Aggregation theory is devoted to the fusing of several values into a unique output that summarizes the given information. Typically, the aggregation process is formalized in terms of an increasing mathematical function that maps the input values to the result, fulfilling some boundary conditions. However, this formalization can be too restrictive for some scenarios. In some cases, the inputs can be seen as observations of random variables, the aggregation result being also a random variable. In others, the aggregation process can be identified as a program that performs the aggregation rather than a mathematical function. In this direction, the concepts of aggregation of random variables and computable aggregation have been defined in the literature. This paper is devoted to the definition of computable aggregation of random variables, which are computer programs, not functions, that aggregate random variables, not numbers. Special attention is given to different possible alternatives to modelize random variables and monotonicity. The implementation of some examples is also provided.

The idea of computable aggregation operators was introduced as a generalization of aggregation operators, allowing the replacement of the mathematical function usually considered for aggregation, by a program that performs the aggregation process. There are different reasons to justify this extension. One of them is the interest in exploring some computational properties not directly related to the aggregation itself but to its implementation (complexity, recursivity, parallelisation, etc). Another reason, the one driving to the present paper, is the need to define a framework where the quite common process of first sampling (over a large data set) and then aggregating the sample, could be analysed as a formal aggregation process. This process does not match with the idea of an aggregation function, due to its non-deterministic nature, but could easily be adapted to that of a (non-deterministic) computable aggregation. The idea of non-deterministic aggregation requires the extension of the concept of monotonicity (a key aspect of aggregation operators) to this new framework. The present paper will explore this kind of non-deterministic aggregation processes, first from an empirical point of view and then in terms of populations, adapting the idea of monotonicity to both of them and finally defining a common framework for its analysis.

A new method is given for a finite reflexive relation with a fixed set of properties and a t-norm T, so that a different, but similar, T-transitive relation could be found. The transitive closure is greater than the given relation; however, the T-transitivized relation defined in this article is less than or equal to the original one, while reaching or maintaining, in this case, the T-conditionality. The algorithm also verifies if a relation is T-transitive. ©1999 John Wiley & Sons, Inc.