Free ideals and real ideals of the ring of frame maps from \mathcal P(\mathbb R) to a frame

Let \mathcal F_{\mathcal P}( L) (\mathcal F_{\mathcal P}^{*}( L)) be   the f-rings   of all (bounded) frame maps from \mathcal P(\mathbb R) to a frame L. \mathcal F_{{\mathcal P}_{\infty}}( L) is  the family of all f\in \mathcal F_{\mathcal P}( L) such that  {\uparrow}f(-\frac ۱n, \frac ۱n) is compact for any n\in\mathbb N and the subring  \mathcal F_{{\mathcal P}_{K}}( L) is the family of all   f\in \mathcal F_{\mathcal P}( L) such that {{\,\mathrm{coz}\,}}(f) is compact. We  introduce  and study  the concept of   real ideals in \mathcal F_{\mathcal P}( L) and \mathcal F_{\mathcal P}^*( L). We  show  that every maximal ideal of \mathcal F_{\mathcal P}^{*}( L) is   real, and also  we study the relation between the conditions ``L is compact" and ``every maximal ideal of \mathcal F_{\mathcal P}(L) is real''. We prove  that for every   nonzero real Riesz map \varphi \colon \mathcal F_{\mathcal P}( L)\rightarrow \mathbb R,  there is an element  p in \Sigma L such that \varphi=\widetilde {p_{{{\,\mathrm{coz}\,}}}}  if L is a zero-dimensional frame for which B(L) is a sub-\sigma-frame  of   L and every maximal ideal of \mathcal F_{\mathcal P}( L) is real. We show  that \mathcal F_{{\mathcal P}_{\infty}}(L)  is equal to the intersection of all  free maximal ideals of \mathcal F_{\mathcal P}^{*}(L) if B(L) is a sub-\sigma-frame  of a zero-dimensional frame  L   and also,  \mathcal F_{{\mathcal P}_{K}}(L) is equal to the intersection of all free ideals \mathcal F_{\mathcal P}( L)   (resp.,  \mathcal F_{\mathcal P}^*( L)) if L is a zero-dimensional frame.  Also, we study free ideals and fixed ideals of    \mathcal F_{{\mathcal P}_{\infty}}( L) and  \mathcal F_{{\mathcal P}_{K}}( L).

Keywords:

Lattice-ordered ring , Zero-dimensional frame , F_{mathcal P}-realcompact , Real Riesz map , Free ideal , Real ideal