A General Principle of Isomorphism: Determining Inverses

Article A General Principle of Isomorphism: Determining Inverses

Vladimir S. Kulabukhov

Uspensky Avionica Moscow Research and Production Complex JSC; 7, Obraztsov St. 127055 Moscow, Russia; [email protected] Received: 18 July 2019; Accepted: 9 October 2019; Published: 15 October 2019

Abstract: The problem of determining inverses for maps in commutative diagrams arising in various problems of a new paradigm in algebraic system theory based on a single principle—the general principle of isomorphism is considered. Based on the previously formulated and proven theorem of realization, the rules for determining the inverses for typical cases of specifying commutative diagrams are derived. Simple examples of calculating the matrix maps inverses, which illustrate both the derived rules and the principle of relativity in algebra based on the theorem of realization, are given. The examples also illustrate the emergence of new properties (emergence) in maps in commutative diagrams modeling (realizing) the corresponding systems.

Keywords: mapping; the principle of isomorphism; realization theorem; system; system eﬀect (emergence); the principle of relativity in algebra; observability; controllability; checkability; identiﬁability of systems; integration of systems

1. Introduction The problem of determining isomorphic transformations typical for group theory and other sections and applications of algebra is considered. Thus, in the algebraic theory of systems based on the general principle of isomorphism [1–3], the problem of determining the inverses for maps forming commutative diagrams characterizing isomorphic transformations of groups, other sets with operations, and in general, algebraic categories often arise. Many problems of system theory come to the construction and analysis of such commutative diagrams [3–27]: modeling of systems and identiﬁcation of models, synthesis of regulators and observers, control and integration of systems. Based on the general provisions of the realization theory [8,16], works [1–3] formulate and prove the realization theorem of the isomorphism by a composition of the mappings asserting the existence of the inverse of the mappings that constitute the composition. The problem lies in ﬁnding these inverses. The fact is that the mappings that make up the composition in general are not isomorphisms in the usual sense outside the commutative diagram. In this regard, the problem of determining the inverses becomes nontrivial. The realization theorem solves this problem: inverses of the mappings, constituting the composition, are possible to identify up to isomorphism, closing the composition in the commutative diagram. This article presents a method of solving the problem, illustrated by examples. The realization theorem of isomorphism by mapping composition is used as the main Lemma within the framework of a new paradigm in algebraic system theory [1–3,25–27], but has a higher generality and is an independent result that can be claimed in other sections of algebra. The very concept of isomorphism implies symmetry of the properties of the corresponding algebraic categories. Based on this, the demonstration of the application of the theorem in the problem of determining the inverses can be useful in the theory of symmetry groups. Thus, the subject of the article is the presentation of a previously unknown method for determining the inverse for maps non-invertible in the usual sense, forming a composition commuting some

Symmetry 2019, 11, 1301; doi:10.3390/sym11101301 www.mdpi.com/journal/symmetry Symmetry 2019, 11, x FOR PEER REVIEW 2 of 12 Symmetry 2019, 11, 1301 2 of 12 Thus, the subject of the article is the presentation of a previously unknown method for determining the inverse for maps non-invertible in the usual sense, forming a composition isomorphism—acommuting some map isomorphism invertible— ina themap traditional invertible sense.in the traditional The method sense. is The based method on a is nontrivial based on a phenomenonnontrivial phenomenon previously discovered previously by discovered the author, by described the author, in detail described in [1– 3in]: Indetail a commutative in [1–3]: In a diagramcommutative built on thisdiagram isomorphism, built on therethis isomorphism, exist the only inverses there exist even the of theonly maps inverses which even are individually of the maps non-invertiblewhich are individually outside the framework non-invertible of the outside commutative the framework diagram. Such of the maps commutative behave as isomorphisms diagram. Such withinmaps a behave commutative as isomorphisms diagram and within satisfy a allcommutative formal rules diagram and constraints and satisfy inherent all formal of ordinary rules and isomorphicconstraints maps inherent considered of ordinary in algebra isomorphic and in category maps considered theory. Therefore, in algebra in [and1–3] in the category author called theory. themTherefore, “isomorphisms in [1–3] the defined author up called to isomorphism”, them "isomorphisms on which defined the composition up to isomorphism", of the corresponding on which the mapscomposition is built (implemented). of the corresponding Note, thatmaps this is isomorphismbuilt (implemented). by itself Note, must that be anthis ordinary isomorphism isomorphic by itself mapmust outside be an the ordinary commutative isomorphic diagram. map outside the commutative diagram. TheThe content content of thisof this article article focuses focuses on the on identification the identification of rules of forrules determining for determining the inverse the ofinverse these of particularthese particular isomorphisms. isomorphisms. The rules are The not rules trivial are and not simple trivial examples and aresimple given examples to demonstrate are given them, to allowing,demonstrate firstly, them, to understand allowing, how firstly, these to rules understand work and, how secondly, these rules to make work sure and, that secondly, this interesting to make phenomenonsure that this really interesting objectively phenomenon exists. This definesreally obje thectively boundaries exists. of This the study defines described the boundaries in the article. of the Thestudy obtained described results in are the applicable article. The to a varietyobtained of problemsresults are in applicable the theory to of systemsa variety and, of problems in the opinion in the oftheory the author, of systems can be and, applied in the in opinion algebra (forof the example, author, incan the betheory applied of in categories, algebra (for in theexample, theory in of the symmetrytheory of groups). categories, To understandin the theory this, of symmetry the necessary groups). minimum To understand references th tois, the English-languagenecessary minimum worksreferences of the authorto the English-language are given. works of the author are given. 2. Application of the Realization Theorem in Finding the Inverses 2. Application of the Realization Theorem in Finding the Inverses We will consider the commutative diagram shown in Figure1. Many commutative diagrams We will consider the commutative diagram shown in Figure 1. Many commutative diagrams are brought to such a diagram in problems of algebraic system theory [3]. The realization theorem are brought to such a diagram in problems of algebraic system theory [3]. The realization theorem (Theorem 1) [1–3] is valid for the diagram, the formulation of which is given here for completeness. (Theorem 1) [1–3] is valid for the diagram, the formulation of which is given here for completeness. TheoremTheorem 1. Realization1. Realization theorem: theorem: If the mapIf the f:X mapY isf:X a→ usualY is isomorphism a usual isomorphism and there are and maps there g:X areZ andmaps → → h:Zg:XY→ suchZ and that h:Z the→ compositionY such that fthe= hg composition is true, then f the = hg maps is true, g and then h are the isomorphisms maps g and up h toare isomorphism isomorphisms f. → up to isomorphism f.

