# A General Principle of Isomorphism: Determining Inverses

## Abstract

**:**

## 1. Introduction

## 2. Application of the Realization Theorem in Finding the Inverses

**Theorem**

**1.**

^{−1}and h

^{−1}in this diagram. We define these inverses for two cases:

- 1)
- In the first case, all maps f, g, and h and the composition f = hg in the commutative diagram in Figure 1 are defined initially. In many problems of system theory there is a need to determine the inverses of g
^{−1}and h^{−1}explicitly; - 2)
- In the second case, the commutative diagram is converted to the form shown in Figure 2. Here, only the isomorphic map f, some map designated as g
^{−1}, and their composition fg^{−1}are initially known. It is clear that under these conditions the map h commuting the specified composition exists, is uniquely and immediately determined by the formula fg^{−1}= h. In this case, the mapping of h^{−1}is unknown and needs to be determined. The inverse of g^{−1}, which also needs to be determined, is also unknown. However, maps g and h^{−1}, inverses to g^{−1}and h, respectively, can be defined only up to class, that is, they are not unique. This is because, based on the requirements of the realization theorem [3], in this case a pair of mappings (h, g) and their composition hg = f, which implement the isomorphism of f, or a pair of mappings (g^{−1}, h^{−1}) and their composition g^{−1}h^{−1}= f^{−1}, which implement the isomorphism f^{−1}, are not determined initially and at the same time. Thus, in this second case it is required to find classes of admissible maps h^{−1}and g that satisfy the condition fg^{−1}= h and other conditions of the realization theorem, which will be given below. It is obvious that the required classes of maps h^{−1}and g will be strictly interconnected taking into account these conditions. Once you have defined the classes of valid mappings, you can select one related mappings instance from the corresponding classes, if necessary, and commit those instances to the commutative diagram. Only in this case the only inverses to these instances of mappings will be fixed and can be calculated. Clearly, after fixing specific instances of mappings from valid classes, this case is no different from the first case where all mappings in a commutative diagram are known.

^{−1}, and their composition h

^{−1}f = g are known and defined in advance. The problem is to define maps h and g

^{−1}. By analogy with Figure 2, maps of h and g

^{−1}in Figure 3 can be defined up to class. Due to the equivalence of the problem statements illustrated in Figure 2 and Figure 3, in the future we will solve only the problem reflected in the commutative diagram in Figure 2.

^{−1}and h

^{−1}in the commutative diagram exist. To find them is required. It is clear that the inverses of g

^{−1}and h

^{−1}, based on the conditions of the theorem and the equations obtained on its basis, must be completely determined by the original maps f, g, and h. We try to express the inverses of g

^{−1}and h

^{−1}through the original known maps and thus obtain rules for their computation.

_{x}

^{right}= f

^{−1}f, e

_{y}

^{left}= ff

^{−1}, e

_{x}

^{left}= g

^{−1}g, e

_{y}

^{right}= hh

^{−1}, f

^{−1}= e

_{x}f

^{−1}, f

^{−1}= f

^{−1}e

_{y},

e

_{x}= e

_{x}

^{right}= e

_{x}

^{left}= e

_{y}

^{right}= e

_{y}

^{left}= e

_{y}= e.

_{x}

^{right}, e

_{x}

^{left}are the right and left units on X; e

_{y}

^{right}, e

_{y}

^{left}are the right and left units on Y. By virtue of the uniqueness of the map f

^{−1}(since by the condition f is the usual isomorphism), the maps e

_{x}and e

_{y}are also unique and equal to e, where e is the usual unit map such that e = ee = ee

^{−1}= e

^{−1}e= e

^{-1}. From the auxiliary diagrams constructed on the basis of the initial diagram and given in Figure 6, Figure 7, Figure 8 and Figure 9, the equalities follow

^{−1}= e

_{z}h

^{−1}, h

^{−1}= h

^{−1}e

_{y}, g

^{−1}= g

^{−1}e

_{z}, g

^{−1}= e

_{x}g

^{−1},

_{z}is some unit on Z. From the auxiliary diagrams shown in Figure 10b, it is also possible to obtain equality

^{−1}= g

^{−1}h

^{−1}, e

_{z}

^{left}= gg

^{−1}, e

_{z}

^{right}= h

^{−1}h,

_{z}

^{left}, e

_{z}

^{right}are left and right units on Z.

