  We introduce a class of trainable nonlinear operators based on semirings that
are suitable for use in neural networks. These operators generalize the
traditional alternation of linear operators with activation functions in neural
networks. Semirings are algebraic structures that describe a generalised
notation of linearity, greatly expanding the range of trainable operators that
can be included in neural networks. In fact, max- or min-pooling operations are
convolutions in the tropical semiring with a fixed kernel.
  We perform experiments where we replace the activation functions for
trainable semiring-based operators to show that these are viable operations to
include in fully connected as well as convolutional neural networks (ConvNeXt).
We discuss some of the challenges of replacing traditional activation functions
with trainable semiring activations and the trade-offs of doing so.
