# A Toy Model for Torsorial Nature of Representations

## Abstract

**:**

## 1. Introduction

## 2. Representation as Janus-Faced Closure

a representation belongs neither to the realm of matter, of outside objects, of things-in-themselves, nor to the realm of pure mind or Platonic Ideas: it constitutes an interface, it stands in between Mind and Nature, in between subject and object, in between Self and World. (Emphasis added by the author of this paper)

#### 2.1. Ontology of Representation

- – Representation
- p/o "form":Representational form
- p/o "content":Proposition

- – Representational form
- – Symbol sequence
- – Natural language
- – Spoken language
- – Written language

- – Artificial language
- – Musical symbol
- – Mathematical symbol, etc.

- – Image
- – Speech

- – Proposition
- – Design Proposition
- – Procedure
- – Music
- – Symbol
- – Specification

- – Product Proposition
- – Novel
- – Poem
- – Painting

- – Represented thing
- – p/o "representation":Representation
- – p/o "medium":Representational medium

- – Representation medium
- – Physical thing
- – paper
- – canvas

- – Electronic thing
- – CD-ROM
- – etc.

#### 2.2. Closure

**Definition.**Closure is an operation C on sets, C : A → A

^{∗}, with the following properties:

- A ⊆ A
^{∗}(monotonicity) - (A
^{∗})^{∗}= A^{∗}(idempotence) - A ⊆ B ⇒ A
^{∗}⊆ B^{∗}(inclusion preservation)

^{∗}= A. Intuitively, such a closure of a set means that somehow “missing elements” are added to it, until no more of them are needed.

**Definition.**A Chu space (X, S, R) consists of two sets X, S and a relation R between them. If an element x of X has the relation R with an element s of S, we write

^{∗}, then Y is said to be closed. Similarly, if T = T

^{∗}, then T is also said to be closed. The closed experience set Y can be regarded as a stable articulation of experiences, and the closed property set can be regarded as stable composite idea.

## 3. Torsor

**Definition.**A torsor (G, (·, ·, ·)) is a set G together with a ternary operation G× G× G → G;(x, y, z) → [xyz] satisfying the identities: for all x, y, z, u, v ∈ G

^{-1}h (here juxtaposition denotes composition of functions). This torsor becomes a group once a particular isomorphism by which the two objects are to be identified is chosen.

**Integers**If x, y, z are integers, we can set [xyz] = x − y + z to produce a torsor.

**Miscellaneous**Energies, voltages, indefinite integrals, and many others are torsors (see [11]).

**Proposition 1.**[xy[zuv]] = [ x[uzy] v]

## 4. Toy Model-Interpretation of Torsor

#### 4.1. Interpretations of the Elements and Operation

#### 4.2. Interpretation of the Identity Law

#### 4.3. Interpretation of the Para-associative Law

_{1 }and X

_{2 }are the substrata again, and Y is also the substratum. X

_{1 }and X

_{2 }yield representations, but Y yields views. The view c is Y in the view of b, and the view [abc] is Y in the view of a. The point here is the representation [cde] is yielded in the view of c and in the view of b simultaneously. Starting from that point, we will trace how this diagram is built. From the transcendental standpoint, it is easy to understand.

_{2 }that yields both of [ab[cde]] and [cde].

_{1 }other than X

_{2}. So, more precisely, the situation is: for given (a, )b, c, d, e ∈ H,

_{1}. The diagram is completed.

#### 4.4. Role of the Closure

- the representation as a closure formed by two constituents
- the above mentioned yielding process

#### 4.5. Summary: How to Use the Toy Model

- Identify representations to examine, and assume that they satisfy the axioms of the torsor;
- identify the two constituents of the representations which form a closure;
- fix the meaning of the equality in the algebra;
- compare algebraic consequences with the reality of the phenomena.

## 5. Examples

#### 5.1. Substitution Cipher-Group Operation

^{-1}z [12]. However, this case is explicitly written down here to clarify what we do.

_{5 }of five elements.

_{1}a

_{2}a

_{3}a

_{4}a

_{5}. As for [cde], because e is yielded in the view of d, the substratum (which can be explicitly represented in this case) is

^{-1}14523. Then

#### 5.2. Multiple Views for the Real World

#### 5.2.1. Known Facts on FCA

**Definition 1.**A formal context = (G, M, I) consists of two sets G and M and a relation I between G and M. The elements of G are called the formal objects and the element of M are called the formal attributes of the context. When an object g is in the relation I with an attribute m, we write g ⊨ m or (g, m) ∈ I and read it as “the object ghas the attribute m”.

**Definition 2.**For a set A ⊆ G of objects we defi ne

**Definition 3.**A formal concept of the context (G, M, I) is a pair (A, B) with A ⊆ G, B ⊆ M, µ( A) = B and γ( B) = A. We call A the extent and B the intent of the concept (A, B). (G, M, I) denotes the set of all concepts of the context (G, M, I).

