Essential extensions and injective hulls of double Stone algebras (Q1239179)

A universal algebra \(\langle L;\cup,\cap,{}^*,{}^+,0,1\rangle\) is called a double Stone algebra if \(\langle L;\cup,\cap,{}^*,0,1\rangle\) is a Stone algebra and \(\langle L;\cup,\cap,{}^+,0,1\rangle\) is a dual Stone algebra (see the review of [the author, Algebra Univers. 7, 1--3 (1977; Zbl 0358.06027)]). In the paper [Algebra Univers. 4, 259--267 (1974; Zbl 0302.06022)] the author described the injective double Stone algebras. In the present paper a characterization of essential extensions and of injective hulls of double Stone algebras is given. More precisely: Let \(L\) be a double Stone algebra. Let \(B\), \(B_0\) and \(B_1\) denote the normal completion of the Boolean algebra generated by \(A(L)\), \(A^2 (L)\) and \(D^2 (L)\), respectively. Then the algebra \(B^{[2]} \times B_0^{[2]} \times B_1^{[3]}\) is the injective hull of \(L\) in the class of all double Stone algebras (Theorem 4). We recall that \(B^{[k]}=\{(a_1,\dots,a_k)\in B^k:a_1\leq\ldots\leq a_k\}\), and \(A(L)\), \(A^2(L)\) and \(D^2(L)\) are special sublattices of \(L\).