The least dimonoid congruences on relatively free trioids

A. V. Zhuchok

doi:10.30970/ms.57.1.23-31

A. V. Zhuchok Luhansk Taras Shevchenko National University

DOI: https://doi.org/10.30970/ms.57.1.23-31

Keywords: trioid; free left n-trinilpotent trioid; free abelian trioid; dimonoid; semigroup; congruence

Abstract

When Loday and Ronco studied ternary planar trees, they introduced types of algebras,
called trioids and trialgebras. A trioid is a nonempty set equipped with three binary associative
operations satisfying additional eight axioms relating these operations, while a trialgebra is just
a linear analog of a trioid. If all operations of a trioid (trialgebra) coincide, we obtain the notion
of a semigroup (associative algebra), and if two concrete operations of a trioid (trialgebra)
coincide, we obtain the notion of a dimonoid (dialgebra) and so, trioids (trialgebras) are a
generalization of semigroups (associative algebras) and dimonoids (dialgebras). Trioids and
trialgebras have close relationships with the Hopf algebras, the Leibniz 3-algebras, the Rota-
Baxter operators, and the post-Jordan algebras. Originally, these structures arose in algebraic
topology. One of the most useful concepts in algebra is the free object. Every variety contains
free algebras and free objects in any variety of algebras are important in the study of that
variety. Loday and Ronco constructed the free trioid of rank 1 and the free trialgebra. Recently,
the free trioid of an arbitrary rank, the free commutative trioid, the free n-nilpotent trioid, the
free rectangular triband, the free left n-trinilpotent trioid and the free abelian trioid were
constructed and the least dimonoid congruences as well as the least semigroup congruence on
the first four free algebras were characterized. However, just mentioned congruences on free
left (right) n-trinilpotent trioids and free abelian trioids were not considered. In this paper, we
characterize the least dimonoid congruences and the least semigroup congruence on free left
(right) n-trinilpotent trioids and free abelian trioids.

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