Accurate definitions improve the precision and communication of science.Sloppy definitions lead to the development of sloppy theory.A lack of definitions leads to no science at all.(Gerard Jagers op Akkerhuis).
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Why it is useful to browse this glossary...


The operator theory represents a new point of view. This implies that a number of new concepts had to be introduced and defined. Furthermore, as a consequence of the logic of the operator theory certain existing concepts were re-evaluated and sometimes re-defined. For all these reasons it was considered appropriate to offer the readers of this website an overview of the concepts used.

Analogy

Analogy implies a mapping of knowledge from one domain (the base) to another (the target) such that a system of relations that holds among the base objects also holds among the target objects (Gentner & Jeziorski 1993).
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Building block

I sometimes use the word 'building block' or 'unit systems' for indicating the particles that in the operator hierarchy are named operators. I always have in mind a system in which the elements create a physical and functional unity on the basis of first-next possible closures.
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Catalytic hypercycle

see Hypercycle
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Closure

In the context of the operator hierarchy, closure relates to a state of matter that is the result of a self-organisation process and that shows a closed topology with respect to structure, process or the combination of both. Closure as a state shows a close association with underlying mechanisms because it also is used to refer to the closing process producing the closed state.
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Closure level

In the context of the operator hierarchy, every first-next possible closure adds one closure level.
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Closure, structural and functional

Functional closure is defined as a closed cycle of processes that does not cause a physical mediating layer. Specific forms of functional closure are the catalytic reactions involved in an autocatalytic set, the pion exchange between protons and neutrons in the atom nucleus and the plasma strands connecting the cells of multicellulars. In contrast, the definition of structural closure demands that interactions create a physically closed topology. In the operator hierarchy, the focus is on a special form of structural closure, namely the structural closure that creates a physical boundary mediating a contained, first-next possible, hypercyclic process. Specific examples of structural closure in the operator hierarchy are the electron shell around the atom nucleus, the cell membrane of a single cell, and the connected cell membranes of multicellular organisms and the sensory interface of the memon.
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Closure dimensions

A closure dimension focuses on similarity in structural closure, functional closure or, when present, a combination of both. The naming of a closure dimension is directed after the most complex organisational property of the system as allowed by structural or functional closure or, when possible, the combination of both. The following closure dimensions are acknowledged in the operator hierarchy: 1. the interface dimension, introduced by the fundamental particles, 2. the hypercyclic dimension, introduced by the quark-gluon plasma, 3. the multi-state dimension, introduced by the hadrons, 4. the hypercycle mediating interface dimension (HMI), introduced by the atoms, 5. the structural copying of information dimension (SCI), introduced by the cell, 6. the structural auto-evolution dimension, introduced by the hardwired memon
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Complexity

Complexity is a concept that refers to our inability to give a straightforward description of the properties of an entity. Complexity relates to the number and inter-relatedness of the elements of an entity and to the patterns and sub-patterns in its states and dynamics.
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Complexity, functional

I regard as the functional complexity of an entity 'all forms of internal or external interactive processes that are supported by the internal organisation of the entity’. Both a large, badly organised structure and a minute, well-integrated structure may allow an entity to show the same functional complexity.
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Complexity of a closure dimension

The complexity of a closure dimension is based on the number of major transitions that were required to construct it. According to the operator theory, the complexities of the closure dimensions that are known to exist are: 1 for the fundamental particles, 2 for the first-next possible closure creating hypercyclic interaction systems, 3 for the hadrons, 4 for the atoms, 5 for the cell and 6 for the memon.
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Complexity of operators

The operator hierarchy measures the complexity of an operator by means of the number of first-next possible closure levels that underlie its construction.
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Confinement

Confinement binds quarks in hadrons in a condensation-like process. It is the result of the strong force, also called the color force, which is conveyed by the exchange of gluons between quarks. At higher temperatures/energies the quarks become increasingly heavy and the confinement relatively weaker, which is referred to as the asymptotic freedom of the quarks.
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Corrupt hierarchy

