A note on drastic product logic

The drastic product $*_D$ is known to be the smallest $t$-norm, since $x *_D y = 0$ whenever $x, y < 1$. This $t$-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product $t$-norm based many-valued logics, in the sense of [EG01]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called ${\rm S}_{3}{\rm MTL}$ in [NOG06]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the $\Delta$ projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure

