THE ROLE OF THE MODULAR PAIRS IN THE CATEGORY OF COMPLETE ORTHOMODULAR LATTICE D. AERTS Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium and C. PIRON Department of TheoreticaIPhysics, University of Geneva, CH-1211 Geneva 4, Switzerland ABSTRACT. We study the modular pairs of a complete orthomodular lattice i.e. a CROC. We propose the concept of m-morphism as a mapping which preserves the lattice structure, the ortho- gonality and the property to be a modular pair. We give a characterization of the m-morphisms in the case of the complex ttilbert space to justify this concept. INTRODUCTION There are many formulations of quantum physics, the Hilbert-space formulation, the phase-space formulation, the C*-algebra formulation, etc. All these formulations are equivalent and which one to choose for formulating and solving a particular problem depends on the nature of the problem at hand. It is thus important to be able to formulate the problem in terms independent of this choice. In pure mathematics we encounter the same situation. For example in algebraic geometry we like to define the objects independent of the coordinate field. This kind of difficulty was solved by the use of the concept of the category. So we will define the proper category correspon- ding to the objects describing a physical system. It is well known that these objects are the com- plete orthomodular lattices. The point is, however, to choose the best notion of morphism, such that we are able to describe in this category all the physical notions that we encounter in quantum physics and not more if possible. In this paper, we propose a notion of morphism called m-morphism to play this role and we discuss why such a notion is particularly well adapted to the physical problems. THE ORTHOMODULAR LATTICES AND THE MODULAR PAIRS Let us first recall the definition of the complete orthomodular lattices (see [1 ] ), which are the Letters in MathematicalPhysics 3 (1979) 1-10. 0377-9017/79/0031-0001 $01.00. Copyright 9 1979 by D. Reidel PublishingCompany, Dordrecht, Holland, and Boston, U.S.A.