, Volume 30, Issue 2, pp 363–401 | Cite as

Associativity, Commutativity and Symmetry in Residuated Structures

  • Maria Emilia Della Stella
  • Cosimo Guido


In this paper we develop the study of extended-order algebras, recently introduced by C. Guido and P. Toto, which are implicative algebras that generalize all the widely considered integral residuated structures. Particular care is devoted to the requirement of completeness that can be obtained by the MacNeille completion process. Associativity, commutativity and symmetry assumptions are characterized and their role is discussed toward the structure of the algebra and of its completion. As an application, further operations corresponding to the logical connectives of conjunction negation and disjunction are considered and their properties are investigated, either assuming or excluding the additional conditions of associativity, commutativity and symmetry. An overlook is also devoted to the relationship with other similar structures already considered such as implication algebras (in particular Heyting algebras), BCK algebras, quantales, residuated lattices and closed categories.


Lattice-ordered structures Residuated lattices Extended-order algebras MacNeille completion Galois connections Logical connectives 

Mathematics Subject Classifications (2010)

03B50 03G10 03G25 06B23 06F99 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly
  2. 2.Department of Mathematics “E. De Giorgi”University of SalentoPiazza TancrediItaly

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