Kryklya Y. Free left n-trinilpotent trioids

The thesis is devoted to the study of free algebras in the variety of trioids. The object of the research of the thesis is free left (right) n-trinilpotent trioids and free trioids. The subject of the research is the structure and properties of free left (right) n-trinilpotent trioids and free trioids. The methods of the algebraic theory of semigroups and universal algebra are used in the process of researching the thesis issues. In the thesis left (right) n-trinilpotent trioids are introduced, which are analogs of left (right) nilpotent semigroups of rank n considered by B. M. Schein, and the free left (right) n-trinilpotent trioid of an arbitrary rank is constructed. The cardinality of the free left (right) n-trinilpotent trioid in the finite case is calculated. All idempotent and all regular elements of free left (right) n-trinilpotent trioids are described. All maximal subtrioids of free left (right) n-trinilpotent trioids (n >1) are found. It is shown that a free left n-trinilpotent trioid contains a subtrioid, which can be represented as a left band of subdimonoids. The cardinality of the endomorphisms semigroup of a free left (right) n-trinilpotent trioid and the number of all idempotent and regular elements of free left (right) n-trinilpotent trioids in the finite case are calculated. The results of the thesis can be applied to the theory of trioids and trialgebras, the theory of dimonoids and dialgebras, semigroup theory, and universal algebra. Key words: trioid, left n-trinilpotent trioid, free left n-trinilpotent trioid, congruence, semigroup, maximal subtrioid, regular element, idempotent element, automorphism group, endomorphism semigroup, cardinality.