# A category of transition systems and its relations with orthomodular posets

## Abstract

Two categories are defined and their relationships are studied. The objects of the first category, PCOS, are prime coherent orthomodular posets, which have been mainly studied in connection with quantum logic. Morphisms in PCOS are homomorphisms in the usual sense, preserving order and a binary relation, named compatibility.

The second category, denoted by LETS, comprises the class of labelled transition systems that can be generated, up to isomorphism, by case graphs of CE systems. Two contravariant functors linking the two categories are defined. The functor from LETS to PCOS is given by the calculus of regions, according to Ehrenfeucht and Rozenberg. The functor from PCOS to LETS defines a procedure which builds a labelled transition system, given an abstract set of regions with their order relation. We show that the two functors form an adjunction.

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