By Garrett P.

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Amer. Math. Soc. 272(2) (1982) 501–526 8. : Relation Algebras. Volume 150 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam (2006) 9. : Contributions to the theory of models. III. Nederl. Akad. Wetensch. Proc. Ser. A. 58 (1955) 56–64 = Indagationes Math. 17, 56–64 (1955) 10. : Some suﬃcient conditions for the representability of relation algebras. Algebra Universalis 8(2) (1978) 162–172 11. : Varieties of relation algebras. Algebra Universalis 15(3) (1982) 273– 298 Finite Symmetric Integral Relation Algebras with No 3-Cycles 29 12.

8. The transformer model of programs is the subspace consisting of strict, positively conjunctive and continuous transformers. X models the appropriate laws is of course standard and well documented [D76, H92, N89]. The transformer semantics is given in Fig. 8. But for implicit consistency with the relational semantics we prefer to deduce it— as much as is possible—from the Galois connection between the relational and transformer models (in the next section). For now, we record: Theorem (transformer model).

W. Sanders Lemma (coercions). b = skip ✁ b ✄ magic is injective and satisﬁes 1. true = skip ; 2. c) and, in particular, coercions commute; 3. b | b ∈ B } and, in particular, coer is antitone; 4. false = magic . ¬b o9 B ) = A ✁ b ✄ B . ¬b o9 magic) . We must be careful not to allow such simply duality to raise our hopes concerning the degree to which there is a transformer-like dual on the space of (relational) computations. 3 Models In this section we recall the relational and transformer models of computations (and programs in particular) and the Galois connection between them.