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 International Conference on Complex Systems (ICCS2007)

Retracts And Fixed Points In Theory Of Ordered Sets. Towards Combinatorial Computer Science

Robert Lisek
Fundamental Research Lab and Department of Logic Wroclaw Uni

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     Last modified: October 22, 2007

In this brief paper we research systems with respect to their abstract properties as structure and organization. In our approach important significance has theory of ordered sets and fixed points of morphism. We present state of art in this field and some new results based on retracts of posets. We point also the importance of methods to represent, manipulate and measure poset.
Retracts and fixed points have a crucial significance for recursion and computation.Well know is that fixed points are important because they exactly characterize solutions to recursive definitions. Itís convenient to describe functions using recursion, certainly in programming languages, also tempting in semantics. The problem is, are they well defined? Idea is successive approximations. The approximation process yields a fixed point, that give us a solutions to the recursive equations.
We use morphisms for processing posets. In our paper we check out different measures of poset morphisms. We define a new concept: the energy of a morphism. The energy of the morphism of order set is a scale-invariant of morphism: function from morphism to rational numbers. Intuitively, the connection between the complexity of the morphism of an order set and its energy is simple: the more complicated morphism, the higher energy.
This kind of research have many application: for networking (portable knowledge management environments), cybernetic, for AGI and many other novel problem area: appearance of large, combinatorial data objects.

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