When an algebraic frame is regular (Q2496171)

A frame is a complete lattice \(L\) satisfying the distributive law \(a\wedge \bigvee S=\bigvee \{ a\wedge s\: s\in S\}\) for each \(a\in L\), \(S\subseteq L\). For each \(a\in L\), let \(a^{\perp }=\bigvee \{ x\in L; x\wedge a=0\}\). An element \(a\in L\) is called regular if \(a=\bigvee \{x\in L; x^{\perp }\vee a=1\}\). \(L\) is called regular if each \(a\in L\) is regular. The main theorem of the paper states that an algebraic frame \(L\) is regular iff its compact elements are complemented (i.e.\ if \(a\vee a^{\perp }=1\) for each \(a\in L\)). More generally, it is shown that each pseudocomplemented element is regular iff each \(c^{\perp \perp }\), with \(c\) compact, is complemented. With a mild assumption on \(L\), each \(c^{\perp }\), with \(c\) compact, is regular precisely when \(p\vee q=1\) for any two minimal primes \(p,q\in L\). These results are then interpreted in various frames of subobjects of \(l\)-groups and \(f\)-rings.