The endomorphism monoid of a free trioid of rank 1 (Q334976)

Trioids were introduced by \textit{J.-L. Loday} and \textit{M. Ronco} [Contemp. Math. 346, 369--398 (2004; Zbl 1065.18007)], as algebras with three binary operations \(x\dashv y\), \(x\vdash y\), \(x\perp y\) satisfying 8 identities of the form \((x*y)\bigstar z = x*'(y\bigstar' y)\) for some operations \(*,*',\bigstar,\bigstar'\in \{\dashv , \vdash , \perp\}\). The first main result of the paper shows an explicit form of a free triod. Let \(X\) be a set and \(\overline{X}\) a copy of \(X\). Suppose that \(\mathrm{Ft}^+(X)\) is a free monoid on \(X\cup \overline X\). Denote by \(\mathrm{Ft}(X)\) the subsemigroup of \(\mathrm{Ft}^+(X)\) consisitng of words containg at least one element from \(\overline{X}\). It is shown that \(\mathrm{Ft}(X)\) is a free trioid with a base \(X\) under the operations \(u\dashv v= u\overline v\), \(u\vdash v= \overline u v\), \(u\perp v=uv\). It is shown that every endomorphism of a free trioid of rank 1 is a monomorphism. A classification of all these endomorphisms and an abstract characterization of the endomorphism monoid are given. The last main result shows that free trioids are isomorphic if and only if their endomorphism monoids are isomorphic.