We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the “powerset monad” on categories, one is the preservation by T of “exactness” of certain squares. Both characterisations ...

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.

A minimal permutation representation of a finite group G is a faithful G-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In these situations (except when central involutions intervene) all minimal permutation representations have the same set of orbit sizes. Using the same ideas w...

In the final chain of the countable powerset functor, we show that the set at index ω1, regarded as a transition system, is not strongly extensional because it contains a “ghost” element that has no successor even though its component at index 1 is nonempty. For Jirka Adámek on his 70th birthday, with thanks for his many contributions and his inspiration.

Björner [1] showed how a construction by von Neumann of examples of continuous geometries can be adapted to construct a continuous analogue of finite partition lattices. Björner's construction realizes the continuous partition lattice abstractly, as a completion of a direct limit of finite lattices. Here we give an alternative construction realizing a continuous partition lattice concretely as ...

Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a G-invariant...

We extend RBPP, the state-of-the-art, translation-based planner for conformant probabilistic planning (CPP) with deterministic actions, to handle a wide set of CPPs with stochastic actions. Our planner uses relevance analysis to divide a probabilistic ”failure-allowance” between the initial state and the stochastic actions. Using its ”initial-state allowance,” it uses relevance analysis to sele...

framework for powerset semantics. We now cast the powerset semantics in our general relation-based framework, which enables us to establish when naı̈ve evaluation works for it. For a set D of database objects and a set C of complete objects, we have a pairR “ pRval,Rsemq of relations withRval Ď Dˆ2 C andRsem Ď 2 ˆC. The first relation corresponds to applying multiple valuations (e.g., relating D...