Associations have (or seem to have) different meaning depending on whether one is talking about cognition or mathematics. In mathematics the associations correspond to different bracketings of mathematical expressions involving symbols denoting mathematical objects and operations between them. The meaning of the expression  in the case that it has meaning  depends on the bracketing of the expression. For instance, one has a(b+c)≠ (ab)+c , that is ab+ac≠ ab+c). Note that one can change the order of bracket and operation but not that of bracket and object.
For ordinary product and sum of real numbers one has associativity: a(bc)=(ab)c and a+(b+c) = (a+b)+c. Most algebraic operations such as group product are associative. Associativity of product holds true for reals, complex numbers, and quaternions but not for octonions and this would be fundamental in both classical and quantum TGD.
The building of different associations means different groupings of n objects. This can be done recursively. Divide first the objects to two groups, divide these tow groups to to two groups each, and continue until you jave division of 3 objects to two groups  that is abc divided into (ab)c or a(bc). Numbers 3 and 2 are clearly the magic numbers.
This inspires several speculative questions related to the twistorial construction of scattering amplitudes as associative singlets, the general structure of quantum entanglement, quantum measurement cascade as formation of association, the associative structure of manysheeted spacetime as a kind of linguistic structure, spin glass as a strongly associative system, and even the tendency of social structures to form associations leading from a fully democratic paradise to cliques of cliques of ... .
 In standard twistor approach 3gluon amplitude is the fundamental building brick of twistor amplitudes constructed from onshellamplitudes with complex momenta recursively. Also in TGD proposal this holds true. This would naturally follow from the fact that associations can be reduced recursively to those of 3 objects. 2 and 3vertex would correspond to a fundamental associations. The association defined 2particle pairing (both associated particles having either positive or negative helicities for twistor amplitudes) and 3vertex would have universal structure although the states would be in general decompose to associations.
 Consider first the spacetime picture about scattering (see this). CD defines interaction region for scattering amplitudes. External particles entering or leaving CD correspond to associative spacetime surfaces in the sense that the tangent space or normal space for these spacetime surfaces is associative. This gives rise to M^{8}H correspondence.
These surfaces correspond to zero loci for the imaginary parts (in quaternionic sense) for octonionic polynomial with coefficients, which are real in octonionic sense. The product of ∏_{i}P_{i}) of polynomials with same octonion structure satisfying IM(P_{i})=0 has also vanishing imaginary part and spacetime surface corresponds to a disjoint union of surfaces associated with factors so that these states can be said to be noninteracting.
Neither the choice of quaternion structure nor the choice of the direction of time axis assignable to the octonionic real unit need be same for external particles: if it is the particles correspond to same external particle. This requires that one treats the space of external particles (4surfaces) as a Cartesian product of of single particle 4surfaces as in ordinary scattering theory.
Spacetime surfaces inside CD are nonassociative in the sense that the neither normal nor tangent space is associative: M^{8}M^{4}× CP_{2} correspondence fails and spacetime surfaces inside CD must be constructed by applying boundary conditions defining preferred extremals. Now the real part of RE(∏_{i}P_{i}) in quaternionic sense vanishes: there is genuine interaction even when the incoming particles correspond to the same octonion structure since one does not have union of surfaces with vanishing RE(P_{i}). This follows from s rather trivial observation holding true already for complex numbers: imaginary part of zw vanishes if it vanishes for z and w but this does not hold true for the real part. If octonionic structures are different, the interaction is present irrespective of whether one assumes RE(∏_{i}P_{i})=0 or IM(∏_{i}P_{i})=0. RE(∏_{i}P_{i})=0 is favoured since for IM(∏_{i}P_{i})=0 one would obtain solutions for which IM(P_{i})=0 would vanish for the i:th particle: the scattering dynamics would select i:th particle as noninteracting one.
 The proposal is that the entire scattering amplitude defined by the zero energy state  is associative, perhaps in the projective sense meaning that the amplitudes related to different associations relate by a phase factor (recall that complexified octonions are considered), which could be even octonionic. This would be achieved by summing over all possible associations.
 Quantum classical correspondence (QCC) suggests that in ZEO the zero energy states  that is scattering amplitudes determined by the classically nonassociative dynamics inside CD  form a representation for the nonassociative product of spacetime surfaces defined by the condition RE(∏_{i}P_{i})=0. Could the scattering amplitude be constructed from products of octonion valued single particle amplitudes. This kind of condition would pose strong constraints on the theory. Could the scattering amplitudes associated with different associations be octonionic  may be differing by octonionvalued phase factors  and could only their sum be real in octonionic sense (recall that complexified octonions involving imaginary unit i commuting with the octonionic imaginary units are considered)?
One can look the situation also from the point of view of positive and negative energy states defining zero energy states as they pairs.
 The formation of association as subset is like formation of bound state of bound states of ... . Could each external line of zero energy state have the structure of association? Could also the internal entanglement associated with a given external line be characterized in terms of association.
Could the so called monogamy theorem stating that only twoparticle entanglement can be maximal correspond to the decomposing of n=3 association to one and twoparticle associations? If quantum entanglement is behind associations in cognitive sense, the cognitive meaning of association could reduce to its mathematical meaning.
An interesting question relates to the notion of identical particle: are the manyparticle states of identical particles invariant under associations or do they transform by phase factor under association. Does a generalization of braid statistics make sense?
 In ZEO based quantum measurement theory the cascade of quantum measurements proceeds from long to short scales and at each step decomposes a given system to two subsystems. The cascade stops when the reduction of entanglement is impossible: this is the case if the entanglement probabilities belong to an extension of extension of rationals characterizing the extension in question. This cascade is nothing but a formation of an association! Since only the state at the second boundary of CD changes, the natural interpretation is that state function reduction mean a selection of association in 3D sense.
 The division of n objects to groups has also social meaning: all social groups tend to divide into cliques spoiling the dream about full democracy. Only a group with 2 members  Romeo and Julia or Adam and Eve  can be a full democracy in practice. Already in a group of 3 members 2 members tend to form a clique leaving the third member outside. Jules and Catherine, Jim and Catherine, or maybe Jules and Jim! Only a paradise allows a full democracy in which nonassociativity holds true. In ZEO it would be realized only at the quantum critical external lines of scattering diagram and quantum criticality means instability. Quantum superposition of all associations could realize this democracy in 4D sense.
A further perspective is provided by manysheeted spacetime providing classical correlate for quantum dynamics.
 Manysheeted spacetime means that physical states have a hierarchical structure  just like associations do. Could the formation of association (AB) correspond basically to a formation of flux tube bond between A and B to give AB and serve as spacetime correlate for (negentropic) entanglement. Could ((AB)C) would correspond to (AB) and (C) "topologically condensed" to a larger surface. If so, the hierarchical structure of manysheeted spacetime would represent associations and also the basic structures of language.
 Spin glass is a system characterized by so called frustrations. Spin glass as a thermodynamical system has a very large number of minima of free energy and one has fractal energy landscape with valleys inside valleys. Typically there is a competition between different pairings (associations) of the basic building bricks of the system.
Could spin glass be describable in terms of associations? The modelling of spin glass leads to the introduction of ultrametric topology characterizing the natural distance function for the free energy landscape. Interestingly, padic topologies are ultrametric. In TGD framework I have considered the possibility that spacetime is like 4D spin glass: this idea was originally inspired by the huge vacuum degeneracy of Kähler action. The twistor lift of TGD breaks this degeneracy but 4D spin glass idea could still be relevant.
See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article From amplituhedron to associahedron.
