ISOMORPHOLOGICAL ANALYSIS: THE THEORY OF IT ALL

Isomorphological Analysis: The Theory of It Al

Copyright, American Society for Engineering Management, 2013

ISOMORPHOLOGICAL ANALYSIS: THE THEORY OF IT ALL

Mario Beruvides, Ph.D., P.E.,

Texas Tech University

[email protected]

Jaime Cantu,

Texas Tech University

[email protected] ____________________________________________________________________________________________

Abstract Complex systems concepts can be traced back to the philosophy of G.W. Leibiniz and Nicholas Cusa, and during the

19th century a systems approach was introduced into the hard sciences of energy transformation and

thermodynamics. Current General Systems Theory was developed from the work of Ludwig von Bertalanffy and

others. Due to the existence of general system properties the appearance of structural similarities or isomorphism’s

in different fields occur. In many cases, isomorphic laws hold for certain classes or subclasses of 'systems', irrespective of the nature of the entities involved. There appear to be general system laws which apply to any system

of a certain type, irrespective if the particular properties of the system and of the elements involved. The reality of

today’s environment is one of complexity, self-organizations, connectionism and adaptive systems in all fields of

research in General Systems Theory. Bertalanffy understood that identifying isomorphologies between complex

systems improved the probabilities of better understanding systems complexities in any one system. The purpose of

this paper is to identify and present a methodological structure for systemic isomorphological analysis to assist

systems theory and engineering management researchers, students and practioners, to address the identification,

application and analysis of complex systems isomorphologies.

Keywords Isomorphism, Systems Theory, Systems Dynamics, Organized Complexity

Introduction “In one way or another, we are forced to deal with complexities, with ‘wholes’ or ‘systems’ in all fields of

knowledge.” (Bertalanffy, 1968, p. 5) Truly, one of the major issues faced in modern society, and the management

of complex technical endeavors specifically, is how to best deal with large scale systems that grow in complexity at

faster rates than ever seen before. The problem is not that complex systems are foreign to our sciences, engineering,

or understanding, (see Weaver, 1948). In fact the study of the basic sciences is a venture into the most complex of

all systems we are aware of, the workings of our world and the universe in which it resides. The real problem is

centered about the reality that many of the current technical and scientific models we possess, no longer hold

valence or are far less equipped than they were in the past to deal with the new challenges and changes that our

societal and technical systems have undergone. Collapses like the ones we have recently experienced in our economic system and the housing and banking markets, present a picture of societal and technical systems that seem

broken, and broken to the point of being a thick bundle of unorganized chaos and complexity. Seemingly, it is a

complex beyond our repair. The nightly news, print media, and some scholarly voices (see Davie, 2012; Faiola,

2008; and Henninger, 2009) have declared that capitalism has run its course. “The theory of unorganized

complexity is ultimately rooted in the laws of chance and probability and in the second law of thermodynamics. In

contrast, the fundamental problem today is that of organized complexity,” (Bertalanffy, 1968,p. 34). With these

points in mind, the question begs to be asked, are we ultimately in an entropic process bound to end in disorder with

respect to our current technical systems? Or, is the problem, as Bertalanffy alludes to, a “fundamental problem” of

“organized complexity’?

These points can be debated philosophically at infinitum, but a cursory look at history reveals that

Bertalanffy was quite prophetic when he first penned those words over 50 years ago. We as engineering managers

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are squarely in the middle of what might be termed ‘super-saturated complexity’ with respect to our technical

systems. This is self evident in one of the biggest problems facing society, business, science, and engineering today,

that being the problem of ‘big data,’ (Rosenbush and Totty, 2013). Our advanced and highly sophisticated and

technical means of communication systems to generate data streams so large that in a matter of days we surpass the

entire data inventories of past centuries. Again, the problems are seemingly insurmountable, but it is of value to

remember though, that the development of systems theory (specifically General Systems Theory), was to address this very problem. (Bertalanffy, 1968; Forrester, 1968) The development of our sciences was a venture into trying

to explore, explain, and better understand observable phenomenon by the methodology/approach we know as the

scientific process. This scientific process uses a logical-positivist philosophical approach (Wittgenstein, 2011) in

which observable phenomena are reduced to the interplay of elementary physical units that can be analyzed

independently. The summation of that independently gathered knowledge would sum, much like in a mathematical

equation, to the respective whole. “The whole is the sum of its parts.” Though this method, the scientific approach,

is unquestionably one of human kinds greatest inventions, it cannot provide answers to all the problems we face

