*4.1. Associativity*

In possibilistic information fusion, the fusion process is rarely considered to be distributed. As a consequence, possibilistic fusion rules are often not associative, which heavily alters the fusion results in differently structured topologies. However, in works regarding possibilistic fusion, associativity has been considered with low priority at best and neglected at worst. For instance, associativity is described as a useful property by Dubois et al. [6]; however, its absence is not considered to be a fatal flaw.

As a first step in discussing associativity, the fusion rules presented in the previous section are summarised in Table 1.



The table also shows whether the rules satisfy the following two properties:

**Definition 6** (Associativity)**.** *A fusion operator fu is associative if the fusion outcome is independent of the sequence in which information items are fused, i.e., fu*(*I*1, *I*2, *I*3) = *fu*(*fu*(*I*1, *I*2), *I*3) = *fu*(*I*1, *fu*(*I*2, *I*3))*.*

**Definition 7** (Quasi-associativity)**.** *A fusion operator fu is quasi-associative if it can be expressed as a sequence of associative steps and a final operation acting on the results of the previous associative steps [47]. Let f be an associative function and g be a function not restricted to the associativity property, then fu is quasi-associative if fu*(*I*1, *I*2, *I*3) = *g*(*f*(*f*(*I*1, *I*2), *I*3)) = *g*(*f*(*I*1, *f*(*I*2, *I*3)))*.*

**Proposition 2.** *If a fusion operator is associative, then it is also quasi-associative.*

**Proof.** Let **I** be a set of information items, let *f* = *fu* and *g* be an identity function: *g*(**I**) = **I**. Then, *g*(*f*(**I**)) = *fu*(**I**)—that is, by making use of an identity function, an associative fusion operator becomes quasi-associative.

From this, it follows that, if a fusion rule is not quasi-associative, then it is also not associative. Associative rules allow unrestricted topology design in the sense that sources can be freely assigned to fusion nodes without changing the overall fusion result. Quasiassociative rules require a final centralised fusion step in which the nonassociative part is computed. The associative part can be distributed to fusion nodes.
