Product of Graphs and Hybrid Logic

Left and right commutativity and the Church-Rosser and reverse Church-Rosser properties are necessary conditions for a graph (frame) to be a (non-trivial) product of two other graphs, but their conjunction is not a sufficient condition. This work presents a fifth property, called H-V intransitivity, that, when added to the four previous properties, results in a necessary and sufficient condition for a finite and connected graph to be a product. Then, we show that although the first four properties can be defined in a modal logic (the reverse Church-Rosser property requires a converse modality), H-V intransitivity is not modally definable. We also show that no necessary and sufficient condition for a graph to be a product can be modally definable. Finally, we present a formula in a hybrid language that defines H-V intransitivity.