Bireachability and final multialgebras

  • Authors:

  • Michał Walicki

  • Affiliations:

  • Department of Informatics, University of Bergen

  • Venue:
  • CALCO'05 Proceedings of the First international conference on Algebra and Coalgebra in Computer Science
  • Year:
  • 2005

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Abstract

Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational structures, representing relations as multivalued functions by selecting one argument as the “result”. This leads to strong algebraic properties missing in the case of relational structures. However, such strong properties can be obtained only by first choosing appropriate notion of homomorphism. We summarize earlier results on the possible notions of compositional homomorphisms of multialgebras and investigate in detail one of them, the outer-tight homomorphisms which yield rich structural properties not offered by other alternatives. The outer-tight homomorphisms are different from those obtained when relations are modeled as coalgebras and the associated congruence is the converse bisimulation equivalence. The category is cocomplete but initial objects are of little interest (essentially empty). On the other hand, the category does not, in general, possess final objects for the usual cardinality reasons. The main objective of the paper is to show that Aczel's construction of final coalgebras for set-based functors can be modified and applied to multialgebras. We therefore extend the category admitting also structures over proper classes and show the existence of final objects in this category.