Power structures

Abstract

Power structures epitomise the attempt to lift some existing structure from elements of a set to subsets of that set. For example, any operation f over the elements of a set A can in a natural way be extended to a power operation f+ over subsets of A, and hence there is for any algebra A a corresponding power algebraP(A) (also called the "complex algebra" or "global" of A), 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 relation R over A an n-ary operation R↑ over P(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 relation R over a set A its power relation R+ over P(A), and hence to form the power 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. © 1993 Birkhäuser Verlag.