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Abstract:
Greechie diagrams are a well known graphical representation of orthomodular partial algebras, orthomodular posets and orthomodular lattices. Kalmbach and Dichtl gave some characterisations of Greechie diagrams of orthomodular posets and of orthomodular lattices under some assumptions, for example, that the family of blocks is pasted, or that the intersection of each pair of blocks contains less or equal than four elements. In this paper I present a generalisation of these characterisations for orthomodular partial algebras (or equivalently orthomodular posets). Here we consider arbitrary hypergraphs with finite lines. A Greechie diagram can be seen as a special hypergraph: Different points of the hypergraph have different interpretations in the corresponding partial algebra of type (2,1,0) and each line in the hypergraph has a maximal Boolean subalgebra as interpretation, in which the points are the atoms. A diagram is complete if each maximal Boolean subalgebra is induced by a line of the hypergraph. Every nontrivial orthomodular partial algebra with finite blocks is the interpretation of a Greechie diagram. The characterisation theorems in this paper provide conditions to check, whether a hypergraph is a complete diagram of an orthomodular partial algebra. This poperty can be checked without having to compute the interpretation. We just have to consider the lines in the hypergraph.

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Verantwortliche Autorin: Ute Fahrholz