If, in any hierarchy, one or more steps/layers do not comply with the hierarchy rule used for creating the ranking, I consider this a corrupt hierarchy. For example the hierarchy 1, 2, 4, 5, 6 ... is corrupt in relation to the rule that for every N, the next element in the ranking must be exactly N+1 (it lacks the 3). For the same reason the ranking 1, 2, 3, 3.2457, 4 ... is a corrupt hierarchy (the number 3.2457 is superfluous). And also the ranking atom, molecule, bacterium, population is a corrupt ranking. The reason is that the population is not a particle, like the other elements, and cannot be ranked in the same hierarchy for this reason.
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Dissipative system

The organisation of a dissipative system results from selforganisation processes caused by the degradation of free energy gradients in the environment. Examples of dissipative systems are waves (degrading wind energy), whirlwinds (degrading a pressure gradient in the atmosphere) and organisms (degrading a chemical and/or radiation energy gradient). As a thermodynamic equilibrium implies a random distribution of elements and interactions, the organisation of elements in a dissipative system is regarded as to be 'far from equilibrium'. Dissipative systems do not contradict with the laws of thermodynamics, because any increase in organisation inside a dissipative system (which implies a decrease in entropy) is driven by a free energy gradient and causes at least as much entropy in the system as is lost during the organisation process.
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Efficiency

I regard efficiency as the lowest resource use, when two processes are compared leading to a functionally equivalent product or achieving a functionally equivalent goal.
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Elementary closure

An elementary closure represents the lowest complexity realisation of its type with respect to the system’s structure, process, or when possible, the combination of both. Elaboration and repetition of the same closure type does not affect the elementary closure.
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Elementary particle

In physics, the concepts of elementary particles and fundamental forces are used to indicate matter particles and force carrying particles of the standard model, respectively. The operator hierarchy suggests that one may also regard all operators as elementary particles with respect to their highest level first-next possible closure. Defined this way, only the particles of the standard model are truly elementary (which I prefer to consider fundamental) in the sense that they presumably show no substructure of smaller particles.
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Emergent property

An emergent property is a group-property that results from interactions between entities which individually do not show this property. For example water flow and waves are emergent properties of interacting water molecules (a single molecule cannot form a multi-molecule wave). Used in this general way, emergent properties arise from almost any interaction between separate elements.
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Entropy

Entropy is the opposite of free energy. Because in a closed system the average free energy can never increase, the entropy can only in crease. An increase in entropy corresponds with the natural pathway of systems from a high to a low energy state. Examples are the reaction of chemicals to low energy products, the transformation of light to heat, the falling of particles in a gravitational or electric field and the change of a system towards the state that shows the highest number of possible microstates. For closed systems Boltzmann has shown that the chance on a decrease in entropy becomes infinitesimally small for systems consisting of many elements. It is an important aspect of nature that in local parts of open systems entropy may decrease, the local system becoming more organised. This does not violate the laws of thermodynamics, as long as the entropy decrease in the local system is driven by a free energy gradient which is decreased in the process.
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Evolution

Any process based on the repeated alternation of diversification and selection steps. Diversification implies the production of entities the structure of which differs from that of their precessors (productive diversification). Selection implies a form of structure-dependent performance of the entities causing a gradient in their evolutionary success (the cause of selection). Evolutionary dynamics can be observed at all levels of the operator hierarchy (e.g. elementary particles, prions and viruses, organisms, technical memons) but also things that are not operators may show evolution, e.g. strings of computer code, bee colonies, or neuron states produced in relation to the invention and weighing of different scenarios in a decision making process. The evolution of neuron states precedes that of tools (cars, windmills and the like) with which it co-evolves.
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Evolutionary success

Evolutionary success is a measure that depends on the relative performance of entities with respect to: 1. the stability of their internal organisation, 2. the stability of their functioning in interactions with other entities and/or forces, 3. their capacity for productive diversification, which in relation to operators ultimately includes the capacity to produce the following first-next possible closure. Without the latter, evolution cannot proceed towards higher level operators.
The evolutionary success of dissipative systems, such as living beings, is related to the relative success with respect to resource dominance, when compared to other systems that take part in the process. The longer an organism exists and the better it functions (which may include reproduction), the more resources it (and its offspring) will use/dominate. Organisms which at a given moment are the best resource dominators, may rapidly lose from others which get slowly better over time, whilst both strategies will be beaten by organisms that can increase the rate of their improvement, especially if they can increase the acceleration of this rate. Strength in competition as well as strength in cooperation may increase evolutionary success. The duration of observations determines which of the above aspects of evolutionary success can be observed.
A higher evolutionary success not necessarily implies a more complex structure. For example, in dynamic environments with a high random mortality, simple, rapidly reproducing entities will prevail.
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Exhaustive closure

