Quotient normed cones

Given a normed cone $(X,p)$ and a subcone $Y,$ we construct and study the quotient normed cone $(X/Y,\tilde{p})$ generated by $Y$. In particular we characterize the bicompleteness of $(X/Y,\tilde{p})$ in terms of the bicompleteness of $(X,p),$ and prove that the dual quotient cone $((X/Y)^{*},\|\cdot \|_{\tilde{p},u})$ can be identified as a distinguished subcone of the dual cone $(X^{*},\|\cdot \|_{p,u})$. Furthermore, some parts of the theory are presented in the general setting of the space $CL(X,Y)$ of all continuous linear mappings from a normed cone $(X,p)$ to a normed cone $(Y,q),$ extending several well-known results related to open continuous linear mappings between normed linear spaces.