Certain verbal congruences on the free trioid (Q6585991)

The notion of a trioid was introduced by \textit{J.-L. Loday} and \textit{M. Ronco} [Contemp. Math. 346, 369--398 (2004; Zbl 1065.18007)]. A trioid is a nonempty set \(T\) equipped with three binary associative operations \(\dashv\), \(\vdash\) and \(\bot\) satisfying additional 8 axioms. A congruence \(\theta\) on an algebra \(A\) is called verbal if there exists a variety \(V\) such that \(\theta\) is the least congruence on \(A\) such that \(A/\theta \in V\). This article continues a large series of articles by the author himself and his co-authors devoted to the description of congruences on triodes. \N\NThe content of the article is quite fully set out in its abstract: ``We characterize the least abelian dimonoid congruences, the least \(n\)-nilpotent dimonoid congruences, the least left (right) \(n\)-trinilpotent dimonoid congruence, the least \(n\)-nilpotent semigroup congruence and the least left (right) \(n\)-nilpotent semigroup congruence on the free trioid. The obtained results can be useful in trialgebra theory.'' \N\NA trioid \(T\) can be considered as a dimonoid if two operations \( \dashv \) and \(\bot \), or \(\vdash\) and \(\bot \) of \(T\) coincide.