_{z}

^{left}= gg

^{−1}, e

_{z}

^{right}= h

^{−1}h, e

_{z}

^{left}= e

_{z}

^{right}= e

_{z}, e

_{z}

^{left}e

_{z}

^{right}= e

_{z}, e

_{z}

^{right}e

_{z}

^{left}= e

_{z}, (e

_{z})

^{−1}= e

_{z}, e

_{z}e

_{z}= e

_{z}.

_{z}

^{left}e

_{z}

^{right}= e

_{z}

^{right}e

_{z}

^{left},

^{−1}h

^{−1}h = h

^{−1}hgg

^{−1}.

^{−1}= g

^{−1}h

^{−1}and f = hg, we obtain

^{−1}h = h

^{−1}f g

^{−1}.

^{−1}and each of the parts of this equality. “Multiply” the equation by g

^{−1}on the left side, we get

^{−1}g f

^{−1}h = g

^{−1}h

^{−1}f g

^{−1}.

^{−1}h

^{−1}= f

^{−1}, the right part of equality (3) gets the form

^{−1}f g

^{−1}= 〈e

_{x}

^{right}= f

^{−1}f〉 = e

_{x}

^{right}g

^{−1}= 〈e

_{x}

^{right}= e

_{x}

^{left}= e〉= g

^{−1}.

_{x}

^{left}= g

^{−1}g we write the left side of the equality (3) in the form

_{x}

^{left}f

^{−1}h = 〈e

_{x}

^{left}f

^{−1}= f

^{−1}〉= f

^{−1}h = g

^{−1}, ⇒ f

^{−1}h = g

^{−1}.

^{−1}= f

^{−1}h.

^{−1}can be easily computed, since f is also a known ordinary isomorphic map for which the inverse of f

^{−1}is known to exist. Similarly, it can be shown that

^{−1}= g f

^{−1}.

^{−1}and h

^{−1}can be quite simply calculated from the known map included in the original commutative diagram. Making sure that the g

^{−1}and h

^{−1}mappings are indeed inverses to the corresponding original mappings in the commutative diagram shown in Figure 1, it is possible by their direct substitution in equality (2), the validity of which must be ensured in accordance with the conditions of the realization theorem [3]. Indeed

_{z}

^{left}= g g

^{−1}= 〈g

^{−1}= f

^{−1}h〉 = g f

^{−1}h and e

_{z}

^{right}= h

^{−1}h = 〈h

^{−1}= g f

^{−1}〉 = g f

^{−1}h ⇒ e

_{z}

^{left}= e

_{z}

^{right}= e

_{z}.

e

_{z}

^{left}e

_{z}

^{right}= gf

^{−1}hgf

^{−1}h = 〈hg = f〉 = gf

^{−1}f f

^{−1}h = 〈f

^{−1}f = e

_{x}

^{right}, ff

^{−1}= e

_{y}

^{left}〉 = gf

^{−1}e

_{y}

^{left}h =

= g e

_{x}

^{right}f

^{−1}h = 〈f

^{−1}e

_{y}

^{left}= f

^{−1}, e

_{x}

^{right}f

^{−1}= f

^{−1}〉 = g f

^{−1}h = e

_{z}.

^{−1}and h

^{−1}are true and can be applied inside the commutative diagram shown in Figure 1. For example, in the simple case where the original isomorphism f—is the usual unit map (in the case of matrices—the usual unit matrix), that is

^{−1}= e,

^{−1}and h

^{−1}are trivial and do not require computation, since in this case g

^{−1}= h and h

^{−1}= g are always true. This can be seen from the auxiliary diagram in Figure 10b, from which it follows that eh = h and ge = g.