f X Y

g h Z

Figure 1. Commutative diagram to the realization theorem. Figure 1. Commutative diagram to the realization theorem. Concerning the composition of hg of maps g and h, we can say that it implements, models Concerning the composition of hg of maps g and h, we can say that it implements, models the the map-isomorphism f. It should be emphasized that the map f must be isomorphic outside the map-isomorphism f. It should be emphasized that the map f must be isomorphic outside the commutative diagram, that is, it must be an ordinary, “unconditional” isomorphism. On maps g and h commutative diagram, that is, it must be an ordinary, "unconditional" isomorphism. On maps g and the theorem states that even if these maps separate, outside the composition and the commutative h the theorem states that even if these maps separate, outside the composition and the commutative diagram, they are not isomorphisms, then as part of the composition f = hg, closed by the isomorphism diagram, they are not isomorphisms, then as part of the composition f = hg, closed by the f in the commutative diagram, become isomorphisms up to the isomorphism f, that is, become “relative isomorphism f in the commutative diagram, become isomorphisms up to the isomorphism f, that is, isomorphisms”—isomorphisms with respect to (up to) the isomorphism f. This means that each of become “relative isomorphisms”—isomorphisms with respect to (up to) the isomorphism f. This the maps g and h is an isomorphism only and strictly within the commutative map diagram shown means that each of the maps g and h is an isomorphism only and strictly within the commutative in Figure1. Under these conditions, the maps g and h in the commutative diagram satisfy all the map diagram shown in Figure 1. Under these conditions, the maps g and h in the commutative properties of isomorphic maps deﬁned in algebra [28]. The proof is given in [3]. However, there is a diagram satisfy all the properties of isomorphic maps defined in algebra [28]. The proof is given in need to reproduce here a considerable part of the proof of the implementation theorem from [3] for a [3]. However, there is a need to reproduce here a considerable part of the proof of the more understandable presentation of the rules of deﬁnition of the inverse mappings g and h in the implementation theorem from [3] for a more understandable presentation of the rules of definition commutative diagram. This is done below directly when considering these rules. of the inverse mappings g and h in the commutative diagram. This is done below directly when The simplest explanation of the realization theorem is as follows. The composition of the mappings considering these rules. g and h forms a system in which these mappings have a new property, the property of having a The simplest explanation of the realization theorem is as follows. The composition of the reverse mapping. This property may not be present in each of them individually, outside of this mappings g and h forms a system in which these mappings have a new property, the property of