(scientifically or non-scientifically). The whole at times is more than the sum of its parts. These are problems

concerned with what Bertalanffy called “wholeness.” These are “problems of organization, phenomena not

resolvable into local events, dynamic interactions manifest in the difference of behavior of parts when isolated or in

a higher configuration, etc.; in short, ‘systems’ of various orders not understandable by investigation of their

respective parts in isolation.” (Bertalamffy, 1968, p. 37). There exists characteristics of living systems and of organization (complex of elements), such as a society, that are concerned with such issues as wholeness, structure,

hierarchical order, competition, control, dominance, growth and differentiation that are not discernible by our most

basic sciences (such as physics). Examples of these are the Malthusian law (population growth exceeding its

resources), or in the economic realm, the law of oligopoly, where in competing organizations, the probability of an

increase in conflict between the entities (organizations/actors) increases with the decrease in the number of entities

(organizations/actors).

Isomorphology Systems theory has developed and evolved over the last half century, with many tools and subsequent off-shoot

theories. As a field of study and practice, systems theory has been very successful, prolific, and is still a growth area

in the academic and professional worlds. But like most areas of study, it is not without its limitations and problems.

Systems theory, by its nature encompasses many areas of study and thus it lacks the tight-knit methodologies found

in other fields of scientific endeavor. The “Systems approach – is a broad term, which takes account of the fact that

there is no single systems approach, but a range of different ones.” (Schaninger 2006, p. 583) There is much truth

in that statement. Schwaninger, goes on to say, “[t]he common denominator of the different systems approaches in

our day is that they share a worldview focused on complex dynamic systems and an interest in describing,

explaining and designing or at least influencing them.” (p. 583) The vast variety of roots and tools, subject matter and impact of the field over the last sixty plus years can be seen in Exhibits 1and 2.

Exhibit 1: Systems Roots/Tools (Adapted from Schwaninger, 2006)

- Mathematics (e.g.Newton, Poincare´, Lyapunov, Lotka, Volterra, Rashevsky);

- Logic (e.g. Epimenides, Leibniz, Boole, Russell and Whitehead, Goedel,

Spencer-Brown);

- Biology, including general physiology and neurophysiology (e.g. Hippocrates,

Cannon, Rosenblueth, McCulloch, Rosen);

- Engineering, including its physical and mathematical foundations (e.g. Heron,

Kepler, Watt, Euler, Fourier, Maxwell, Hertz, Turing, Shannon and Weaver, von

Neumann, Walsh) and

- Social and human sciences, including economics (e.g. Hume, Adam Smith,

Adam Ferguson, John Stuart Mill, Dewey, Bateson, Merton, Simon, Piaget).



- Systems Dynamic Modeling (e.g. Forrester, work done at the MIT Systems

Lab, etc.)



Exhibit 2: Mathematical & Non-Mathematical Fields of Systems Theory Influence (Adapted from Schwaninger, 2006)

Mathematical Roots/Tools Non-Mathematical Roots/Tools

Engineering, namely information and communication

theory and technology (e.g. Kalman filters, Walsh

functions, hypercube architectures, automata, cellular

automata, artificial intelligence, cybernetic machines,

neural nets)

Economics, namely its institutional/evolutionist

strand (Veblen, Myrdal, Boulding)

Operations research (e.g. modelling theory and

simulation methodologies, Markov chains, genetic

algorithms, fuzzy control, orthogonal sets, rough sets)

Sociology (e.g. Parsons’ and Luhmann’s social

system theories, Hall’s cultural systems theory)

Social sciences, economics in particular (e.g. game

theory, decision theory) and

Political sciences (e.g. Easton, Deutsch, Wallerstein)

Ecology (e.g. E. and H. Odum’s systems ecology). Anthropology (e.g. Levi Strauss’s structuralist

functionalist anthropology, Margaret Mead)

