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Algebra universalis

, Volume 73, Issue 2, pp 143–155 | Cite as

Compatible operations on commutative weak residuated lattices

  • Hernán Javier San MartínEmail author
Article

Abstract

Compatibility of functions is a classical topic in Universal Algebra related to the notion of affine completeness. In algebraic logic, it is concerned with the possibility of implicitly defining new connectives.

In this paper, we give characterizations of compatible operations in a variety of algebras that properly includes commutative residuated lattices and some generalizations of Heyting algebras. The wider variety considered is obtained by weakening the main characters of residuated lattices (A, ∧, ∨, ·, →, e) but retaining most of their algebraic consequences, and their algebras have a commutative monoidal structure. The order-extension principle a ≤  b if and only if ab ≥ e is replaced by the condition: if a ≤  b, then ab ≥ e. The residuation property c ≤  ab if and only if a · c ≤ b is replaced by the conditions: if c ≤  ab , then a · c ≤ b, and if a · c ≤  b, then ec ≤  ab. Some further algebraic conditions of commutative residuated lattices are required.

2010 Mathematics Subject Classification

Primary 06B10 Secondary 03G10 03G25 

Key words and phrases

commutative residuated lattices weak Heyting algebras congruences compatible functions 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Conicet and Departamento de Matemática, Facultad de Ciencias ExactasUniversidad Nacional de La PlataLa PlataArgentina

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