On a new algorithm for representations of semichain bands (Q2730524)

A partially ordered set \(P\) is called a semichain if it has the form \(P=\bigcup_{i=1}^m P_i\), where each link \(P_i\) is either a single point or consists of two noncomparable points and \(P_1<\cdots<P_m\). Let \(S=\{A_1,\dots,A_n,B_1,\dots,B_n\}\), \(n\geq 1\), be a family of mutually disjoint semichains and let \(A=\bigcup_{i=1}^n A_i\), \(B=\bigcup_{i=1}^n B_i\). The couple \(\overline S=(S,*)\) is called the band of the semichains \(A_1,\dots,A_n,B_1,\dots,B_n\), where \(*\) is an involution on \(A\cup B\) such that \(x^*=x\) for any point \(x\) belonging to a two-point link of a semichain. The triple \((U,V,\varphi)\) is called a representation of the band of semichains \(\overline S\) over the field \(k\). Here \(U=\{U_1,\dots,U_n\}\), \(V=\{V_1,\dots,V_n\}\) are collections of \(k\)-spaces such that \(U_i\in\text{mod}_{A_i}k\), \(V_i\in\text{mod}_{B_i}k\), \((\bigoplus_{i=1}^n U_i)\oplus(\bigoplus_{i=1}^n V_i)\) belongs to the subcategory \(\text{mod}_{(A\cup B,*)}k\subset\text{mod}_{A\cup B}k\), \(\varphi=\{\varphi_1,\dots,\varphi_n\}\) is a family of linear mappings \(\varphi_i\in\Hom_k(U_i,V_i)\), \(i=1,\dots,n\). The author proposes some algorithm for representations of bands of semichains which is simpler than the one proposed earlier by the author.