Biorder structure of the bilinear form semigroup (Q1579875)

Let \(X\) and \(Y\) be two finite-dimensional vector spaces over a field \(K\) of characteristic zero with conjugate spaces \(X^*\) and \(Y^*\), respectively. For each bilinear form \(B\) from \(X\times Y\) to \(K\), denote by \(B_X\) the map from \(X\) to \(Y^*\) which is defined by \(B_X(x)=B(x,-)\) and denote by \(B_Y\) the map from \(Y\) to \(X^*\) which is defined by \(B_Y(y)=B(-,y)\). Denote by \(L(X)\) and \(L(Y)\) the semigroups of all linear transformations on \(X\) and \(Y\), respectively. Then \[ L_B'(X)=\{f\in L(X):\text{Ker}(B_X)\subseteq\text{Ker}(f)\text{ and }\text{Im}(f)\cap \text{Ker}(B_X)=\{0\}\} \] is a regular subsemigroup of \(L(X)\) and similarly, \[ L_B'(Y)=\{g\in L(Y):\text{Ker}(B_Y)\subseteq\text{Ker}(g)\text{ and }\text{Im}(g)\cap\text{Ker}(B_Y)=\{0\}\} \] is a regular subsemigroup of \(L(Y)\). A pair \((f,g)\in L(X)\times L(Y)\) is referred to as an adjoint pair with respect to the bilinear form \(B\) if \(B(xf,y)=B(x,gy)\) for all \((x,y)\in X\times Y\). It was shown by \textit{D. Rajendran} and \textit{K. S. S. Nambooripad} [Bilinear forms and semigroups of linear transformations, Southeast Asian Bull. Math. (to appear)] that the set \(S(B)\) of all adjoint pairs in \(L_B'(X)\times L_B'(Y)^{\text{op}}\) is a regular semigroup where the multiplication in \(S(B)\) is given by \((f,g)(f',g')=(ff',g'g)\). Any semigroup which is isomorphic to \(S(B)\) for some bilinear form \(B\) is referred to as a bilinear form semigroup and these are the semigroups which are investigated in the present paper. For example, it is shown that the maximal idempotents in a bilinear form semigroup are all \(\mathcal D\)-equivalent and this \(\mathcal D\)-class is referred to as the maximal \(\mathcal D\)-class. It is shown further that the maximal \(\mathcal D\)-class of a bilinear form semigroup is a completely simple orthodox subsemigroup. The author concludes with an investigation of the idempotents of a bilinear form semigroup.