Semiotics (e.g. general semantics (Korzybski,

Hayakawa, Rapoport))

Psychology and psychotherapy (e.g. systemic

intervention (Bateson, Watzlawick, F. Simon), and

fractal affect logic (Ciompi))

Ethics and epistemology (e.g. Vickers, Churchman,

von Foerster, van Gigch)

Interesting enough of the many concepts found within systems theory (i.e. open v. closed systems, neg-entropy,

equifinality, dynamic equilibrium, etc.), the concept that has been least explored is the idea of the existence of

isomorphisms is systems. Isomorphology, as a concept, is present is several sciences. In biology, geology and plant

science, isomorphology is a drawing based method used for enquiry into the shape of the morphologies of plants (vegetables), minerals, and animals. Though there are some similarities in the overall concept, systems

isomorphology is quite different. Here Bertalanffy noticed that certain patterns of behavior (quantitative and

qualitative) seem to repeat in differing fields of study. For example, when one looks at growth, be it in biological

systems, economic systems, population growth, etc., patterns emerge that show consistent similarities to the point

that even are found in the mathematical models that describe these seemingly disparate fields. Bertalanffy states as

follows:

“A consequence of the existence of general system properties is the appearance of structural

similarities or isomorphisms in different fields. There are correspondences in the principles that

govern the behavior of entities that are, intrinsically, widely different. … To take a simple

example, an exponential law of growth applies to certain bacterial cells, to populations of bacteria, of animals or humans, and to the progress of scientific research measured by the number of

publications in genetics or science in general. The entities in question, such as bacteria, animals,

men, books, etc., are completely different, and so are the causal mechanisms involved.

Nevertheless, the mathematical law is the same (Bertalanffy, 1968, p. 33).



This characteristic of systems provide researchers and practitioners with a powerful tool/concept to better

understand complex system’s behavior, in that if confronted with a complexity in one system that is yet unresolved,

and knowledge of another system (possibly in another field of study) that has similar systems characteristics [iso

(same) morphology (structure)], the transfer of knowledge from one field or system to the other is possible and can

assist in the resolution of the problem. When an extensive search in the literature of isomorphology was conducted

(see Exhibit 4), an interesting result was found. The vast majority of articles are found in the quantitative realm. Very little research has been done in the qualitative area. In addition, little to no work has been done to explore

more fully the idea as presented by Bertalanffy in his seminal text General Systems Theory. To explore this concept

further and see how it might assist systems researchers and practitioners, let us look at both the quantitative and

qualitative forms of isomorphisms.

Exhibit 4: Literature Review of Isomorphology

Isomorphism = 17,520 results

Mathematics 10,613

Computer Science 2,079

Physics 2,072

Chemistry 694

Science Technology 137

Crystallography 118

Materials Science 111

Sociology 74

Biochemistry Molecular Biology 74

Geochemistry, Geophysics, & Geology 56

Polymer Science 42

Construction & Building Technology 2

Closed System Totals 16,072

Interdisciplinary 599

Business Economics 442

Engineering 266

Psychology 141

Open System Totals 1,448

With respect to quantitative isomorphology, abstract algebra is a sub-area of algebra which studies algebraic

structures such as, algebras, fields, groups, modules, rings and vector spaces. Algebraic structures along with

homomorphisms form categories; category theory is a formalism for studying and comparing different algebraic

structures. Textbooks in abstract algebra start with axiomatic definitions of algebraic structures and then establish

their properties. Historically collections of facts from various branches of mathematics acquired a common theme

and then became a part of abstract algebra.

Heinrich M. Weber in 1882 realized a connection in Leopold Kronecker work of ideal class groups and

Carl Friedrich Guass’s work on the structure of multiplicative groups which led to his work on finite groups. Joseph-

Louis Lagrange studied permutations in his 1770 paper “Thoughts on Solving Algebraic Equations”, his goal was

the understanding of solvability of algebraic equations. Paolo Ruffini then came along to study and develop the theory of permutation groups but the next step of formalizing the concept of a permutation group was Evariste

Galois in 1832. In 1854 Arthur Cayley wrote several papers on the abstract notion of a group, what he realized is



that a group need not be a permutation or even finite and can consist of matrices. Cayley would later establish that

any group is isomorphic to a group of permutations (Kleiner, 2007)

Abstract algebra became modern algebra around the start of the 20th century, mathematicians were no

longer interested in establishing properties of concrete objects, and intellectual rigor was now the driving force.

