Yashchuk V. Algebraic structures related to lattices

The dissertation is a study of some properties of lattice groups and lattice rings, and their relations to the corresponding fuzzy structures; as well as the study of the structure of various types of Leibniz algebras. In the first section of the dissertation considered new algebraic structures – lattice groups and lattice rings. The origins of their appearance are lie in the L-fuzzy groups theory and in the L-fuzzy rings theory. But if the definition of L-fuzzy structures is not algebraic (rather it is functional), then lattice groups and lattice rings are already purely algebraic structures. A fairly important part of the work was to establish ties between L-fuzzy groups and L-fuzzy rings with lattice groups and lattice rings. In the process of elucidating such connections, a clear overall picture of the structure of lattice groups and lattice rings was obtained. Only associative rings were considered. Therefore, the question of considering similar lattice structures for non-associative rings naturally arose. One of the important types of such rings is Leibniz rings and their partial case – Leibniz algebras. But unlike the theory of associative rings and associative algebras – the theory of Leibniz rings and Leibniz algebras is not well developed. Therefore, before proceeding to the construction of lattice structures over Leibniz rings, it is necessary to answer important questions about the structure of Leibniz rings and Leibniz algebras. These questions are covered in the second part of the dissertation