Development of an Algorithm for Constructing Multioperation Lattices Based on Indecomposable Algebras

• Main Author: Dmitriy Eremenko
• Format: Article
• Language: Russian
• Published: The Fund for Promotion of Internet media, IT education, human development «League Internet Media» 2021-04-01
• Series: Современные информационные технологии и IT-образование
• Subjects: operations; multioperations; lattice of algebras of multioperations; indecomposable algebras of operations
• Online Access: http://sitito.cs.msu.ru/index.php/SITITO/article/view/674

The theory discussed in the article refers to the theory of functional systems. This branch of mathematics explores functions defined on finite sets, as well as the composition of these functions. Such functions are used in mathematical logic and in universal algebra, and in particular, in the clone theory. The traditional research objects in universal algebra are the algebras of operations and multioperations. One of the main problems in the theory of multioperations is the classification of algebras. To solve this problem, it is necessary to construct a lattice of algebras. The article presents an algorithm for constructing lattices of multioperations based on indecomposable algebras. To implement this algorithm, all indecomposable algebras of unary multioperations of rank 3 were found; they were presented in the form of an inclusion graph. The vertices of the graph are indecomposable algebras, and the edges of the graph reflect the connection between the inclusion algebras. If there is a path between two vertices of the graph, then one algebra is a subalgebra for the other. Using the resulting graph, an algorithm for constructing a lattice of unary multioperations of rank 3 was implemented. The results obtained agree with the results described in the article "Algebras of unary multioperations". This algorithm can be used to construct lattices of multioperations of higher ranks or large areas.