Figure 1: The operator hierarchy
In the preceding paragraph it was shown how topological rules (structural and functional closure) allow for the construction, from the ground up, of a hierarchy of so called 'operator kinds'. The particles that are linked to these kinds are called 'operators'. Interestingly, the hierarchy of operator kinds suggests an internal logic. This logic may well represent a novel discovery in science (and a new paradigm), which is the reason that we discuss it in detail in this page.

For more references and information about this topic see:
The operator hierarchy. A chain of closures linking matter, life and artificial intelligence.
Jagers op Akkerhuis G.A.J.M. (2010). Alterra Scientific Contributions 34

The operators and the operator hierarchy

'Operator' is a generic name for any object, e.g. an atom or bacterium, that correspond to one of the organisational kinds in Figure 1. Every such kind is based on the presence of closed ('circular') topology, which can be of a structural and/or functional kind. Typical for every such topology is that it represents the first occuring next opportunity for the new kind of closure. Accordingly one obtains a mechanism by which the highest level operators so far allow for the next, new kind of closure. Because this mechanism is repetitive, the scheme steps from one closed organisation to the next. As a closured topology cannot be half-present, the mechanism does not allow for 'intermediate' kinds. However, the mechanism does allow for 'branching' of the ranking. For example in the case where a bacterial cell can either lead to an endosymbiont cell or can lead to a multicellular organisation. Because the overall scheme ranks the structural complexity of different kinds of operators in a hierarchical way, it is called the 'operator hierarchy'. Because of confusion with the mathematical operator concept, I have since 2014 sometimes replaced 'operator' by 'closon', and 'operator hierarchy' by 'closon hierarchy'.

If one views the many small evolution steps as representing real numbers, the closure types represent the integers

The relationship between the small evolution steps and the (large) closure steps shows an analogy with the real and integer numbers. Small evolution steps are like real numbers. If one looks at the real numbers, there are innumerous small steps between 1 and 2, such as 1.00397, 1.02492, 1.2748 and so forth. Also the number 1.00000000... is just another real number in the row. Using this analogy, closures are similar to the integers. In order to recognize that a number is an integer, such as 1, 2, 3, etc. one needs a separate methodology for the distinction of kinds. Similarly, the recognition of closure requires a specific own logic. At the same time, both approaches remain connected, because the integer 1 corresponds to (but is not the same as) the real number 1.0000000......

Internal structure in the operator hierarchy

Initially, the operator hierarchy was constructed as a methodology to, from the gound up, define kinds of entities and levels of structural organisation in ecotoxicology. Later it was found that the resulting kinds seem to comply with a higher order logic. The higher oder logic of the operator hierarchy suggests that it can also be viewed as a novel 'periodical system'. In fact, the ranking of the operator hierarchy goes beyond that of a mere periodical system, because every step in the operator hierarchy is process based, as the result of all the subsequent closing processes. However, for simplicity reasons we will use the analogy with a periodical system.

Explaining the operator hierarchy step by step

Fundamental particles

To construct the operator hierarchy, and to introduce the concept of a closure dimension, I start from the ground up. In high energy physics, even at very high energies, there are currently no measurements suggesting that the particles in the standard model show inner structure based on still smaller particles. This does not exclude the possibility, however, that some of the particles that in all measurements appear as fundamental, may require lower level theoretical descriptions based on theoretic units, which alone, or in combination determine their structures (e.g. Greben arXiv:1208.5406). While science does not yet have all the answers in relation to questions at this level of organisation, there may well exist some kind of particles which are truly fundamental. Such 'fundamental particles' either represents a single particle, such as a quark, or a combination of theoretical units that combine to a unity which in all physical processes behaves as a single particle. In both cases the result must represent the most fundamental closure of a first level 'interface' around one or more (theoretical) units, creating a single 'particular' unity. This first level of interfacing is a structural closure.

Quark-gluon plasma

A single fundamental particle (e.g. a quark) can produce and absorb gluons. And two quarks can mutually exchange the gluons they produce, and in this way combine their production-absorption cycles to a new closure configuration: a hypercycle. We view the hypercycle as a functional closure.

Quark confinement

Next, the exchange of gluons between quarks allows for the formation of a new type of interface, which -unlike the preceding level of interfacing- includes several fundamental particles which already obtained a fundamental interface during the above formation step. The new, second level interface takes the form of the exchange of gluons, and creates a strong field that 'confines' the quarks into small 'bundles', called hadrons. Here we discuss this 'bundle' as if it exists separately. This should be viewed as an abstract way to represent the structural closure that surrounds two or more fundamental particles.


If one combines the quark-gluon hypercycle and the gluon confinement, one obtains a hadron. Operators of the kind 'hadron' are the first particles that consist of fundamental units with interface which are bound by an interface. Accordingly their novel closure dimension was called 'multi-particle-ness'. If a hadron, such as a proton, would now attract an electron, this would imply a structural closure. But structural closure alone does not lead to a new operator type, because for this, we need the combination of a structural and a functional closure.

Atom nuclei (or 'nucleids')

If we now imagine a novel functional closure first, the most direct way to realise this is by the exchange of small two-quark particles (hadrons called 'pions)' between hadrons. This exchange creates a new kind of hypercycle.


