Injective and projective regular double Stone algebras (Q1104954)

A double Stone algebra \(L=(L;\vee,\wedge,\quad *,\quad +,0,1)\) is a bounded distributive lattice such that L and its dual \(\bar L\) are pseudocomplemented (i.e. \(x\leq a\) * iff \(x\wedge a=0\) and \(x\geq a\) \(+\) iff \(x\vee a=1)\) and it satisfies the identities x \(*\vee x^{**}=1\), x \(+\wedge x^{++}=0\). A double Stone algebra is called regular, if it satisfies the condition: x \(*=y\) * and x \(+=y\) \(+\) imply \(x=y\). This is a continuation of the author's paper in Houston J. Math. 9, 455-463 (1983; Zbl 0531.06007). The author gives a characterization of injective regular double Stone algebras [see also the reviewer's paper in Algebra Univ. 4, 259-267 (1974; Zbl 0302.06022)] and a characterization of projective regular double Stone algebras. In particular, he describes those regular double Stone algebras which are also projective in the category of double Stone algebras.