^{−1}and h

^{−1}in the commutative diagram, partly repeating the proof of the realization theorem [3], and strictly derive Equations (4) and (5) for their calculation.

**Example**

**1.**

^{−1}= H and H

^{−1}= G, that is,

^{−1}and H

^{−1}in (7) are indeed the only inverses to matrices G and H in the commutative diagram, since they satisfy all conditions of (1) and (2).

**Example**

**2.**

^{−1}, and their composition FG

^{−1}= H. Moreover, H is unique, since maps F and G

^{−1}are completely (up to an instance) determined. However, maps G and H

^{−1}to be defined, which are inverse to G

^{−1}and H, respectively, are not unique in this case and can only be defined up to class precision. This, as mentioned above, follows from the realization theorem [3]: In this case, a pair of maps (H, G) and their composition HG = F, which implements the isomorphism F, or a pair of maps (G

^{−1}, H

^{−1}) and their composition G

^{−1}H

^{−1}= F

^{−1}, which implements the isomorphism F

^{−1}, are not defined initially and simultaneously.

^{−1}and G admissible under the condition FG

^{−1}= H and conditions in form (1), (2). After definition of classes, proceeding from additional considerations, it is possible to choose one instance of mappings per class and to fix these instances in the commutative diagram. For fixed instances of mappings, the only inverses can be calculated. After fixing specific instances of mappings from a valid class, this case is no different from the first case where all mappings in a commutative diagram are known initially.

^{−1}and G by known conditions in the form (1), (2), and the condition FG

^{−1}= H. These conditions are sufficient to determine the classes of maps H

^{−1}and G, and, if necessary, their specific instances. Let us demonstrate this with an example.

**Example**

**3.**

^{−1}= H. The size and structure of matrices H

^{−1}and G are determined, taking into account the size of matrices G

^{−1}and H in (9) from equality

_{1}… k

_{6}must be calculated from conditions in the form (1), (2). Given the fact that F is an ordinary identity matrix, FG

^{−1}= H, H

^{−1}F = G ⇒ G

^{−1}= H and H

^{−1}= G, we can write

_{1}+ 0.5k

_{5}= 1; 0.5k

_{2}+ 0.5k

_{6}= 0; k

_{3}= 0; k

_{4}= 1.

^{−−1}= G up to a class in the form (10) is solved. The relationships of the matrix elements in (10) are determined by Equations (12) and (13), in which there is freedom of choice for the two elements. Equality (11) can be used to check the feasibility of conditions in the form (2). Set k

_{1}= 2, k

_{2}= 1. Then, from (13) we get k

_{5}= 0, k

_{6}= −1, and a specific instance of the matrix will be in the form

^{−1}in (9) was specifically chosen to be equal to matrix G

^{−1}in (7) from the first example. It is easy to verify that matrices H

^{−1}= G from the first example belong to class (10). Indeed, if we set (13) k

_{1}= 1, k

_{2}= −1, we get k

_{5}= 1, k

_{6}= 1. Then from (10) and (13) we get other instances of matrices of class (10)

^{−1}= G in commutative diagrams in Figure 11 and Figure 12 there are the only inverses H = G

^{−1}defined by relations (6) and (7) from the first example.

## 3. Discussion

## 4. Summary

## Funding

## Conflicts of Interest

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Kulabukhov, V.S.
A General Principle of Isomorphism: Determining Inverses. *Symmetry* **2019**, *11*, 1301.
https://doi.org/10.3390/sym11101301

**AMA Style**

Kulabukhov VS.
A General Principle of Isomorphism: Determining Inverses. *Symmetry*. 2019; 11(10):1301.
https://doi.org/10.3390/sym11101301

**Chicago/Turabian Style**

Kulabukhov, Vladimir S.
2019. "A General Principle of Isomorphism: Determining Inverses" *Symmetry* 11, no. 10: 1301.
https://doi.org/10.3390/sym11101301