If we now combine a hypercycle of hadrons (the least complex system that complies with this rule is Deuterium) and the capturing of an electron shell (as a structural interface), we can combine a functional and a structural closure. The resulting new particle bears the name of an "atom". As the electron shell mediates the interactions of the nucleus with the surrounding environment, the new closure dimension of the atom was named 'hypercycle mediating interface', or HMI.


The electron shells of an atom allow for a novel functional property, which is the exchange of paired electrons between the neighbouring electron shells of two atoms. The exchange of paired electrons causes an attractive force between the atoms, called 'covalent bonding'. Meanwhile, the paired electrons also create a structural property, namely a common electron shell connecting both atoms. A molecule is composed of multiple atoms. For this reason a molecule complies with the already existing closure dimension of the multi-particle-ness.

Autocatalytic sets

Another way in which molecules can create a new closure is by means of catalytic interactions. Now, a whole group of catalytic molecules may digest substrate molecules and transform these into catalytic molecules. If all the molecules in the set produce product molecules which are identical to those already in the set, such a set of molecules can maintain itself. This is, because the degradation of catalytic molecules is compensated for by the production of new catalytic molecules. When the catalytic cycles of several molecules are linked in this way into a cycle of cycles, this can be viewed as a hypercycle.

Cell membranes

The condensation of molecules can lead to a membraneous sheet. Such a sheet may take the form of a globule, and in this way show structural closure.


The next new closure dimension emerges from the combination of an autocatalytic set and a surrounding membrane. The cell membrane functions as as an interface that mediates the transport of molecules to and from the autocatalytic set.The new closure dimension of the cell was named structural auto-copying of information (SCI), because the cell is the first system capable of autonomously copying all its 'information' by simply copying its structures (please check the glossary for the specific interpretation of information in the operator theory).

Endosymbiont cells (here: the host cell that is inhabited by an endosymbiont)

Similar to our example of the balloon in the preceding page, a cell can form a new closure by engulphing another cell. In this case the structural closure takes the form of a cell withing another cell (while both are bounded by their membranes). The functional closure takes the form of the mutual obligatory dependence of the cells for their functioning.


The example of the balloon (in the preceding page) also showed another option for structural closure: that of attached cells. However, simple attachment would be a property that single cells also possess when interacting as cells. To define multicellularity, additional properties are required, including a common new type of functional relationship between the cells, and a new type of common interface. A common interface is formed automatically, when the attached cells form plasma connections. Now, from a topological viewpoint the cells are contained in one large membrane, while the plasmas are connected and allow for a new type of functional interactions between cells.

CALM closures

Instead of multicellular organisms interacting as a population or other group, the operator hierarchy demands that the shortest route towards the next closure must be sought for the operator hierarchy. In this case, this shortest route is that from cells in a multicellular to groups of cells which are physically connected by means of a new kind of connection. Cell extensions in the form of dendrites can connects distant cells, and can form a new structure. New structures of this kind have e.g. been named categorising and learning modules, or CALM's. Special about a CALM is that it can function as a flexible memory. while in a CALM the neurons interact in a circular way, several CALM's can interact in a higher order circle. In this way the neural hypercycle is formed.

Neural interface

If one would place a brain on a table, without connections to the outer world, such a brain could not learn anything, because it could neither sense nor respond to stimuli. The outer limit of a neural network must for this reason be formed from sensory nerve cells, which form the interface.


The combination of the neural hypercycle and the interface of sensors introduces the neural network organism, or 'memon'. The memon introduces the closure dimension that has been named 'sructural auto-evolution'. This implies that based on sensory imputs, the system is capable of autonomously adapting its learned patterns. Strictly, the concept auto-evolution requires the testing of scenarios, and is an overly demanding concept for primitive memons.

Between layer transitions in the operator hierarchy

In Figure 1, any turn of the blue line to the left, indicates a (major) step towards a group of systems possessing a new kind of property which opens up a new closure dimension. The steps to the left are called 'between operator layer (BOL) transitions'. One can recognise six of such transitions: 1. Towards fundamental (matter-)particles; the particles in the standard model. These particles supposedly do not consist of smaller units. It is their interface which constructs the particle. 2. Towards hypercyclic interations in particle plasm, 3. Towards hadrons (heavy examples of which are the protons and neutrons), 4. Towards the atoms, 5. Towards cells, and 6. Towards memons (the neural network organism). From the level of the hadron and up, every BOL transition combines an emergent hypercycle with an emergent interface. Operators that are formed via BOL transitions are also called 'primary' operators.

Layers in the operator hierarchy

All transitions that fall between two BOL transitions are considered to belong to the same layer in the operator hierarchy. Accordingly, a layer starts always with an operator that is formed by a BOL transition and ends with the separately evaluated hypecycle and interface (before they combine and form the next major transition). The first layer is that of the fundamental particles. The second layer is that of the interacting fundamental particles. The third layer is that of the hadrons and nuclei. The fourth layer is that of atomic systems. This includes: the atoms, the molecules, and the autocatalytic sets. It ends with the interfacting of an autocatalytic set by a membrane. The fifth layer is that of cellular systems. It runs from the bacterial cell, to the endosymbiont cell, to (bacterial and endosymbiontic) multicellular organisms and the neural hypercycle. It ends with the sensory interface. The sixth layer is that of the memons. This is the layer of organisms with neural networks. Of course the above is based on theoretic limits. In reality, sensors in biological organisms, for example, will not arise separately from their neural network (although this is different for technical neural network organisms!).

A graphical representation of various subsets

This can be found by following this link.