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 ...

We note that Fujikawa’s proposal of generalization of the GinspargWilson relation is equivalent to setting R = (aγ5D) 2k in the original Ginsparg-Wilson relation Dγ5 + γ5D = 2aDRγ5D. An explicit realization of D follows from the Overlap construction. The general properties of D are derived. The chiral properties of these higher-order ( k > 0 ) realizations of Overlap Dirac operator are compared...

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...

This paper explores the set theoretic assumptions used in the current published proof of Fermat’s Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions. Does the proof of Fermat’s Last Theorem (FLT) go beyond Zermelo Fraenkel set theory (ZFC)? Or does it merely use Peano Arithmetic (PA) or some weaker fragmen...

It is shown how Lawvere’s one-to-one translation between Birkhoff’s description of varieties and the categorical one (see [6]) turns Hu’s theorem on varieties generated by a primal algebra (see [4], [5]) into a simple reformulation of the classical representation theorem of finite Boolean algebras as powerset algebras.

Papert Strauss (Proc. London Math. Soc. 18(3), 217–230, 1968) used Pontryagin duality to prove that a compact Hausdorff topological Boolean algebra is a powerset algebra. We give a more elementary proof of this result that relies on a version of Bogolyubov’s lemma.