Symmetry 2019, 11, x FOR PEER REVIEW 3 of 12

Symmetryhaving2019 a ,reverse11, 1301 mapping. This property may not be present in each of them individually, outside3 of 12 of this system. This is a manifestation of emergence—the main sign of the emergence of any new system. Consider, for example, two children. Together they can play hide-and-seek, and each child system. This is a manifestation of emergence—the main sign of the emergence of any new system. separately, outside the “children” system, can not hide from himself. The “hide” property is a Consider, for example, two children. Together they can play hide-and-seek, and each child separately, systemic property of the “children” system, a manifestation of its emergence. At the same time, the outside the “children” system, can not hide from himself. The “hide” property is a systemic property bystander can see the hidden child: the property of “hiding” belongs only to the system “children”. of the “children” system, a manifestation of its emergence. At the same time, the bystander can see So the property of reversibility of maps g and h is manifested only in the system-composition, the hidden child: the property of “hiding” belongs only to the system “children”. So the property of realizing or commuting some unconditional isomorphism. The realization theorem describes reversibility of maps g and h is manifested only in the system-composition, realizing or commuting (formalizes) in the most General form the occurring of the emergence effect and in this sense is the some unconditional isomorphism. The realization theorem describes (formalizes) in the most General most General mathematical definition of the system as a mathematical abstraction and a form the occurring of the emergence effect and in this sense is the most General mathematical definition phenomenon of real reality. of the system as a mathematical abstraction and a phenomenon of real reality. Since the maps g and h in the commutative diagram are isomorphisms, they must have the only Since the maps g and h in the commutative diagram are isomorphisms, they must have the only ‒1 ‒1 inverses 1g and h1 in this diagram. We define these inverses for two cases: inverses g− and h− 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 (1) In the first case, all maps f, g, and h and the composition f = hg in the commutative diagram in in Figure 1 are defined initially. In many problems of system theory there is a need to Figure1 are defined initially. In many problems of system theory there is a need to determine the determine the inverses of g‒1 and h‒1 explicitly; 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. (2) In the second case, the commutative diagram is converted to the form shown in Figure2. Here, Here, only the isomorphic map f, some map designated as g‒1, and their composition fg‒1 are only the isomorphic map f, some map designated as g 1, and their composition fg 1 are initially initially known. It is clear that under these conditions− the map h commuting− the specified known. It is clear that under these conditions the map h commuting the specified composition composition exists, is uniquely and immediately determined by the formula fg‒1 = h. In this exists, is uniquely and immediately determined by the formula fg 1 = h. In this case, the mapping of case, the mapping of h‒1 is unknown and needs to be determined.− The inverse of g‒1, which h 1 is unknown and needs to be determined. The inverse of g 1, which also needs to be determined, is − also needs to be determined, is also unknown. However,− maps g and h‒1, inverses to g‒1 and also unknown. However, maps g and h 1, inverses to g 1 and h, respectively, can be defined only up h, respectively, can be defined− only up to class,− that is, they are not unique. This is because, to class, that is, they are not unique. This is because, based on the requirements of the realization based on the requirements of the realization theorem [3], in this case a pair of mappings (h, h g hg f theoremg) [and3], in their this casecomposition a pair of mappingshg = f, which ( , ) andimplement their composition the isomorphism= , which of f, implementor a pair of the isomorphism of f, or a pair of mappings (g 1, h 1) and their composition g 1h 1 = f 1, which mappings (g‒1, h‒1) and their composition− g‒−1h‒1 = f‒1, which implement −the− isomorphism− f‒1, f 1 implementare not the determined isomorphism initially− , are and not at determined the same time. initially Thus, and in atthis the second same time.case it Thus, is required in this to second case it is required to find classes of admissible maps h 1 and g that satisfy the condition find classes of admissible maps h‒1 and g that satisfy− the condition fg‒1 = h and other fg 1 h − = conditionsand other of conditions the realization of the realizationtheorem, which theorem, wi whichll be given willbe below. given It below. is obvious It is obvious that the that the required classes of maps h 1 and g will be strictly interconnected taking into account required classes of maps h‒1 and− g will be strictly interconnected taking into account these these conditions.conditions. OnceOnce you you have have defined defined the the classes classes of of valid valid mappings, mappings, you you can can select select one one related related mappingsmappings instance instance from the from corresponding the corresponding classes, if necessary,classes, if andnecessary, commit thoseand commit instances those to the commutativeinstances to diagram. the commutative Only in this diagram. case the Only only in inverses this case to the these only instances inverses of to mappings these instances will be fixedof andmappings can be will calculated. be fixed Clearly, and can after be calculated. fixing specific Clearly, instances after offixing mappings specific from instances valid of classes,mappings this case isfrom no divalidfferent classes, from this the firstcase caseis no where different all mappingsfrom the first in a case commutative where all diagram mappings are known.in a commutative diagram are known.

f X Y

g -1 h Z

Figure 2. Commutative diagram for the 2nd case. Figure 2. Commutative diagram for the 2nd case. The case shown in Figure2, is typical, for example, for the application of the realization theorem The case shown in Figure 2, is typical, for example, for the application of the realization theorem to the problem of determining regulators in the theory of systems [3,25]. The case of determination to the problem of determining regulators in the theory of systems [3,25]. The case of determination of of observers—see Figure3—considered in [ 3,26] is also equivalent to it. In defining observers, the observers—see Figure 3—considered in [3,26] is also equivalent to it. In defining observers, the difference is that the isomorphic map f, some map designated as h 1, and their composition h 1f = g difference is that the isomorphic map f, some map designated as− h‒1, and their composition− h‒1f = g are known and defined in advance. The problem is to define maps h and g 1. By analogy with Figure2, are known and defined in advance. The problem is to define maps h and− g‒1. By analogy with Figure maps of h and g 1 in Figure3 can be defined up to class. Due to the equivalence of the problem 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 Figures2 and3, in the future we will solve only the problem reflected in the commutative diagram in Figure2.

