A. V. Litavrin, E. A. Potolovskaya

Siberian Federal University (Krasnoyarsk, Russian Federation)

Abstract. Additive subnetwork groupoids, multiplicative subnetwork groupoids, and corresponding generalized subnetwork groupoids of a multilayer feedforward neural network were previously introduced to study the relationships between the algebraic properties of these groupoids and the structural properties of the multilayer feedforward neural network upon which these groupoids are constructed. In particular, it was established that the associative nature of these groupoids imposes a strict constraint on the structure of the neural network graph. In this paper, generating sets are found for the additive subnetwork groupoid, the multiplicative subnetwork groupoid, and the additive generalized subnetwork groupoid. For such neural networks that each their layer has strictly more than one neuron, a generating set is found for the multiplicative groupoid of generalized subnetworks. The generating sets for these groupoids are further proposed for use in elementwise description of the monoid of all endomorphisms for these groupoids. Finding the monoid of all endomorisms for a groupoid is generally a nontrivial problem and yields some properties of this groupoid. Furthermore, the generating sets of these groupoids may be used to study their algebraic properties. Generating sets for these groupoids are obtained by finding a decomposition for each element of the groupoid in terms of elements of special sets. These generating sets were constructed by adding missing elements and removing the elements generated by the remaining ones. The results of this work may be used to study the properties of multilayer feedforward neural networks. Thus, they have practical significance.

Key Words: groupoid, generating set of groupoid, additive groupoid of subnets, multiplicative groupoid of subnets, additive groupoid of generalized subnets, multiplicative groupoid of generalized subnets, multilayer neural network

For citation: A. V. Litavrin, E. A. Potolovskaya. Generating sets for additive and multiplicative groupoids of generalized subnetworks of a multilayer feedforward neural network. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:3(2026), 212–224. DOI: https://doi.org/10.15507/2079-6900.28.202603.212-224

Information about the authors:

Andrey V. Litavrin, Ph.D. (Phys. and Math.), Associate Professor of the Department of Higher Mathematics No. 2, Siberian Federal University (79 Svobodny Ave., Krasnoyarsk 660041, Russia), ORCID: https://orcid.org/0000-0001-6285-0201, anm11@rambler.ru

Eva A. Potolovskaya, Student at the Institute of Mathematics and Computer Sciences, Siberian Federal University (79 Svobodny Ave., Krasnoyarsk 660041, Russia), ORCID: https://orcid.org/0009-0002-1990-5275, eva.potolovskaya@gmail.com