To assist in identifying the new type of elementary closure, I suggest identifying the ‘exhaustive closure’ in the system at the one lower closure level. A system shows exhaustive closure, if there is no remaining potential for elaborating the structural and dynamic aspects of the elementary closure type, because any further development will cause the construction of a new elementary closure type. While the elementary closure of the system remains the same, the state of exhaustive closure allows for a new functionality.
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First-next possible closure (FNPC)

First-next possible closure is defined as follows: given any system A that shows first-next possible closure, the next first-next possible closure creates the least complex system type above A that shows a new type of elementary closure based on A and, when required, any highest level system type possible below A that shows first-next possible closure. This definition is inherently recursive because a system showing first-next possible closure produces a more complex system showing first-next possible closure. Because a first-next possible closure is always built from systems showing a preceding first-next possible closure, the recursive definition does not lead to logical loops.
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Force

A force is the result of an interaction. For example the repulsion between electrons (electromagnetism) is caused by the interaction of two electrons by means of the exchange of virtual photons. Likewise, the exchange of gluons causes the attractive force between quarks. In principle the equating of force and interaction works al all levels of the operator hierarchy. For example the interaction between a predator and its prey can be regarded as a ‘predation force’.
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Fundamental-particle

A physical particle that shows no substructure of smaller particles. In the operator hierarchy: the system showing the lowest level first-next possible closure (see also elementary particle).
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Hardwiring

This concept is used to describe a neural architecture that is based on physical 'wires' between neurons, for example in the shape of dendrites. The difference with 'softwiring' is that softwired connections between neurons do not exist as physical wires, but as entries in computer memory telling which neuron is connected to which other neurons and what is the strength of the connection.
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Hierarchy

Hierarchy assumes a ranking of entities. Formally, elements in a hierarchy meet the following two demands (Simon 1973): Irreflexivity: A can never hold a hierarchical position below itself. Transitivity or non-transitivity: if A has a lower hierarchical position than B, and B has a lower position than C, then A does have, or does not have a lower position than C. From irreflexivity and (non-)transitivity follows Antisymmetry: if A has a lower hierarchical position than B, then B cannot hold a lower position than A. An example of a transitive hierarchical relationship is: A is ancestor of B, B is ancestor of C. In this case A is also ancestor of C. An example of a non-transitive hierarchy is: A is a child of B, B is a child of C. In this case A is not a child of C.
Bottom-up hierarchies arise when the behaviour of lower level elements integrates them in larger assemblies (e.g. when atoms create a molecule). Top-down hierarchies may have different origins, for example 1. when social agreements define a top figure (marshal/manager) and allow it to direct groups of (now) lower level figures (soldiers/workers) or 2. when after the formation of a functional unit, internal differentiation leads to internal hierarchies of functional relationships (see also 'Penultimate level').
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Hierarchical layer

Operators and pre-operator interaction systems that are based on the same primary system and do not yet represent the next major transition, are regarded as to belong to the same hierarchical layer.
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Hypercycle

The use of this concept in the literature generally refers to the enzymatic hypercycle as discussed by Eigen and Schuster (1977). With respect to the operator hierarchy, the concept is used for any second order cyclic process that is related to a first-next possible closure.
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Information

My favorit definition of information (Latin 'informare'='to put into a certain form') is that of 'data with a meaning in a context' (Checkland and Scholes 1990). Next comes 'everything that reduces uncertainty' (Gates). Shannon and Weaver 1949 introduced a structural definition that assumes a well defined context and focuses on positions in a code string, having the value of 0 or 1. Five positions can code for 2x2x2x2x2 =32 states. Every position that is lost halves the information content. One can also focus on the change in a receiving system. Now, a simple message can be highly informative (e.g. an empty letter confirming marryage) whilst complex coding (Chinese, Arabic or Latin scripture) is uninformative to analfabetics and people not grasping the content. Personally I find a natural context for information in the hypercyclic interactions inside certain operators. Every part in a hypercyclic set of interactions represents a piece of information in the context of its contribution to the functioning of the set as a whole.
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Information and complexity