Symmetry 2019, 11, x FOR PEER REVIEW 4 of 12

Symmetry 2019, 11, x FOR PEER REVIEW 4 of 12 statements illustrated in Figures 2 and 3, in the future we will solve only the problem reflected in the commutative diagram in Figure 2. Symmetrystatements2019, 11 illustrated, 1301 in Figures 2 and 3, in the future we will solve only the problem reflected4 of in 12 the commutative diagram in Figure 2. f X f Y

X -1 Y g h Z -1 g h Z Figure 3. Equivalent commutative diagram for the 2nd case. Figure 3. Equivalent commutative diagram for the 2nd case. Figure 3. Equivalent commutative diagram for the 2nd case. We will start with the first case and Figure 1, when in the commutative diagram the We will start with the first case and Figure1, when in the commutative diagram the composition compositionWe will f =start hg andwith all the maps first f , caseg, and and h Figureare initially 1, when defined. in the And commutative f is such that diagram it is anthe f = hg and all maps f, g, and h are initially defined. And f is such that it is an isomorphism outside isomorphismcomposition outside f = hg theand commutative all maps f, diagram.g, and h Baseare dinitially on the defined.realization And theorem, f is such it is thatknown it is[3] an thethat commutative in this case diagram.the only Basedinverses on g the‒1 and realization h‒1 in the theorem, commutative it is known diagram [3] that exist. in this To casefind the them only is isomorphism1 outside1 the commutative diagram. Based on the realization theorem, it is known [3] inverses g and h in the commutative diagram‒1 ‒ exist.1 To find them is required. It is clear that the required.that in −thisIt is caseclear− the that only the inverses ofg‒ 1g and and h ‒1h in, basedthe commutative on the conditions diagram of theexist. theorem To find and them the is g 1 h 1 inversesequations of obtained− and −on, basedits basis, on themust conditions be completely‒1 of the‒ 1determined theorem and by the the equations original obtainedmaps f, g on, and its basis,h. We required. It is clear that the inverses of g and h , based on the conditions of the theorem and 1the must be completely determined‒1 by the original‒1 maps f, g, and h. We try to express the inverses of g tryequations to express obtained the inverses on its of basis, g and must h be through completely the original determined known by mapsthe original and thus maps obtain f, g, rulesand h for−. We and h 1 through the original known maps and thus obtain rules for their computation. theirtry− computation.to express the inverses of g‒1 and h‒1 through the original known maps and thus obtain rules for Directly from the commutative diagram shown in Figure1 (see also the supporting diagrams in theirDirectly computation. from the commutative diagram shown in Figure 1 (see also the supporting diagrams in Figures4–9, by virtue of isomorphism in the usual sense of the map f, the equalities follow FiguresDirectly 4–9, by fromvirtue the of commutativeisomorphism diagramin the usual shown sense in ofFigure the map 1 (see f, alsothe equalities the supporting follow diagrams in right ‒1 left ‒1 left ‒1 right ‒1 ‒1 ‒1 ‒1 ‒1 Figures 4–9, rightby virtue1e xof isomorphismleft= f f, ey1 = leftff in, ethex 1usual= g gright ,sense ey =of hh 1the, mapf1 = e fx,f the1, f equalities=1 f ey, 1 follow ex = f − f, ey = ff − , ex = g− g, ey = hh− , f − = exf − , f − = f − ey, exright = f‒1f, erightyleft = ff‒1left, exleft = gright‒1g, eyrightleft = hh‒1, f‒1 = exf‒1, f‒1 = f‒1ey, (1) ex = ex = exright= e =x exleft= =e eyyright == eeyyleft = e=y =ey e=. e. (1) right left right left rightright leftleft ex = ex = ex = ey right=right ey left= leftey = e. HereHeree exx ,, eex areare the the right right and and left left units units on on XX; ;eyey , e,y ey areare the the right right and and left left units units on onY. ByY. ‒1 virtue of the uniquenessright left of the map f (since1 by the conditionright f leftis the usual isomorphism), the maps By virtueHere of theex uniqueness, ex are the of right the map andf left− (since units byon theX; e conditiony , ey aref is the the right usual and isomorphism), left units on theY. By x y ‒1 ‒1 1 -1 mapse virtueande xeand ofare the ealsoy areuniqueness unique also unique and of equalthe and map equalto ef,‒ 1where (since to e, where e by is thee isusualcondition the usualunit fmap is unit the such map usual that such isomorphism), e = that ee =e ee= ee== e theeee−= maps e=. From1 the-1 auxiliary diagrams constructed on the basis of the initial diagram and given in Figures 6–9, e− eex =ande .ey From are also the unique auxiliary and diagrams equal to constructede, where e is onthe the usual basis unit of map the initialsuch that diagram e = ee and= ee‒ given1 = e‒1 e in= e-1. FigurestheFrom equalities6 –the9, theauxiliary follow equalities diagrams follow constructed on the basis of the initial diagram and given in Figures 6–9,

the equalities follow ‒1 z ‒1 ‒1 -1 y ‒1 ‒1 z ‒1 x ‒1 1 h 1= e h1 , h 1= h e 1, g = 1g e , g1 = e g 1, h− = ezh− , h− = h− ey, g− = g− ez, g− = exg− , h‒1 = ezh‒1, h‒1 = h-1ey, g‒1 = g‒1ez, g‒1 = exg‒1, where ez is some unit on Z. From the auxiliary diagrams shown in Figure 10b, it is also possible to where e is some unit on Z. From the auxiliary diagrams shown in Figure 10b, it is also possible to obtain equalityz obtainwhere equality ez is some unit on Z. From the auxiliary diagrams shown in Figure 10b, it is also possible to obtain equality 1 ‒1 1‒1 ‒left1 left 1 ‒1 rightright ‒1 f − = gf− h=− g, hez , ez= gg = −gg, e,z ez == hh− hh,, ‒1 ‒1 ‒1 left ‒1 right ‒1 left right f = g h , ez = gg , ez = h h, wherewheree zezleft,, eezzright areare left left and and right right units units on on ZZ..

where ezleft, ezright are left and right units on Z. eX

ffeX XYX ff XYX eY f -1 f -1 eY f -1 Y f -1 Symmetry 2019, 11, x FOR PEER REVIEW 5 of 12 FigureFigure 4. 4.Auxiliary AuxiliaryY diagram diagram 1. 1.

