Some basic notions and results in topological dynamics are extended to continuous groupoid actions spaces. We focus mainly on recurrence properties. Besides that analogous the classical case of group actions, but which have be put right setting, there also new phenomena. Mostly for groupoids whose source map is not open (and many), some properties were equivalent become distinct this general fr...

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

Minkowski in 1908 used space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector. We propose consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept of a binary axiomatic relative velocity made possible the replace...

Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until M. Stronkowski and D. Stanovský’s recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes t...

Dependent type theory associates to any elements x, y of a type A an identity type x =A y, the type of proofs of equality of these elements. In PER Martin-Löf’s extensional type theory, the identity type is a subsingleton inhabited precisely when x and y are judgmentally equal. Semantically, types are thus sets equipped with equivalence relations given by their identity types. Groupoid Martin-L...

We give a simple combinatorial model for plethysm. Precisely, the bialgebra dual to plethystic substitution is realised as homotopy cardinality of incidence an explicit simplicial groupoid, obtained from surjections by construction reminiscent Waldhausen S and Quillen Q-construction.