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

Document Type: Research Paper

Authors

1 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Postal Code 9617976487, Sabzevar, Iran

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

10.29252/as.2020.1798

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 $f\in \mathcal F_{\mathcal P}( L)$ such that  ${\uparrow}f(-\frac 1n, \frac 1n)$ 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


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