On Endomorphisms of the Additive Monoid of Subnets of a Two-layer Neural Network

Abstract

Previously, for each multilayer neural network of direct signal propagation (hereinafter, simply a neural network), finite commutative groupoids were introduced, which were called additive subnet groupoids. These groupoids are closely related to the subnets of the neural network over which they are built. A grupoid is a monoid if and only if it is built over a two-layer neural network. Earlier, endomorphisms and their properties were studied for these groupoids. Some endomorphisms were constructed, but an exhaustive element-by-element description was not received. It was shown that every finite monoid is isomorphic to some submonoid of the monoid of all endomorphisms of a suitable additive subnet groupoid for some suitable neural network. In this paper, we study endomorphisms of additive groupoids of subnets of twolayer neural networks. The main result of the work is an element-wise description of the monoid of all endomorphisms of additive monoids of subnets built over a two-layer neural network. The item-by-item description is obtained by constructing a general form of endomorphism. The general view of an endomorphism is parameterized by the endomorphisms of suitable booleans with respect to the union operation. Therefore, endomorphisms of these Booleans were studied in this work. In particular, the semirings of endomorphisms of these Booleans with respect to the union were studied. In addition, to describe the general form of the endomorphism of the additive monoid of subnets, homomorphisms of one Boalean into another (with respect to union) were used.