Complete Categorical Deduction for Satisfaction as Injectivity

  • Grigore Roşu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4060)


Birkhoff (quasi-)variety categorical axiomatizability results have fascinated many scientists by their elegance, simplicity and generality. The key factor leading to their generality is that equations, conditional or not, can be regarded as special morphisms or arrows in a special category, where their satisfaction becomes injectivity, a simple and abstract categorical concept. A natural and challenging next step is to investigate complete deduction within the same general and elegant framework. We present a categorical deduction system for equations as arrows and show that, under appropriate finiteness requirements, it is complete for satisfaction as injectivity. A straightforward instantiation of our results yields complete deduction for several equational logics, in which conditional equations can be derived as well at no additional cost, as opposed to the typical method using the theorems of constants and of deduction. At our knowledge, this is a new result in equational logics.


Inference Rule Factorization System Deduction System Free Algebra Partial Algebra 


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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Grigore Roşu
    • 1
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUSA

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