The concepts of free modules, projective modules, injective modules, and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injective L-modules. In this paper, we give an alternate definition for injective L-modules and prove t...

We prove a max-min theorem for weak containment in the context of algebraic actions. Namely, we show that given an action $G$ on $X,$ there is maximal, closed $G$-invariant subgroup $Y$ $X$ so weakly contained Bernoulli shift. This also minimal any shift $G\curvearrowright X/Y$-ergodic presence X$. give several applications, including major simplification proof measure entropy equals topologica...

for α ∈ Ak−e+iX. Thus there are canonical isomorphisms θE : ⊕ e i=0Ak−e+iX ∼ → AkPE. θE : ⊕αi 7→ ∑e i=0c1(OPE(1)) pαi. Proof. Our plan was to prove this in the following order: π surjective, θE surjective, θE injective, π injective. The proof is a delicate interplay between E and PE. We had done all but the last step, and we had reduced the last step to the case where E is a trivial bundle, i.e...

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional $1$-injective Banach space contains a hyperplane is $(2+\varepsilon )$-injective for $\varepsilon \gt 0$, yet not $2$-injective, rem

LetR be a ring andM a rightR-module with S= End(MR). The moduleM is called almost principally quasi-injective (or APQ-injective for short) if, for any m∈M, there exists an S-submodule Xm of M such that lMrR(m) = Sm ⊕ Xm. The module M is called almost quasiprincipally injective (or AQP-injective for short) if, for any s∈ S, there exists a left ideal Xs of S such that lS(ker(s)) = Ss ⊕ Xs. In thi...

Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts random graphs generated by a general graphon and independent vertex labels. We interested these statistics because they key to understanding fluctuations regular — cornerstone dense graph limit theory. also identify resulting limiting Gaussian stochastic measures means theory generalised...