Essential extensions and injective hulls of double Stone algebras
type: Item
instance of: scholarly article
title: Essential extensions and injective hulls of double Stone algebras
zbMATH Open document ID: 0358.06028
DOI: 10.1007/BF02485414
published in: Algebra Universalis
publication date: 1977-01-01T00:00:00Z
review text: 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 \(L\).
Mathematics Subject Classification ID: 06D15
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 3557850
Wikidata QID: Q114233813
author: Tibor Katriňák
reviewed by: Tibor Katriňák
MaRDI profile type: Publication
cites work: Categorial characterization of the MacNeille completion
cites work: Stone Lattices. I: Construction Theorems
cites work: Injectives in non-distributive equational classes of lattices are trivial
cites work: Congruence Relations in Direct Products
cites work: Q5639727
cites work: The structure of distributive double p-algebras. Regularity and congruences
cites work: Injective double Stone algebras
cites work: Construction of modular p-algebras
cites work: Injective Hulls of Stone Algebras
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description: scientific article; zbMATH DE number 3557850