Abstract.The standard operation refinement ordering is a kind of "meet of opposites": non-determinism reduction suggests "smaller" behaviour while increase of definition suggests "larger" behaviour. Groves' factorization of this ordering into two simpler relations, one per refinement concern, makes it more mathematically tractable but is far from fully exploited in the literature. We present a pointfree theory for this factorization which is more agile and calculational than the standard set-theoretic approach. In particular, we show that factorization leads to a simple proof of structural refinement for arbitrary parametric types and exploit factor instantiation across different subclasses of (relational) operation. The prospect of generalizing the factorization to coalgebraic refinement is discussed.

The paper discusses the role of interpretations, understood as multifunctions that preserve and reflect logical consequence, as refinement witnesses in the general setting of π-institutions. This leads to a smooth generalization of the "refinement by interpretation" approach, recently introduced by the authors in more specific contexts. As a second, yet related contribution a basis is provided to build up a refinement calculus of structured specifications in and across arbitrary π-institutions.

Software connectors are external coordination devices which ensure the flow of data and enforce synchronization constraints within a component's network. The specification of software connectors through which context dependent behaviour is correctly propagated remains an open, non trivial issue in their semantics. This paper, building on previous work by the authors, revisits this problem and introduces a model in which context awareness is suitably handled.

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