A characterization of primitive Boolean algebras (Q1179768)

Heyting* algebras (i.e. Heyting algebras with a unary operation \(^*\) such that \(a\leq a^*\), \((a\land b)^*=a^*\land b^*\), \((a\to b)^*=a\lor(a\to b))\) were introduced and extensively studied by the author [ibid. 24, 21-31 (1987; Zbl 0644.06004)]. In this paper he proves that primitive Boolean algebras (in the sense of Hanf) can be characterized in terms of Heyting* algebras of ideals, i.e. Heyting algebras closed under the operation \(^*\) defined by \(I^*=\{a: a/I\) is a finite sum of atoms\}; namely, a denumerable Boolean algebra is primitive iff its smallest complete Heyting* algebra of ideals has the property that each element is the sum of completely join-irreducible elements.