Rees semigroups of digraphs for classification of data. (Q284651)

Let \(D=(V,E)\) be a digraph whose vertex set \(V=I\cup\Lambda \) can be split into two non-empty subsets \(I,\Lambda\); \(G\) a group, \(P=[{p_{\lambda i}}]\), \({p_{\lambda i}}\in G\) a \((\Lambda\times I)\)-matrix. Define multiplication on the set \(M_D^0(G;I,\Lambda;P)=\{\theta\cup(g;i,\lambda),\;i\in I,\;\lambda\in\Lambda\}\) by \[ (g_1;i_1,\lambda_1)\cdot(g_2;i_2,\lambda_2)=\begin{cases} (g_1p_{\lambda_1}i_2g_2;i_1,\lambda_2),\text{ if }(i_1,\lambda_2)\in E\\ \theta.\end{cases} \] If this multiplication is associative, \(M_D^0(G;I,\Lambda;P)\) is called the Rees semigroup of the digraph \(D\). For a finite subsemigroup \(S\) of the Rees semigroup of a digraph and for a zero-divisor-free idempotent semiring \(F\) with identity are described all ideals of the semigroup semiring \(F_0[S]\) which have largest possible weight; the weight of a semiring element is the number of its nonzero coefficients; the weight of a subset of elements the lower bound of the weights of nonzero elements in the subset.