Information has a structural and a semantic aspect. The structural aspect relates to the number of elements and the different states they can code for (e.g. Shannon 1948). The semantic aspect relates to the interpretation of a coding. In principle, there is no limit to the semantic complexity of a message, because even a simple coding may relate to a highly complex interpretation. The functional elements of a coding (the ‘bites’, 'characters', 'genes', 'symbols', etc.) normally show an intermediate complexity. Too low complexity would allow too little differences between the elements. Too high complexity would make the coding/decoding difficult.
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Intelligence

In the context of the operator theory intelligence is an emergent property brought about by the functioning of an operator with a complex enough memic architecture. Intelligence is associated with the autonomous capacities to observe, learn, make internal representations, predict, evaluate and act. Intelligence requires sensors for contacting the world and a neural network for creating internal multiple-channel representations, the latter offering a natural source for association and creativity. Intelligence shows a gradual increase with increasing neural complexity. For intentional behaviour, the representations must be available for the evaluation of which actions are most effective for reaching the mental concept of a certain goal. Intelligence, therefore, has much to do with the art of defining the right multi-factorial problems and choosing appropriate aspects for evaluation.
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Interaction

An interaction is an event that involves at least two entities and causes a change in one or more parameters of both systems. This excludes as interactions all events in which one element remains unchanged. For example, if someone listens to a radio program, this seems an interaction between the person speaking on the radio and you. But the person speaking is not affected in this process, and therefore it is not an interaction according to the above definition.
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Interaction system

An interaction system consists of interacting operators but lacks the closure(s) that would allow it to be regarded as an operator. The world is swarming with interaction systems, including footballs, stars, cars, water, populations, society, companies, etc. Of course also operators consist of interacting elements. Yet, like Orwell made his pigs advocate that 'all animals are equal, but some are more equal than others', I advocate that all systems are interaction systems, but that the operators show such special interactions, that it is wise to regard them as 'more equal than others'. Accordingly I assign them to a special subset. Although they play an important role in the operator hierarchy, the pre-operator hypercyclic sets and interfaces are not operators, but interaction systems.
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Life as a concept that refers to the presence of the closures defining the operators of the kind organism (a system-based definition).

Where 'living' focuses on dynamics, 'life' focuses on organisation. In this context, life can be defined as a generic concept that focuses on the presence of the typical closures that define all the different types of organisms (as I have explained in "The pursuit of complexity" 2012) This is not a circular definition, (life->organisms and organisms->life) because the definition of the organism that is used, is based on the ladder of the operator hierarchy (so the definition runs from ladder->organisms->life, or also, from ladder->life). The different subsets of organisms that can be recognised, are associated with different subsets of the definition of life, for example autocatalytic life (uni- and multicellular plants, fungi, sponges, protozoa, etc.) or memic life (including all memic operators, regardless whether they are based on a cellular or a technical construction). One may also distinguish the subsets of organic life and technical life.
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Life, the meaning of

The meaning of life is in the first place a personal subject. Besides this, the operator theory offers two general ways of thinking about the meaning of life. The first meaning is associated with the functioning of organisms as individual operators. Every organism has to stay alive and memic organisms additionally care about their daily satisfaction. Survival is a quite objective criterion. The validation of daily satisfaction depends on the individual judgment of what is 'satisfying'. The second meaning of life is associated with the contribution of all operators to evolution. On the one hand, this implies an involuntary act. Whatever you do, you are always part of the large organization process that is driven by entropy production. On the other hand, increasing insight in the overall structure of evolution may have the implication that a person decides to realise a directed contribution to evolution, for example by wilfully constructing a technical memon
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Living

Living can be defined as the dynamic activity of systems that comply with the definition of life. A bacterium, for example, may or may not be living, depending on whether it is active or frozen/dried.
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Major transition

Any first-next possible closure creating a primary system is regarded as a major transition (not to be confused with the major evolutionary transitions of Maynard Smith and Szathmáry, 1995a, 1995b)
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Meme

