Even though all the systems in the operator hierarchy are characterized by a first-next closure they are not identical. Their closures show differences that can be recognized as various types. Below I discuss how I grouped the different closure types. Besides this I decided that it was practical to recognize evolutionary layers and specific operator types. Inspiration for names was sometimes found in the literature, if not, I had to define new names. The above choices are explained in more detail in the following sections:
Every single closure step in the operator hierarchy that repeats an existing closure type is considered a 'minor transition'. The name ‘minor’ is chosen because these transitions involve but a single first-next possible closure. The following steps are considered minor transitions: the formation of the quark confinement and the formation of the atomary nucleus, the electron shell, the molecules, the autocatalytic sets, the cell membranes, the eukaryotic cells, the bacterial and eukaryotic multicellulars, the neural hypercycles and the sensory interfaces. Please note that the minor transitions that create the hypercyclic systems and interfaces represent transitions (and system states) that not always occur independently in nature. The separation of these steps in the operator hierarchy represents a theoretical construction that is a consequence of the focus on first-next closure and corresponding closure types.
Major transitions are all transitions in the operator hierarchy that create a new closure type. With the exception of the superstring and quark-gluon plasma, all major transitions involve a special pair of minor transitions. The first closure of such a pair creates a new pre-operator hypercyclic set, and the second closure gives the new operator structural unity by creating an interface. The subsequent closures in these pairs are regarded in the operator hypothesis as the first and second step of a major transition. In the operator hierarchy six major transitions can be recognised.
The first major transition creating the superstrings. Required are two closures: the rolling up of a field-sheet to a long tube and the closure of the tube to a finite system; the torus of the closed string.
The second major transition creating the superstring hypercycle. Required are two closures: the superstring cycle (large superstring splitting off and reabsorbing a small superstring) and the hypercyclic arrangement of minimally two of such cycles in a hypercyclic arrangement.
The third major transition creating the hadron. Required are two closures: the superstring hypercycle and the confinement of the quarks at low energies by gluon interactions.
The fourth major transition creating the atom. Required are two closures: the nuclear hypercycle and the electron shell.
The fifth major transition creating the cell. Required are two closures: the catalytic hypercycle and the cell membrane.
The sixth major transition creating the memon. Required are two closures: the neural hypercycle and the sensory interface.
Please note that although both steps of a major transitions are regarded as separate events from the point of view of system organisation, they may –and at elevated levels in the operator hierarchy almost always must- have occurred together during the natural processes that allowed the emergence of these systems.
The operators that have been formed via major transitions are called 'primary systems'. The primary systems in the operator hierarchy are the superstring, the quark-gluon plasma, the hadron, the atom, the cell and the memon. The superstring and the superstring hypercycle are no operators (their construction does not involve a hypercycle with interface) and are regarded as primary interface and primary hypercycle, respectively. All the other primary systems can be regarded as primary operators.
The formation of every primary interface, primary hypercycle and primary operator ends a shorter or longer sequence of minor transitions. In the operator hierarchy, any primary system with all the system types based on it until the next primary system, are regarded as to belong to the same hierarchical layer. As a consequence the operator hierarchy shows the following layers:
Individual superstring layer. At the super string layer we only find individual super strings.
Superstring hypercycle layer. At the super string hypercycle layer, we find the quark plasma and the gluon interface.
Hadron layer. At the hadron layer we find the protons, the neutrons and a large number of unstable mesons (two interacting hadrons) and baryons (three interacting hadrons). Furthermore the hadron layer includes the nucleus and the envelope of the electron shell that forms a mediating interface around it.
Atom layer. At the atom layer we find the atoms, the molecules, the autocatalytic set and the chemical envelope that forms a mediating interface around it. Cell layer. At the cell layer we find the pro- and eukaryotic cells, the multicellular forms hereof, the hypercyclic neural CALM networks and the sensory organs that form a mediating interface around.
Memon layer. At the memon layer we find the cellular memons. In the future also technical memons may become part of this layer.
