What does this research mean for the field?

The class of n-ary T-quasigroups coincides with the class of affine n-ary quasigroups and forms a mathematical variety characterized by heredity, homomorphic, and multiplicative closure. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to explore the properties and relationships of n-ary T-quasigroups, focusing on their classification and structure.

May 28, 2026

Polyadic Т-Quasigroups

Key Points

This research aims to explore the properties and relationships of n-ary T-quasigroups, focusing on their classification and structure.
Proved the isomorphism of derived n-ary groups from two T-groups.
Identified necessary and sufficient conditions for n-ary loops as derivatives of n-ary T-groups.
Established heredity, homomorphic, and multiplicative closure for n-ary T-quasigroups.
Demonstrated that the class of n-ary T-quasigroups coincides with that of affine n-ary quasigroups.
Showed that the class of n-ary T-quasigroups is a variety, confirming closure properties.

Abstract

The paper studies the properties of n -ary Т-quasigroups (n 3). The isomorphism of the derived n -ary groups from two Т-groups for an n -ary Т-quasigroup is proved. A necessary and sufficient condition is found under which an n -ary loop is a derivative of an n -ary group from a Т-group for an n -ary Т-quasigroup. The coincidence of the class of n -ary T-quasigroups with the class of affine n -ary quasigroups is proved. The heredity, homomorphic and multiplicative closure of the class of all n -ary T-quasigroups are established, which means that this class is a variety.

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