Figure 4. AuxiliaryeY diagram 1. ff Y X Y

eX f -1 f -1 X FigureFigure 5. 5. AuxiliaryAuxiliary diagram diagram 2. 2.

ZYZhh

eY h-1 h-1 Y Figure 6. Auxiliary diagram 3.

YZYhh

eZ h-1 h-1 Z Figure 7. Auxiliary diagram 4.

gg ZXZ

e g-1 X g-1

X Figure 8. Auxiliary diagram 5.

gg XZX

e g-1 Z g-1 Z

Figure 9. Auxiliary diagram 6.

Symmetry 2019, 11, x FOR PEER REVIEW 5 of 12 Symmetry 2019, 11, x FOR PEER REVIEW 5 of 12 Symmetry 2019, 11, x FOR PEER REVIEW 5 of 12 e Symmetry 2019, 11, x FOR PEER REVIEW Y 5 of 12 ffeY Y eXY Y ffe Y ffXY Y eX Yf -1 ffX f -1Y e Symmetry 2019, 11, x FOR PEER REVIEW Y -1 X X -1Y 5 of 11 f XeX f f -1 f -1 XeX f -1 e f -1 Figure 5. AuxiliaryXZ diagram 2. Figure 5. AuxiliaryX diagram 2. Figure 5. Auxiliaryhh diagram 2. ZYZeZ Symmetry 2019, 11, 1301 Figure 5. Auxiliary diagram 2. 5 of 12 eZ e ZYZ-1 hheZ Y -1 h e h ZYZhhZ hh ZYZ-1 YeY -1 h hhh ZYZeY Figureh-1 6. Auxiliary diagramh-1 3. -1 YeY -1 h e h -1 Y Y -1 Figureh 6. AuxiliaryeY diagramh 3. Y Figure 6. AuxiliaryY diagram 3. Figure 6. Auxiliary diagram 3. YZYhheY FigureFigure 6. 6.Auxiliary Auxiliary diagram diagram 3. 3. eY hheY eZ YZYh-1 h-1 eY YZYhh hh YZY-1 ZeZ -1 h hhh YZYeZ Figureh -17. Auxiliary diagramh-1 4. -1 ZeZ -1 h e h -1 Z Z h-1 Figureh 7. AuxiliaryeZ diagram 4. Z Figure 7. AuxiliaryZ diagram 4. Figure 7.ggAuxiliary diagram 4. FigureZXZ 7. AuxiliaryeZ diagram 4. Figure 7. Auxiliary diagram 4. eZ gge eX ZXZg-1 Z g-1 ggeZ ZXZ ggX ZXZ-1 eX -1 g ggg ZXZ-1 eX -1 Figureg 8. Auxiliary diagramg 5. -1 XeX -1 g e g g-1 X X g-1 Figure 8. AuxiliaryX eX diagram 5. Figure 8. Auxiliary diagram 5. Figure 8. AuxiliaryX diagram 5. gg FigureXZ 8. AuxiliaryeX diagram 5.X Figure 8. Auxiliary diagram 5. eX gge eZ XZg-1 X g-1X ggeX XZX ggZ XZ-1 eZ -1X g ggg XZ-1 eZ -1X Figureg 9. Auxiliary diagramg 6. -1 ZeZ -1 g g -1 eZ -1 Figureg 9. AuxiliaryZ diagramg 6. Figure 9. AuxiliaryZ diagram 6. Z Figure 9. Auxiliary diagram 6. Figure 9. Auxiliary diagram 6.e Y e X f -1 Figure 9. Auxiliary diagram 6. f -1 -1 f ex X Y ey f f Y X Y X

g h

g-1 h-1 h g h g Z

ez Z

(a) The initial commutative diagram. (b) Equivalent diagram transformations.

FigureFigure 10.10. Auxiliary commutative diagrams.diagrams.

In the proof of the realization theorem in [3] it is shown that in the commutative diagram in Figures 1 and 10a the following equalities are true

left 1 right 1 left right left right right left 1 ez = gg− , ez = h− h, ez = ez = ez, ez ez = ez, ez ez = ez,(ez)− = ez, ez ez= ez. (2) Symmetry 2019, 11, 1301 6 of 12

It follows from (2) that left right right left ez ez = ez ez , that is 1 1 1 1 gg− h− h = h− hgg− .

1 1 1 Then, taking into account equations f − = g− h− and f = hg, we obtain

1 1 1 g f − h = h− f g− .