Dawkins (The selfish gene, 1976) defines the meme as: '... new replicators, which I called memes to distinguish them from genes, can propagate themselves from brain to brain, from brain to book, from book to brain, from brain to computer, from computer to computer.' Dawkins' memes are not structural analogues of genes, because any pattern of replicable information in brains, books, etc. could be a meme, whilst not any pattern of replicable information in cells is regarded a gene. To solve this, I propose the use of different meme concepts: 1. Abstract meme (A-meme): abstractions communicated between memons (melodies, stories, and theories), 2. Physical memes (P-meme): physical models of thoughts (written language, artwork, buildings), 3. Functional meme (F-meme): the actual neural network, or part of it, harboring information, 4. Coding meme (C-meme): C-memes code for neural network architecture just as genes code for catalytic molecules.
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Minor transition

Any first-next possible closure creating a system that shows an already existing closure dimension, is regarded a minor transition.
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Model

I regard as a model any entity that represents another entity. This may involve mental models that represent parts of our environment, scaled physical models that represent larger or smaller physical or mental originals, and simulation models that represent aspects of real world systems.
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Modularity

Modularity refers to the construction of larger systems from small functional and/or structural subassemblies or 'modules'. Herbert Simon illustrated the evolutionary merits of modularity in the parable of the watchmakers Tempus and Hora, making watches consisting of 1000 parts each. Tempus made his watches bit by bit. When disturbed, he had to restart the entire assembling process. Hora put his watches together from subassemblies of 10 parts each. 10 subassemblies were put together in a larger subassembly and 10 larger subassemblies constituted the whole watch. Hence, when Hora was disturbed he only lost the specific subassembly-job he was involved in. Simon calculated that if customers disturbed Hora and Tempus on average once in hundred assembly operations, Tempus would need about four thousand times longer to assemble a watch.
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Natural systems

I regard as natural systems all elements of the universe of which the physical existence allows us to interact with it and describe static and/or dynamic properties (qualities and quantities) in a consistent way. More/better measurements should lead to an increasing detail of our representation of the system. Kragh (Aktuel Naturvidenskab 1999) has said this in the following way. 'It is difficult to understand, how measurements of important scientific parameters normally can become more and more precise, and how knowledge shows a general consolidation and refining, if these measurements and knowledge do not concern something that really exists in nature'. In the context of the operator theory, natural systems include the neural configuration and neural dynamics that 'carry' a thought, for example of the number pi, but do not include the word 'thoughts' or the number ‘pi’.
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Operator

Although other uses of the word exist (for example in mathematics and communication companies), the operator theory uses the word operator as a generic indication of all 'particles', both physical particles and organisms. An operator is defined as a system type that shows first-next possible closure with a closure dimension of at least 3. This implies that operators can also be defined as those systems showing first-next possible closure that are not pre-operator interaction systems. All operators that are currently known to exist are: the hadron, the atom, the molecule, the (bacterial) cell, the eukaryote cell, the bacterial and eukaryote multicellular and the neural network organism, called the ‘memon’ in the operator hierarchy. Frequently, I will use a more liberal interpretation, in which I also accept the quarks as a type of operator. This is done to indicate that the ranking starts with quarks.
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Operator hierarchy

The operator hierarchy is the structuring of all operators and pre-operator interaction systems on the basis of first-next possible closure and closure dimensions.
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Operator hypothesis

The operator hypothesis states that the fundamental limitations imposed by first-next possible closure act as a strict mould for particle evolution. If this hypothesis holds, it implies that evolution at meta-level is a predictable process.
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Operator theory

The operator theory is the theoretical framework that offers the context for the operator hierarchy and the operator hypothesis.
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Organism

The complexity ladder of the operator theory can be used to define the organism concept independently of the definition of life: all operators that show a complexity equal or higher than that of the bacterial cell (as an operator type) are organisms. The definition includes future technical memons as organisms and as life-forms. The complexity type of any organism is determined by its typical (level related) first-next possible cosure. It is the presence of a high enough typical closure that characterises an organism as representing life. The loss of the typical closure implies that the organism has died away from the indicated level. By using the ladder of the operator hierarchy as a basis for the definition of the organism, and using the organism as the basis for the definition of life, any circularity in the definition process is prevented.
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Penultimate level (Law of the branching growth of the …)

