Remarks on annihilators preserving congruence relations. (Q2884001)

A compact totally order-disconnected topological space \(X\) is called a Priestley space. It is known that the system of all clopen up-sets of \(X\) is a bounded distributive lattice. Conversely, if \(A\) is a bounded distributive lattice, then the family \(X(A)\) of all prime filters of \(A\) is a Priestley space. The annihilator of \(a\in A\) is the set \(a^0=\{c\in A: a\wedge c=0\}\). A congruence \(\theta\) of \(A\) is defined to be annihilator preserving (AP) if \((\forall a,b\in A)(a\equiv_\theta b\Rightarrow(\forall x\in a^0)(\exists y\in b^0)(x\equiv_\theta y))\).NEWLINENEWLINE Some equivalent conditions characterizing AP-congruences are proved. There is shown a close relation between AP-closed subsets of \(X(A)\) and the set of the AP-congruences of \(A\).NEWLINENEWLINE Further, the results of the last two theorems are as follows: If \(A\) is a bounded distributive lattice and is quasicomplemented (normal, respectively) and \(\theta\) is an AP-congruence, then \(A/\theta\) is quasicomplemented (normal) as well.