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

Abstract

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 $fin mathcal F_{mathcal P}( L)$ such that ${uparrow}f(-frac 1n, frac 1n)$ is compact for any $ninmathbb N$ and the subring $mathcal F_{{mathcal P}_{K}}( L)$ is the family of all $fin mathcal F_{mathcal P}( L)$ 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 $varphi=widetilde {p_{{{,mathrm{coz},}}}}$<br /> 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 $mathcal F_{{mathcal P}_{K}}( L)$.