This law by Turchin (1977) states that: '...after the formation, through variation and selection, of a control system C, controlling a number of subsystems Si, the Si will tend to multiply and differentiate. The reason is that only after the formation of a mechanism controlling the Si it becomes useful to increase the variety of the Si.
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Pre-operator-interaction-systems

Systems, caused by first-next possible closure, that selectively create an interface or a 2nd order cyclic interaction (the ‘hypercycle’) as closure dimension, are called pre-operator interaction systems.
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Primary system

An operator (or pre-operator interaction system) that shows a new closure dimension is regarded a primary system.
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Quanta of evolution

Quantification normally involves simple variables such as length, weight, color (wavelength), etc. A more complex variable to quantify is the evolutionary complexity of a system, for which quanta of evolution have been proposed (Turchin 1995, Heylighen, Joslyn and Turchin 1995). In a similar way as a yardstick uses numbers as the quanta for measuring length, the operator hierarchy quantifies the level of organisation of a given operator by counting the number of first-next possible closure steps that were needed for its structure.
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Recursion of a definition

A definition is recursive if it explains new entities Xn in terms of old entities X. The operator hierarchy is recursively defined, because lower level operators are used to define higher level operators. Note that a recursive definition does not suffer from a logical loop.
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Resource

A resource is something a living operator needs to enable certain aspects of its existence.
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Resource-dominance

Resource dominance is defined as the capacity of organisms to gain dominance over resources. There are different aspects of resource dominance that are of evolutionary importance. First, there is the amount of resources that an organism can dominate by degrading them. After degradation they are no longer available for other organisms. Second, there are resources that an organism may dominate, without that it uses them. Again they are no longer available for other organisms. There exist various resource dominance strategies by which an organism can increase its evolutionary success.
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Softwiring

In softwired memons, the neural network is based on computer files that keep track of the neurons involved, the interactions they show with other neurons, and the types and connection strengths of the interactions.
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System

The system concept relates to entities which can be endowed group-wise functionality because of past, present or future interactions and/or relationships. The entities that by their relationships define the group are considered as the elements of the system. The system concept applies to the universe or a subset of it (the latter including mental states and via these, all aspects of our imagination). Frequently, the entities and their interactions are chosen in such a way that they define, include or relate to some kind of physical system limit.
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System type of closed systems

Two systems that are created by the same first-next possible closure are considered to be of the same system type.
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Tool

In the context of the operator hierarchy I define as a tool 'any idea or physical object that a memic operator may use to help it manipulating the world to its liking'. Consequently there are mental tools, such as mathematics, and physical tools, such as a stick, a hammer, a car, a house, a factory and a computer. It is interesting to realise that the operator hierarchy predicts that the next level memons are not based on cells but on technical hardware. Consequently, the process of reproduction (giving birth), that is so important for the evolution of organisms based on one or more cells, will be replaced by the process of producing an 'offspring-tool'.
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Topology

Topology studies qualitative aspects of geometrical structures. In stead of asking how big a certain thing is, it focuses on other aspects, such as: does it have any holes in it; is it connected together, or can it be separated into parts.
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Transition

If a system changes from state A to state B this can be regarded as a transition of the system between these states. For any particular transition the property that is going to be looked at as well as the states that can be recognised have to be defined.
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Type

A type is used as a grouping of entities according to a common property. Using types of types leads to the creation of a type-hierarchy in which lower-level types show properties that are grouped into higher-level types. For example the higher-plant-type and the animal-type can be grouped within the multicellular-organism-type. Frequently used types in relation to the operator hierarchy are: 1. process type: a comparable aspect that can be recognised in the dynamics of different entities. 2. System type (general): the type of a system is defined by comparable aspects with respect to a. the rules involved in the selection of elements, b. the properties defining the elements and c. the relationships that may exist between the elements, and 3. Closure type: a closure type is defined by the presence of comparable aspects with respect to the type of elements and the type of mechanisms causing the closure.
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Typical closure

The typical closure is the closure type that is typical for any operator at a given level. A given typical closure equals a given first-next possible closure. For example, the typical closure of the bacteria consists of the combination of the cell membrane and the autocatalytic set of molecules.
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