Every primary system in the operator hierarchy defines a new closure type. This closure type characterizes all the new systems in the associated layer and can be found to recur at higher levels in the operator hierarchy in the form of a minor transition. In order of increasing complexity:
The first closure type: the interface. The superstrings are assumed to show the most basal closure. This involves the interfacing between an interior and an exterior world. This interfacing requires the transformation from endless field-sheets to open or closed superstrings. Open strings are only partly closured. They are assumed to take the form of tubes that stick out from a field-sheet or of arches that are linked at two different places to a field-sheet. Closed strings are fully 'closed' in the sense that they show a second order rolling-up; first from sheets to tubes, than from tubes to toruses.
Without an interface any pre-operator is not more than a sandstorm in a desert. It is the interface that lends a system its spatial limits. The only system type in the operator hierarchy that show the interface as the only closure type is the superstring. The interface closure re-occurs on higher levels in the operator hierarchy in the form of the confinement of quarks, the electron shell of the atoms, the membrane surrounding the cell and the sensory interface mediating the contact of neural networks with their environment.
The second closure type: the hypercyclic interaction structure. Single superstrings create a world full of superstrings. Interesting interactions do not occur until strings start changing each other’s state by the exchange of charge-carrying particles. When a superstring emits a particle and re-absorbs it, this forms a small reaction cycle. When two of such cycles link to form a second order reaction cycle, this marks the emergence of a superstring hypercycle. As long as energy levels in the early universe were high, superstring hypercycles may have formed naturally in the hot primordial superstring-soup. Such hypercycles did not show a physical boundary, but existed as temporary whirlwinds. Real physical units were formed later, when expansion of the universe caused lower temperatures and the quark hypercycles were confined by the forces transmitted by the particles that were exchanged in the hypercycle.
In the operator hierarchy, systems in which first-next closure involves the emergence of a hypercycle are called ‘pre-operator hypercyclic sets’ or, in short, 'pre-operators'. Examples of pre-operators are: the superstring hypercycle, the atom nucleus, the autocatalytic set of the cell, and the CALM network of the memon.
The third closure type: the multi-particle system. The system that is meant here shows recurrent interactions between same-type operators. In the operator hierarchy, operators showing this property are called 'multi-operators' (derived from multi-unit operators).
The multi-operator closure type occurs for the first time in the hadrons. It re-occurs in the multi-atoms and the pro- and eukaryote multicellular organisms.
Operators. In the operator hierarchy, the systems that show a closure type with the minimum complexity of multiness are called operators. The operators are: the hadrons, the atoms, the molecules, the cells, the multicellular prokaryotes, the multicellular eukaryotes and the memons. The word operator was chosen to indicate that this kind of systems operates in the environment on the basis of system own dynamics that include a specific hypercycle type and interface. This interface either surrounds a single unit or creates structural unity in the case of multiness.
The fourth closure type: the hypercycle mediating interface (HMI). HMI operators show an interface layer that mediates the interactions of the –now- internal hypercyclic information, with -what has just become- the outside world.
The HMI-closure occurs for the first time in the atoms as the result of a major transition. It can be found to re-occur at higher levels, where it occurs as the result of a minor transition. Examples of HMI operators are the atoms and the eukaryote cells.
The fifth closure type: the structural (auto-)copying of information (SCI). SCI operators show the capacity of autonomously re-recreating the major aspects of their structure and in this way replacing degraded elements (maintenance), producing more elements (growth) and –if enough material and energy is available- creating an offspring (reproduction).
This property occurs for the first time in the cells, where the autocatalytic set of enzymes defines and maintains the structure of the cell and its membrane. If every enzyme in the autocatalytic set does what it 'has to do', a full cycle of enzyme reactions re-creates every enzyme in the set at least once. In this way, the cell has created a full structural copy of its basal enzymes, which can be regarded as the 'information' of the cell. The SCI property may re-occur in the future in the soft-wired memon.
The sixth closure type: the structural auto-evolution (SAE). SAE operators show the capacity to change the structure of their hypercyclic information autonomously. In the memon this closure type is reflected in by the continuous changes of the settings of a neural network as the result of thinking. The hardwired memon is the only example so far of an operator with the SAE property.