On some commutative and idempotent finite groupoids associated with subnets of multilayer feedforward neural networks

The work studies commutative and idempotent finite groupoids that are associated with subnetworks of multilayer feedforward neural networks (hereinafter simply neural networks). Previously, the concept of a neural network subnet was introduced. This paper introduces the concept of a generalized subnetwork of a neural network. This concept generalizes the previously introduced concept. The resulting groupoids are called additive and multiplicative groupoids of generalized subnets of a given neural network. These groupoids model the union and intersection of generalized subnets of a neural network. The conditions that the neural network architecture must satisfy in order for the additive groupoid of generalized subnets to be associative are identified. The conditions that the neural network architecture must satisfy in order for the multiplicative groupoid of generalized subnets to be associative are obtained. Subgroupoids of the constructed groupoids are studied.