1 It is obvious that there exists a composition of map g− and each of the parts of this equality. 1 “Multiply” the equation by g− on the left side, we get

1 1 1 1 1 g− g f − h = g− h− f g− . (3)

1 1 1 Since g− h− = f − , the right part of equality (3) gets the form

1 1 right 1 right 1 right left 1 f − f g− = ex = f − f = ex g− = ex = ex = e = g− . h i h i left 1 Taking into account ex = g− g we write the left side of the equality (3) in the form

left 1 left 1 1 1 1 1 1 ex f − h = ex f − = f − = f − h = g− , f − h = g− . h i ⇒ That is, ﬁnally 1 1 g− = f − h. (4)

1 Here h is initially known, and f − can be easily computed, since f is also a known ordinary 1 isomorphic map for which the inverse of f − is known to exist. Similarly, it can be shown that

1 1 h− = g f − . (5)

The equality (5) can also be directly veriﬁed by the commutative diagram shown in Figure 10a. Here also all the mappings are known on the right side. 1 1 Thus, the inverses of g− and h− can be quite simply calculated from the known map included in 1 1 the original commutative diagram. Making sure that the g− and h− mappings are indeed inverses to the corresponding original mappings in the commutative diagram shown in Figure1, 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

left 1 1 1 1 right 1 1 1 1 left right ez = g g− = g− = f − h = g f − h and ez = h− h = h− = g f − = g f − h ez = ez = ez. left right h 1 1i 1 1 h1 right i 1 left⇒ 1 left ez ez = gf − hgf − h = hg = f = gf − f f − h = f − f = ex , ﬀ − = ey = gf − ey h = right h1 i 1 left 1 righth 1 1 1 i = g ex f h = f ey = f , ex f = f = g f h = ez. − h − − − − i − 1 1 Thus, Equations (4) and (5) for calculating the inverses of g− and h− are true and can be applied inside the commutative diagram shown in Figure1. 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 f = f − = e,

1 1 1 the inverses g− and h− are trivial and do not require computation, since in this case g− = h and 1 h− = 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. At ﬁrst glance, Equations (4) and (5) could be written directly from the commutative diagrams shown in Figures1 and 10a. In fact, these equations are not as trivial as they may seem, because Symmetry 2019, 11, 1301 7 of 12 the inverses of maps g and h outside the commutative diagram may not exist. These arguments just 1 1 allowed us to prove the existence of inverses g− and h− in the commutative diagram, partly repeating the proof of the realization theorem [3], and strictly derive Equations (4) and (5) for their calculation. Consider the examples of calculating the inverses for the ﬁrst case when matrices are used as mappings in the original diagram in Figure1. The commutative diagram for matrices is shown in Symmetry 2019, 11, x FOR PEER REVIEW 7 of 11 Figure 11. Example 1. Let the matrices have the following structure: Example 1. Let the matrices have the following structure: 1 −1 1 0   0 .5 0 0 .5  х   у  =    1  1 =   = 1 = 1 F " , #G = 0 1 , H " # , X" #  , Y" #  . (6) 011 0   −   00.5 01 0.5 0  x1 х y1 у F =   , G =  0 1 , H = , X =  , 2Y= .2  (6) 0 1 1 1   0 1 0 x y   1 1  2 2

F X Y

G H Z

FigureFigure 11. 11.Commutative Commutative diagram diagram for for matrices. matrices. It is clear that F isomorphism and composition F = HG are true, which corresponds to the It is clear that F isomorphism and composition F = HG are true, which corresponds to the conditions of the realization theorem. Hence, as follows from this theorem, despite the fact that conditions of the realization theorem. Hence, as follows from this theorem, despite the fact that matrices H and G are rectangular and in the usual sense irreversible, inside the commutative diagram matrices H and G are rectangular and in the usual sense irreversible, inside the commutative they must have the only inverses up to isomorphism F. Since F is a usual unit matrix, then according to diagram they must have the only inverses up to isomorphism F. Since F is a usual unit matrix, then (4) and (5) in this case G 1 = H and H 1 = G, that is, according to (4) and (5)− in this case −G‒1 = H and H‒1 = G, that is,   " # 1 1 − 0.5 0 0.5  1 1 1 − 0.5 0 0.5 1  −  G− =1 = , H−, = −1 =0 1 . . (7) G 0 1 0  H  0 1 (7)  0 1 0   1 1   1 1  You can verify that matrices G 1 and H 1 in (7) are indeed the only inverses to matrices G and H You can verify that matrices− G‒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). in the commutative diagram, since they satisfy all conditions of (1) and (2).

ExampleExample 2. 2.Here Here the the matrices matrices have have the the following following structure structure [3 ]:[3]: " # " # " # " # 1 1 0 0 h i  11  a   a  F = = 1 , G, =G = []11 0 , , H == 1 , X == ,, Y == . (8) Fba  −10  H  ba −1  X b  Y  b  (8) −ba 0 ba −  b  b

