We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions. Following the automorphism case, we introduce two notions for endomorphisms of type III factors: modular endomorphisms and Connes-Takesaki modules. Several applications to compact groups of automorphisms and subfactors of type III factors are give...

This article describes the use of permutation representations to obtain a partial solution to the Extensible Automorphisms Problem. It introduces a notion of permutation-extensibility and proves the equivalence of this notion with the notion of being subgroup-conjugating. 1. The problem we want to solve 1.1. Three big problems. A quick recall of three definitions: Definition. (1) An automorphis...

This article gives the proof of results announced in [Bla07a] and some description of automorphisms of rational surfaces. Given a finite abelian subgroup of the Cremona group of the plane, we give a way to decide whether this one is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the pla...

It is already shown that a Boolean function for a NP-complete problem can be computed by a polynomial-sized circuit if its variables have enough number of automorphisms. Looking at this previous study from the different perspective gives us the idea that the small number of automorphisms might be a barrier for a polynomial time solution for NP-complete problems. Here I show that by interpreting...

A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C 1 and C 2 are structures in C, C 1 nite, C 1 C 2 , and p 1 , p 2 , : : :, p n are partial automorphisms of C 1 extending to automorphisms of C 2 , then there exist a nite structure C 3 in C and automorphisms 1 , 2 , : : :, n of C 3 extending the p i. We will prove that some classes of ...

The aim of this work is to introduce an approach for interval-valued R-implications, which satisfy some analogous properties of R-implications. We show that the best interval representation of an R-implication that is obtained from a left continuous tnorm coincides with the interval-valued R-implication obtained from the best interval representation of such t-norm, whenever this is an inclusion...

This paper establishes decomposability for composition operators induced on the Hardy spaces H (1 ≤ p < ∞) by parabolic linear fractional self-maps of the unit disc that are not automorphisms. This result, along with a recent theorem of Miller and Miller [15], shows that no such composition operator is supercyclic. The work here completes part of a previous investigation [1] where the author an...

The transition graph of a cellular automaton (CA) represents the CA’s global dynamics. The automorphisms of the transition graph are its self-isomorphisms or “symmetries”, and in a sense these are precisely the symmetries of the CA’s dynamics. Previously we have argued that studying how the total number of automorphisms varies with the number of cells on which the CA operates yields a partial c...

A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1, p2, . . . , pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and automorphisms α1, α2, . . . , αn of C3 extending the pi. We will prove that some classes of structur...

Polster and Steinke [Result. Math., 46 (2004), 103–122] determined the possible Kleinewillinghöfer types of flat Laguerre planes. These types reflect transitivity properties of groups of certain central automorphisms. We exclude three more types from the list given there with respect to Laguerre homotheties. This yields a complete determination of all possible single types with respect to Lague...