Abstract A split system $$\mathcal S$$ S on a finite set X , $$|X|\ge 3$$ | X ≥ 3 is of bipartitions or splits which contains all the form $$\{x,X-\{x\}\}$$ { x , <mml:...

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P , RP is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreove...

We introduce an equivariant version of the weak expectation property (WEP) at level operator modules over completely contractive Banach algebras [Formula: see text]. prove a number general results — for example, characterization text]-WEP in terms appropriate text]-injective envelope, and also those text] which implies WEP. In case text], we recover text]-[Formula: text]-algebras recent work Bu...

Injective Pure Type Systems form a large class of Pure Type Systems for which one can compute by purely syntactic means two sorts elmt(?jM) and sort(?jM), where ? is a pseudo-context and M is a pseudo-term, and such that for every sort s, ? ` M : A ^ ? ` A : s) elmt(?jM) = s ? ` M : s) sort(?jM) = s By eliminating the problematic clause in the (abstraction) rule in favour of constraints over el...

We establish a strengthening of Jordan separation, to the setting of maps f : X → S, where X is not necessarily a manifold, and f is not necessarily injective.

We prove that if S is a properly embedded π1-injective surface in a compact 3-manifold M , then π1S is separable in π1M .

Given a collection C of subsets of a finite set X, let S C = ∪S∈CS. Philip Hall’s celebrated theorem [2] concerning ‘systems of distinct representatives’ tells us that for any collection C of subsets of X there exists an injective (i.e. one-to-one) function f : C → X with f(S) ∈ S for all S ∈ C if and and only if C satisfies the property that for all non-empty subsets C ′ of C we have | S C | ≥...