Endomorphisms of finite commutative groupoids related with multilayer feedforward neural networks

In this paper, we introduce commutative, but generally not associative, groupoids $\mathrm{AGS}(\mathcal{N})$ consisting of idempotents. The groupoid $(\mathrm{AGS}(\mathcal{N}),+)$ is closely related to the multilayer feedforward neural networks $\mathcal{N}$ (hereinafter just a neural network). It turned out that in such neural networks, specifying a subnet of a fixed neural network is tantamount to specifying some special tuple composed of finite sets of neurons in the original network. All special tuples defining some subnet of the neural network $\mathcal{N}$ are contained in the set $\mathrm{AGS}(\mathcal{N})$. The rest of the tuples from $\mathrm{AGS}(\mathcal{N})$ also have a neural network interpretation. Thus, $\mathrm{AGS}(\mathcal{N})=F_1\cup F_2$, where $F_1$ is the set of tuples that induce subnets and $F_2$ is the set of other tuples. If two subnets of a neural network are specified, then two cases arise. In the first case, a new subnet can be obtained from these subnets by merging the sets of all neurons of these subnets. In the second case, such a merger is impossible due to neural network reasons. The operation $(+)$ for any tuples from $\mathrm{AGS}(\mathcal{N})$ returns a tuple that induces a subnet or returns a neutral element that does not induce subnets. In particular, if for two elements from $F_1$ the operation $(+)$ returns a neutral element, then the subnets induced by these elements cannot be combined into one subnet. For any two elements from $\mathrm{AGS}(\mathcal{N})$, the operation has a neural network interpretation. In this paper, we study the algebraic properties of the groupoids $\mathrm{AGS}(\mathcal{N})$ and construct some classes of endomorphisms of such groupoids. It is shown that every subnet $\mathcal{N}'$ of the net $\mathcal{N}$ defines a subgroupoid $T$ in the groupoid $\mathrm{AGS}(\mathcal{N})$ isomorphic to $\mathrm{AGS}(\mathcal{N}')$. It is proved that for every finite monoid $G$ there is a neural network $\mathcal{N}$ such that $G$ is isomorphically embeddable into the monoid of all endomorphisms $\mathrm {AGS}(\mathcal{N}))$. This statement is the main result of the work.