Implicative twist-structures

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• 作者：Umberto Rivieccio (1)
  
• 关键词：Primary ; 06D05 ; Secondary ; 18A23 ; 06D20 ; 03G25 ; 03G27 ; twist ; structure ; implicative bilattice ; N4 ; lattice ; Nelson lattice ; representation ; subreducts ; algebraic logic
• 刊名：Algebra Universalis
• 出版年：2014
• 出版时间：April 2014
• 年：2014
• 卷：71
• 期：2
• 页码：155-186
• 全文大小：328 KB
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• 作者单位：Umberto Rivieccio (1)
  
  1. Delft University of Technology, 2600 AA, Delft, The Netherlands
  
• ISSN：1420-8911

文摘

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.