**Definition 4.**If (G, M, I) is a formal context and if H ⊆ G and N ⊆ M, then (H, N, I ∩ H × N) is called a subcontext of (G, M, I).

**Definition 5.**A subcontext (H, N, I ∩ H × N) is called compatible if the pair ( A ∩ H, B ∩ N) is a concept of the subcontext for every concept (A, B) ∈ B(G, M, I).

**Definition 6.**If (G, M, I) is a context, g ∈ G an object, and m ∈ M an attribute, we write

**Definition 7.**A subcontext (H, N, I ∩ H × N) of a clarified context (G, M, I) is arrow-closed if the following holds: h ↗ m and h ∈ H together imply m ∈ N, and g ↙ n and n ∈ N together imply g ∈ H.

**Proposition 2.**Every compatible subcontext is arrow-closed. Every arrow-closed subcontext of a finite context is compatible.

#### 5.2.2. A Model

_{0 }of the formal object G. Similarly, an operation : M → G × M is defined as extracting the minimum arrow-closed subcontext containing any specified subset N

_{0 }of the formal attribute M. These operations, combined with projection, are defined for later convenience:

_{H}

_{0 }generated with any specified subset H

_{0 }of the formal object G is defined as

_{N}

_{0 }generated with any specified subset N

_{0 }of the formal attribute M is defined as

_{H }◦ π

_{H }= π

_{H}, ρ

_{N>}◦ ρ

_{N }= ρ

_{N }, hence each of them is a kind of projection.

_{0 }of M. But as a view, it works as π

_{H}

_{0 }or ρ

_{N}

_{0 }, that is, it induces an arrow-closed subcontext including the associated subset.

#### 5.2.3. A Concrete Example

_{1},g

_{2},g

_{3},g

_{4}}, M = {m

_{1}, m

_{2}, m

_{3}, m

_{4}, m

_{5}} and I is defined by the Table 1 (Figure 3.2 of [9]).

m_{1} | m_{2} | m_{3} | m_{4} | m_{5} | |
---|---|---|---|---|---|

g_{1} | x | x | |||

g_{2} | x | x | x | x | |

g_{3} | x | x | |||

g_{4} | x | x | x |

_{2 }to g

_{1 }and m

_{1 }to g

_{3 }are unidirectional, and the others are bidirectional. Thus if we start with a subset {g

_{1}}, the arrow-closed subset induced by the subset is ({g

_{1}}, {m

_{5}},I ∩{g

_{1}}×{m

_{5}}).

Element | a | b | c | d | e |
---|---|---|---|---|---|

H_{0} | {g_{4}} | {g_{1},g_{2},g_{3}} | - | {g_{1},g_{2}} | |

N_{0} | - | - | {m_{3},m_{5}} | - | |

H | {g_{1},g_{3},g_{4}} | {g_{1}g_{2}g_{3}} | {g_{1}g_{3}} | {g_{1}g_{2}} | Obtained from d |

N | {m_{1},m_{2}m_{3},m_{5}} | {m_{3},m_{4}m_{5}} | {m_{3},m_{5}} | {m_{4},m_{5}} | Obtained from d |

_{1},g

_{2}} to C. Asa result, e is a formal concept lattice of subcontext:

_{1},g

_{2}} and {g

_{2}} and ∅. See Figure 5d.) Recall that there is a canonical correspondence between the view and the formal context and the set of formal concepts.

_{1}}, {g

_{3}}, {g

_{4}}, {g

_{1},g

_{4}}, {g

_{3},g

_{4}}, {g

_{1},g

_{3},g

_{4}}} (see Figure 5a). This corresponds to [ab[cde]].

#### 5.3. Letter Recognition

^{'}E] in the right hand of the equation. It must be considered to be a view for what is a letter “rho” in the view of G. This term is interpreted as the representation of a substratum that yields the representation E in the view of E

^{'}, and the representation [RE

^{'}E] is obtained in the view of R. Extremely roughly speaking, since the view [RE

^{'}E] is something obtained in the view of R,a Cyrillic letter-related view, [RE' E] might be also a Cyrillic letter-related view, and hence [[RE' E] G(rho)] = (er). But a closer examination is required.

## 6. Conclusions

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Yoshitake, M.
A Toy Model for Torsorial Nature of Representations. *Information* **2012**, *3*, 546-566.
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**AMA Style**

Yoshitake M.
A Toy Model for Torsorial Nature of Representations. *Information*. 2012; 3(4):546-566.
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Yoshitake, Makoto.
2012. "A Toy Model for Torsorial Nature of Representations" *Information* 3, no. 4: 546-566.
https://doi.org/10.3390/info3040546