Radical operations on the multiplicative lattice (Q2862251)

The author considers properties of radical and quasi-radical operations on multiplicative lattices. A complete lattice \(L\) is called \textit{multiplicative} if it has a commutative, associative, completely join-distributive product for which a greatest element \(I\) is an identity. For an operation \(F\) on a multiplicative lattice \(L\) and and elements \(a, b\in L\), \(F\) is called \textit{quasi-radical} if it satisfies the conditions (a) \(a\leq F(a)\), (b) \(F(F(F(a))=F(a)\), (c) \(F(A\wedge b)= F(a) \wedge F(b) = F(ab)\), and an element \(a\) is called \(F\)-radical if \(F(a)=a\). An operation \(F\) is defined as \textit{radical} if NEWLINE\[NEWLINEF(a) = \bigwedge_{a\leq p, p\in Q_F} p,NEWLINE\]NEWLINE where \(Q_F = \{p \mid p=F(p)\) and \(p\) is a prime element\(\}\), an element \(p<I\) is \textit{prime} if \(ab\leq p\) implies \(a\leq p\) or \(b\leq p\). It is proved that for a strongly compact multiplicative lattice \(L\), a quasi-radical operation \(F\) is a radical operation if and only if it satisfies the condition \(F(\bigvee_{j\in J} a_j) = \bigvee_{j\in J} a_j\) if \(\{a_j\}_{j\in J}\) is an ordered family of \(F\)-radical elements.