopism groups. J. Dy. Sy. & Geom. Th. 5(1): 19–32, 2007.
u˜
4(5): 47–51, 2015.
1937.
51: 570–576, 1967.
Mechanics for nonconservative and Galilei form-nonivariant systems. Hadronic J. 1: 223-423, 1978.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms
Ra´ ul Falc´
u˜ nez2 rafalgan@us.es
1Department of Applied Mathematics I. University of Seville. 2Department of Geometry and Topology. University of Seville.
Ra´ ul Falc´
u˜ nez An approach to the isotheory by means of extended pseudoisotopisms