Magari and \(\Delta\)-pseudo-Boolean algebras (Q2640631)

The contents of this paper belong to the algebraic theory of those algebras which are studied in connection with the proof-intuitionistic calculus and the modal system GL \((=\) Gödel-Löb). We shall limit ourselves to describing the main result. The Heyting algebra (H,\(\vee,\wedge,\to,0)\) is called \(\Delta\)- enrichable, if there exists an operation \(\Delta\) on H such that \(p\leq \Delta p\), \((\Delta p\to p)=p\) and \(\Delta\) \(p\leq q\vee (q\to p)\) for all p,q\(\in H\). A topological Boolean algebra (B,\(\vee,\wedge,-,I)\) is called Grzegorczyk algebra if for every element \(a\in B\) \(Ia=I(-I(-A\vee Ia)\vee a).\) The main result is that every Grzegorczyk algebra (B,I) is embedding in a Grzegorczyk algebra \((B_ 0,I_ 0)\) in which the Heyting algebra \(H_ 0\) of its open elements in \(\Delta\)-enrichable.