Interest into structure and classification of mathematical objects became the focus of research. Emmy Noether in her

paper “Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern” formulated the isomorphism theorems for homomorphisms in 1927 (Kleiner, 2007). The three isomorphism theorems, called

homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly. The three

isomorphism theorems are within the context of groups. The statements of the theorems for rings are similar, with

the notion of a normal subgroup replaced by the notion of an ideal (Allenby, 1991).

By abstracting away various amounts of detail, mathematicians have created theories of various algebraic

structures that apply to many objects and because of its generality, abstract algebra is used in many fields of

mathematics and science. Algebraic topology uses algebraic objects to study topologies and in physics, groups are

used to represent symmetry operations, and the usage of group theory could simplify differential equations.

The qualitative aspects of isomorphisms was not lost to the early pioneers of systems theory. The works of Forester

(1968) and many others reflect this. Bertalanffy states it clearly, “[s]ystem isomorphisms also appear in problems

which are recalcitrant to quantitative analysis but are nevertheless of great intrinsic interest (1968, p. 34). It must be

noted that much work has been done in the areas of innovation and idea generation. The work done in this area also spans a variety of scientific fields from the study of brain functions to the fields of business, engineering,

psychology, sociology and may more.

An example of this is the very popular work by Johanson (2006) titled “The Medici Effect.” “When you

step into an intersection of fields, disciplines, or cultures, you can combine existing concepts into a large number of

extraordinary new ideas.” (Johansson 2006, p.2) Johanson explored how historically the coming together of

different fields of study and practice generated a cross-feeding of ideas that was beneficial to all and resulted in

innovative solutions. Johanson aptly notes that the sharing of ideas can happen with-in and between fields and terms

these directional and intersectional ideas. “The key difference between a field and an intersection of fields lies in

how concepts within them are combined. If you operate within a field, you primarily are able to combine concepts

within that particular field, generating ideas that evolve along a particular direction – what I call directional ideas.

… When you step into the Intersection, you can combine concepts between multiple fields, generating ideas that leap in new directions – what I call intersectional ideas. (Johansson 2006, p.17) The innovations with-in a field,

Johanson terms “single-disciplinary incrementalism,” which though important, he believes does not generate the

leaps in knowledge and innovation that are seen with intersectional ideas. There is no doubt that what Johanson is

talking about, is closely related to what the systems theorist call isomorphisms.

Another researcher who has looked at the qualitative aspects that are crucial in the transference of ideas

between fields and the overcoming of barriers that impede idea generation, innovation and inventiveness is James

Adams (2005) in his also popular book “Conceptual Blockbusting.” Adams explores the mental, perceptual cultural,

environmental, intellectual, and expressive blocks that can impede the generation of ideas (see Exhibit 5). He

defines conceptual blocks as “[c]onceptual blocks are, mental walls that block the problem-solver from correctly

perceiving a problem or conceiving its solution,” (Adams 2005, p. 13). Though it is beyond the scope of this

research to delve deeply into the myriad of ways the conceptual block-busting , like Johanson’s intersectional ideas

approach, can help in developing isomorphologies, it is sufficient to know that these approaches, along with others not mentioned here, can help engineering managers and researchers.