CompositionCompositionF =FHG= HGis also is valid.also valid. In [3 In] it [3] is shownit is shown that that therealization the realization theorem theorem is also is also valid valid in thein wider the wider case—for case—for isomorphisms isomorphisms that are suchthat notare onlysuch in not the usualonly sense—outsidein the usual sense—outside the commutative the diagram—butcommutativealso diagram—but have special also properties. have special So, in properties. this case, isomorphismSo, in this case,F has isomorphism the peculiarity F has that the itspeculiarity structure dependsthat its structure on the structure depends of on mapping the structure matrices of mappingX, Y. Indeed, matrices it is quiteX, Y. obviousIndeed, thatit is quite the determinantobvious that of the matrix determinantF is zero and of matrix it has noF is inverse zero and in the it has usual no sense.inverse At in the the same usual time, sense. we At can the make same suretime, that we for can matrix makeF ,sure the structurethat for matrix of which F, the is determinedstructure of bywhich the rulesis determined (8), there isby an the inverse rules (8), (in there the algebraicis an inverse sense (in [28 the]) matrix algebraic sense [28]) matrix

− − − − −  1 0 F 1 = FF 1 = F 1F = FF = F 1F 1 ="F = # . 1 1 1 1 1 1ba 0−1 0 F− = FF− = F− F = FF = F− F− = F = 1 .  ba− 0 That is, F is isomorphism. From relations (4), (5) it is possible to find inverse matrices:

−  1  − 1 = , H 1 = []1 0 . G  −1  ba  All conditions (1) and (2) necessary to satisfy the realization theorem are satisfied. Consider the second case with respect to the commutative diagram shown in Figure 12. Here, only the isomorphic map F is known, as well as some map designated as G‒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‒1H‒1 = F‒1, which implements the isomorphism F‒1, are not defined initially and simultaneously.

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That is, F is isomorphism. From relations (4), (5) it is possible to ﬁnd inverse matrices: " # 1 1 1 h i G− = 1 , H− = 1 0 . ba−

All conditions (1) and (2) necessary to satisfy the realization theorem are satisfied. Consider the second case with respect to the commutative diagram shown in Figure 12. Here, 1 only the isomorphic map F is known, as well as some map designated as G− , and their composition 1 1 FG− = H. Moreover, H is unique, since maps F and G− are completely (up to an instance) determined. 1 1 However, maps G and H− to be defined, which are inverse to G− 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 1 1 1 1 1 implements the isomorphism F, or a pair of maps (G− , H− ) and their composition G− H− = F− , Symmetry 2019, 11, x FOR PEER REVIEW 1 8 of 11 which implements the isomorphism F− , are not defined initially and simultaneously. F X Y

G -1 H Z

Figure 12. Commutative diagram for the second case. Figure 12. Commutative diagram for the second case. 1 In this second case it is required to find classes of interconnected mappings H− and G admissible 1 under the condition FG− = H and conditions in form (1), (2). After definition of classes, proceeding In this secondfrom additional case considerations,it is required it is possible to tofind choose classes one instance of of interconnected mappings per class and mappings to fix H-1 and G these instances in the commutative‒1 diagram. For fixed instances of mappings, the only inverses can be admissible undercalculated. the condition After fixing specific FG instances= H and of mappings conditions from a valid in class,form this (1), case is(2). no di Afterfferent from definition of classes, proceeding fromthe firstadditional case where all considerations, mappings in a commutative it is diagrampossible are known to choose initially. one instance of mappings per 1 Thus, the task of the second case is to determine the classes of admissible maps H− and G by class and to fix these instances in the commutative diagram.1 For fixed instances of mappings, the known conditions in the form (1), (2), and the condition FG− = H. These conditions are sufficient to 1 only inverses candetermine be calculated. the classes of After maps H −fixingand G, and,specific if necessary, instances their specific of instances.mappings Let us from demonstrate a valid class, this case is no different fromthis with the an example.first case where all mappings in a commutative diagram are known initially. Thus, the Exampletask of 3. theLet the second matrices have case the following is to structuredetermine on Figure the 12: classes of admissible maps H-1 and G by " # " # " #‒1 " # known conditions in the form (1),1 0(2), and1 0.5the 0condition 0.5 FGx1 = H.y 1These conditions are sufficient to F = , G− = , X = , Y = . (9) determine the classes of maps0 1H-1 and 0G, 1and, 0 if necessary,x2 y 2 their specific instances. Let us demonstrate this with an" example.# 0.5 0 0.5 Then H = can be immediately determined out of the condition FG 1 = H. The size 0 1 0 − 1 1 Example 3. Let andthe structure matrices of matrices have theH− andfollowingG are determined, structure taking on into Figure account 12: the size of matrices G− and H in (9) from equality    k1 k2    G = H 1 = GF 1 =  k k  1 0 −1 0 .5− 0 − 0 .53  4 , х 1   у (10)1  F = , G =  ,  = , = .    k5 k6 X   Y   (9) 0 1  0 1 0   х 2   у 2 

0.5 0 0.5 = ‒1 Then H   can be immediately determined out of the condition FG = H. The  0 1 0  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

k1 k2  G = H −1 = GF −1 = k k  ,  3 4  (10) k5 k6  where unknown elements k1 … k6 must be calculated from conditions in the form (1), (2). Given the fact that F is an ordinary identity matrix, FG‒1 = H, H‒1F = G  G‒1 = H and H‒1 = G, we can write

0.5k1 k2 0.5k1  eleft = GG −1 = eright = H −1H = 0.5k k 0.5k  = e , z z  3 4 3  z (11) 0.5k5 k6 0.5k6 

