J.-L. Loday and M.O. Ronco introduced the concepts of a trialgebra and a trioid, and defined the constructions of a free trialgebra and a free monogenic trioid. Trialgebras are related to the operads associated with chain modules of simplices and Stasheff polytopes. A trioid is the basis of a trialgebra and it is defined as a set with three binary associative operations satisfying the same axioms as a trialgebra, so trialgebras are linear analogs of trioids. If the operations of a trioid coincide, it becomes a semigroup. In this paper, we study the natural relationships between arbitrary semigroups and trioids defined by these semigroups. We present new classes of trioids constructed from various semigroups and show that any semigroup can be embedded into a suitable non-trivial trioid as a subtrioid in which all operations coincide.
Yu. V. Zhuchok (Sun,) studied this question.