Exhibit 5: List of Conceptual Blocks (Adams, 2005)

Perceptual Blocks: 1. Seeing what you expect to see – stereotyping

2. Difficulty in isolating the problem

3. Tendency to delimit the problem area too closely

4. Inability to see the problem from various viewpoints

5. Saturation

6. Failure to utilize sensory input

Cultural Blocks:

1. Taboos

2. Fantasy and reflection are a waste of time, lazy, even crazy

3. Playfulness is for children only



4. Problem-solving is a serious business and humor is out of place 5. Reason, logic, numbers, utility, practicality are good; feeling, intuition, qualitative

judgments, pleasure are bad

6. Tradition is preferable to change

7. Any problem can be solved by scientific thinking and lots of money

Environmental Blocks:

1. Lack of cooperation and trust among colleagues

2. Autocratic boss who values only his own ideas; does not reward others

3. Distractions-phone, easy intrusions 4. Lack of support to bring ideas into action

Intellectual Blocks & Expressive Blocks:

1. Solving the problem using an incorrect language (verbal, mathematical, visual)-as in trying

to solve a problem mathematically when it can more easily be accomplished visually.

2. Inflexible or inadequate use of intellectual problem-solving strategies.

3. Lack of, or incorrect, information.

4. Inadequate language skill to express and record ideas (verbally, musically, visually, etc.)

How to Approach Isomorphisms The knowledge of the existence of isomorphisms in systems assists in handling complexity and bridging knowledge

gaps to improve innovation. But how do we apply/find the isomorphologies that will help us with the complex

systems issues we are facing or might face in the future? The major problem faced by most scientist and engineers

(and engineering managers) is that when they are aware of the concept of isomorphologies, the immediate desire is

to find the mathematical isomorphology. This is not an incorrect desire, because ultimately, if the mathematical

model is found, it will assist in being able to better model the system behavior and thus predict the system’s

performance. As Edwards Deming use to often repeat, management is prediction. But rarely is the mathematical

isomorphology known. A more effective way is first get to know the systems, through the use of building process

flow diagrams, behavior over time graphs, causal loop diagrams etc. This leads to a more rich and profound

understanding of the behavior of the system and the critical characteristics that comprise the system. By doing this, one also will get to know the critical actors, elements, policies, equipment, economics, systems delays, etc. that are

essential to that specific system. Once the systems characteristics are known, they need to be classified as to if they

are summative or constitutive. Bertalanffy defines these terms as follows:

Types of Characteristics

The summative characteristics of an element are those which are the same within and outside the

complex; they may therefore be obtained by means of summation of characteristics and behavior

of elements as known in isolation. (Bertalanffy, 1968, p. 55)

Constitutive characteristics are those which are dependent on the specific relations within the

complex; for understanding such characteristics we therefore must know not only the parts, but also the relations. (Bertalanffy, 1968, p. 55)

This allows the practitioner (or researcher, or manager) to look for analogous behavior in other systems. Note there

are three types/levels in description of phenomena: analogies, homologies, and explanations. Again, Bertalanffy

provides us with some critical definitions that are of practical knowledge:

1.Analogies - i.e., superficial similarities of phenomena which correspond neither in their causal

factors not in their relevant laws.

2. Homologies - Such are present when the efficient factors are different, but the respective laws

are formally identical.

3. Explanation - i.e., the statement of specific conditions and laws that are valid for an individual

object or for a class of objects (Bertalamffy, 1968, p. 84-85).



Here we are faced with an important interim step that many are not aware of when trying to develop

isomorphologies. It is important to start with finding the analogous system to the system being studied/analyzed,

but the analogy is scientifically worthless. What is crucial is to test the analogy to see if in fact the analogous

system is also homologically similar to the system under study. If it is not, the analogy serves only as a useful

descriptor of the analogous characteristics, but will not be helpful in developing the mathematical models

(mathematical isomorphology) needed to better predict and manage the system. Bertalanffy states it this way, “If an object is a system, it must have certain general system characteristics, irrespective of what the system is otherwise.

Logical homology makes possible not only isomorphy in science, but as a conceptual model has the capacity of

giving instructions for correct consideration and eventual explanation of phenomena,” (p. 85).

This observation of analogy to homology, is critical in developing systems isomorphologies. But there is

still another barrier to overcome. How do you know that an analogy is potentially homological? Here we need to

go to the work of another famous thinker, Thomas Kuhn (2012) and his pivotal work, “The Structure of Scientific

Revolutions.” What is critical to look for in two analogous systems, is if they are commensurable. We know from

basic sciences, that two quantities (or systems) are commensurable if they can be measure in the same units. But

this can be extended beyond the units to the theoretical language or the systemic structure – here is the intersectional

leap. We know from the philosophy of science, that theories are incommensurable if they do not have a common

theoretical language. We can say the same if their systems structure is different. But if the systems structure is

similar (i.e. their growth patterns follow similar systems characteristics, it is very well possible that their mathematical forms will be homologous.