()()0.5k + 0.5k 0.5k + 0.5k  1 0 left = −1 = right = −1 = 1 5 2 6 = =   ex G G ey HH   e   . (12)  k3 k4  0 1

From (12) we obtain equality

0.5k1 + 0.5k5 = 1; 0.5k2 + 0.5k6 = 0; k3 = 0; k4 = 1. (13)

Thus, the problem of calculating matrices H-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 k1 = 2, k2 = 1. Then, from (13) we get k5 = 0, k6 = −1, and a specific instance of the matrix will be in the form

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where unknown elements k1 ... k6 must be calculated from conditions in the form (1), (2). Given the fact that F is an ordinary identity matrix, FG 1 = H, H 1F = G G 1 = H and H 1 = G, we can write − − ⇒ − −    0.5k1 k2 0.5k1  le f t right   e = GG 1 = e = H 1H =  k k k  = e z − z −  0.5 3 4 0.5 3  z, (11)   0.5k5 k6 0.5k6 " # " # le f t 1 right 1 (0.5k1 + 0.5k5)(0.5k2 + 0.5k6) 1 0 ex = G− G = ey = HH− = = e = . (12) k3 k4 0 1 From (12) we obtain equality

0.5k1 + 0.5k5 = 1; 0.5k2 + 0.5k6 = 0; k3 = 0; k4 = 1. (13)

1 Thus, the problem of calculating matrices H− = 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 = 2, k = 1. Then, from (13) we get k = 0, k = 1, and a speciﬁc 1 2 5 6 − instance of the matrix will be in the form    2 1  1   G = H− =  0 1 . (14)    0 1  − Checking of matrices (10) and (14) according to the formulas in the form (1) and (2) shows their full compliance with all the conditions applicable in the commutative diagrams in Figures 11 and 12. You may verify that the following matrices are the only inverses for ﬁxed instances of matrices (14) in the commutative diagram in Figure 11. " # 0.5 0 0.5 G 1 = H = . (15) − 0 1 0

1 1 In this example, matrix G− in (9) was specifically chosen to be equal to matrix G− in (7) from 1 the first example. It is easy to verify that matrices H− = G from the first example belong to class (10). Indeed, if we set (13) k = 1, k = 1, we get k = 1, k = 1. Then from (10) and (13) we get other 1 2 − 5 6 instances of matrices of class (10)    1 1  1  −  H− = G =  0 1 . (16)    1 1  Comparison of matrices (16) with the corresponding matrices from (6) and (7) of example 1 shows 1 their coincidence. For fixed instances (16) of matrices H− = G in commutative diagrams in Figures 11 1 and 12 there are the only inverses H = G− defined by relations (6) and (7) from the first example. Examples 1 and 3 are interesting in the way that they demonstrate not only the rules for determining inverses in commutative diagrams satisfying the conditions of the realization theorem [3], but also illustrate the principle of relativity in mathematics formulated in [3]: The same map can have different inverses up to the same isomorphism in different compositions with other maps commuting this isomorphism. Moreover, these inverses are the only ones in their compositions realizing (commuting) the same isomorphism. So, in example 1 a pair of mappings in (6)    1 1  " #  −  0.5 0 0.5 G =  0 1 , H = , (17)   0 1 0  1 1  Symmetry 2019, 11, 1301 10 of 12

" # 1 0 commuting isomorphism F = in the diagram in Figure 11, has inverses (7) 0 1   " #  1 1  1 0.5 0 0.5 1  −  G− = , H− =  0 1 . (18) 0 1 0    1 1 

In Example 3, a pair of mappings (14) and (15)    2 1  " #   0.5 0 0.5 G =  0 1 , H = , (19)   0 1 0  0 1  − " # 1 0 commuting the same isomorphism F = , has inverses 0 1   " #  2 1  1 0.5 0 0.5 1   G− = , H− =  0 1 . (20) 0 1 0    0 1  − 3. Discussion Thus, the same map H in different compositions (17) and (19) commuting the same isomorphism F has different inverses (18) and (20) and these inverses are the only ones in their compositions. The algebraic principle of relativity in relation to the theory of systems is interpreted in [3] as follows: the same element (characterized by some mapping) in combination with different elements in different systems can exhibit different properties and even properties that it did not have outside the system at all—for example, the property of having an inverse map to the map being its model. This allowed us to assert [3] that the theorem of realization and the corresponding mathematical (algebraic) principle of relativity strictly formally reveal and explain the mechanism of occurrence of the emergent property in systems (including physical ones). Emergence as a phenomenon and property is the main feature of the system and a key element of the meaningful definition of “system” [3,29–33]. This property appears due to “system effect”: In the system, created as a result of combining some elements in it, new properties arise and manifest themselves, which are not reduced to the sum of the properties of the elements united in the system, as well as the elements in the system manifest new properties they lack outside the system.

4. Summary In the article, based on the conditions arising from the realization theorem [3], formal rules for determining the inverses of maps forming a composition, commuting (implementing) some isomorphism are obtained. The rules are obtained for commutative diagrams corresponding to diﬀerent cases of setting initial maps and their compositions commuting a known ordinary isomorphism.

Funding: This research received no external funding. Conﬂicts of Interest: The author declares no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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