algebra universalis

, Volume 30, Issue 2, pp 177–216 | Cite as

Power structures

  • Chris Brink


Power structures epitomise the attempt to lift some existing structure from elements of a set to subsets of that set. For example, any operationf over the elements of a setA can in a natural way be extended to apower operation f+ over subsets ofA, and hence there is for any algebraA a correspondingpower algebraP(A) (also called the “complex algebra” or “global” ofA), which is the power set endowed with the power operations. This idea goes back to Frobenius. A natural generalisation is to define for any (n + l)-ary relationR overA ann-ary operationR overP(A), and hence to form also the power algebra of any relational structure. Jónsson and Tarski first made systematic use of this construction in the study of Boolean algebras with operators. A further generalisation is to define for any relationR over a setA itspower relation R+ overP(A), and hence to form thepower structure of any relational structure. Power relations have been used, for example, in denotational semantics, in fixed-point theory, and in the study of verisimilitude. This paper offers an overview of known work and a pilot study of power structures in a universal-algebraic context.


Pilot Study Power Relation Relational Structure Natural Generalisation Power Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Chris Brink
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa
  2. 2.Automated Reasoning Project Centre for Information Science ResearchAustralian National UniversityCanberraAustralia

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