Summary The management of large complex systems is a major issue in modern society, yet the true problem is that the

current technical and scientific models society uses are no longer equipped to deal with these challenges. Historically the scientific process was a summation of individually observed phenomena which would sum to a

respective whole. On many occasions the whole was found to be more than the sum of its parts, these observations

are what Ludwig von Bertalanffy is discussing when he speaks of “wholeness” and the “problems of organization”.

Bertalanffy noticed patterns repeating in wide variety of systems and organizations, these structural similarities or

isomorphism he defines in his text (Bertalanffy, 1968). These isomorphisms could provide researchers and

practitioners a better understanding of the characteristics of systems and their behavior. Yet, when a literature

review was done over 90%o of journal articles fell into the quantitative realm of mathematics. This suggests little

research has been done to more fully flesh out Bertalanffy’s idea of isomorphisms in his General Systems Theory.

Seventeen thousand journal articles deal with isomorphisms, of those ten thousand deal with abstract

algebra where the roots of quantitative side lie. Another six thousand articles deal with the practical applications

isomorphisms from abstract algebra in fields such as; physics, materials science, chemistry and polymer science.

The remaining fourteen hundred fall into the isomorphisms of qualitative problems, but not many tools exist for identifying the characteristics of a system qualitatively and then using tools from the quantitative side of the field.

In this paper, we have presented an initial development of the concept of isomorphological analysis that goes

beyond and builds on the work done by the early systems theory thinkers. There is no doubt that more work need to

be done in the area of isomorphological analysis of systems. The work presented here provides a foundation for

future research.

References Adams, J.L. (2005). Conceptual Blockbusting: A Guide to Better Ideas, New York, Basic Books

Allenby, R.B.J.T. (1991). Rings, Fields and Groups, Butterworth-Heinemann

Anderson, V. & Johnson, L. (1997). Systems Thinking Basics. Waltham MA: Pegasus Communications, Inc.

Bertalanffy, L. von. (1968). General System Theory: Foundations, Development, & Applications. New York:

George Braziller.

Cantu, J. (2013). Isomorphisms within lumber and cotton futures with an empirical analysis of usage announcements

tied to storage. Working Paper Lubbock, Texas, Texas Tech University.



Davie, E. (2012). With The Financial Crisis, Did Capitalism Fail?. Forbes. Retrieved May 2013.

http://www.forbes.com/sites/realspin/2012/02/09/with-the-financial-crisis-did-capitalism-fail/

Faiola, A. (2008). The End Of American Capitalism?. The Washington Post. Retrieved May 2013.

http://articles.washingtonpost.com/2008-10-10/business/36925874_1_free-market-financial-system-capitalism

Forrester, J. W. (1968). Principles of Systems. Cambridge MA: Productivity Press.

Gray, W., & Rizzo, N. D. (1973). Unity through diversity; a Festschrift for Ludwig von Bertalanffy. New York:

Gordon and Breach Science Publishers.

Henninger, D. (2009). Is This The End Of Capitalism?. The Wall Street Journal. Retrieved May 2013.

http://online.wsj.com/article/SB123862779016180329.html

Isomorphism Theorems, (2012). Proofwiki.org. Retrieved May 22, 2013.

http://www.proofwiki.org/wiki/Isomorphism_Theorems

Johansson, F. (2006). The Medici Effect: What Elephants & Epidemics can Teach us about Innovation, Boston,

Harvard Business School Press.

Kleiner, Israel. (2007). A History of Abstract Algebra. BirkHauser

Kuhn, T. S. (2012). The Structure of Scientific Revolutions Chicago, University of Chicago Press..

Rosenbush, S. and M. Totty (2013). “How Big Data Is Changing the Whole Equation for Business,” March 8, The

Wall Street Journal.

Senge, P.M., (2006). The Fifth Discipline: The Art and Practice of the Learning Organisation, New York,

Doubleday Currency.

