The standard operations in mathematical morphology involve erosions, dilations, openings, and closings which have been defined on binary and grey scale images. A generalization of standard morphology is discussed and leads to a new operation: weighted rank order filters. A different type of generalization leads to morphology in a vector space. A combination of the two ideas is developed, and involves a sequence or network of layers of weighted rank order filters on vector spaces which have properties very similar to multi-layer neural networks. A weighted rank order cell has a non-linear soft threshold response at the output. Due to the nature of rank order filtering, a unique supervised training procedure can be defined which allows weights in hidden layers to be trained as quickly and easily as those in the output layer.