Natural partial ordering on \(E(\mathrm{Hyp}_G(2))\) (Q2853415)

Let \(W(X)\) be the free groupoid over \(X=\{x_1,x_2,\dots\}\). The map \(S:W(X)^3\to W(X)\) is defined inductively by the following rules: (i) \(S(x_1,t_1,t_2)=t_1\), \(S(x_2,t_1,t_2)=t_2\), and \(S(x_j,t_1,t_2)=x_j\) if \(j>2\); (ii) \(S(t't'',t_1,t_2)=S(t',t_1,t_2)S(t'',t_1,t_2)\). A generalized hypersubstitution \(\sigma\) of type (2) is specified by fixing a term \(t_\sigma\in W(X)\) giving rise to a map \(\hat\sigma:W(X)\to W(X)\) such that \(\hat\sigma(x)=x\) for all \(x\in X\) and \(\hat\sigma(t_1t_2)=S(t_\sigma,t_1,t_2)\). The set Hyp\(_G(2)\) of all generalized hypersubstitutions of type (2) becomes a monoid if one defines the product \(\sigma\tau\) of \(\sigma,\tau\in{}\) Hyp\(_G(2)\) as the generalized hypersubstitution specified by the term \(\hat\sigma(t_\tau)\). The authors [Int. J. Math. Math. Sci. 2008, Article ID 263541, 8 p. (2008; Zbl 1159.08002)] have described the idempotents of the monoid Hyp\(_G(2)\); here they determine the natural partial order of the idempotents.