Schwaninger, M. (2006). “System Dynamics and the Evolution of the Systems Movement,” Systems Research and

Behavioral Science Journal, v. 23, pp. 583-594.

Sterman, J.D. (2000). Business Dynamics: Systems Thinking and Modeling for a Complex World. New York, McGraw-Hill/Irwin

Weaver, W. (1948). “Science and Complexity,” American Scientist, v. 36, n. 4, October, pp. 536-544.

Wittgenstein, L. (2011) Tractatus Logico-Philosophicus, New York, Taylor Francis.

About the Author(s) Mario Beruvides received his Ph.D. in Industrial and Systems Engineering from Virginia Tech. He holds an M.S.

degree in Industrial Engineering and a B.S. degree in Mechanical Engineering from the University of Miami. Dr.

Beruvides is currently the AT&T Professor of Industrial Engineering at the Department of Industrial Engineering and the Director of the Laboratory for System Solutions at Texas Tech University. His major areas of interest are

advanced economic analysis, management of technology, engineering management, white collar/knowledge work,

productivity and performance measurement, and operations and system engineering. He currently holds a

professional engineering license in the state of Texas. Dr. Beruvides worked for 10 years in private industry as a

project manager, product development manager and CAD/CAM engineer prior to his academic career.

Jaime Cantu is a graduate student and PhD candidate in Systems and Engineering Management in the Department

of Industrial Engineering at Texas Tech University. He received his B.S. in Electrical and Computer Engineering

and a M.S. in Systems and Engineering Management from Texas Tech University.

http://www.forbes.com/sites/realspin/2012/02/09/with-the-financial-crisis-did-capitalism-fail/

http://articles.washingtonpost.com/2008-10-10/business/36925874_1_free-market-financial-system-capitalism

http://online.wsj.com/article/SB123862779016180329.html

http://www.proofwiki.org/wiki/Isomorphism_Theorems



Appendix First Isomorphism Theorem

1. Groups

Let ϕ:G1→G2 be a group homomorphism.

Let ker(ϕ) be the kernel of ϕ.

Then:

Im(ϕ)≅G1/ker(ϕ)

where ≅ denotes group isomorphism.

2. Rings

Let ϕ:R→S be a ring homomorphism. Let ker(ϕ) be the kernel of ϕ.

Then:

Im(ϕ)≅R/ker(ϕ)

where ≅ denotes ring isomorphism.

Second Isomorphism Theorem

1. Groups

Let G be a group, and let:

H be a subgroup of G N be a normal subgroup of G.

Then:

HH∩N≅HNN


2. Rings

Let R be a ring, and let:

S be a subring of R.

J be an ideal of R.

Then: (1):S+J is a subring of R

(2):J is an ideal of S+J

(3):S∩J is an ideal of S

(4):SS∩J≅S+JJ


This result is also referred to by some sources as the first isomorphism theorem.

Third Isomorphism Theorem

1. Groups Let G be a group, and let:

H,N be normal subgroups of G

N be a subset of H.

Then:

H/N is a normal subgroup of G/N

where H/N denotes the quotient group of H by N

G/NH/N≅GH


2. Rings Let R be a ring, and let:

J,K be ideals of R

J be a subset of K.

Then:



K/J is an ideal of R/J

where K/J denotes the quotient ring of K by J

R/JK/J≅RK

where ≅ denotes ring isomorphism.

Fourth Isomorphism Theorem

Let ϕ:R→S be a ring homomorphism.

Let K=ker(ϕ) be the kernel of ϕ.

Let K be the set of all subrings of R which contain K as a subset.

Let S be the set of all subrings of Im(ϕ).

Let fϕ:P(R)→P(S) be the mapping induced on P(R) by ϕ.

Then its restriction fϕ:K→S is a bijection.

Also:

(1):fϕ and its inverse both preserve subsets.

(2):fϕ and its inverse both preserve ideals: If J is an ideal of R, then fϕ(J) is an ideal of S

If J′ is an ideal of S, then f−1ϕ(J′) is an ideal of R

ISOMORPHOLOGICAL ANALYSIS: THE THEORY OF IT ALL

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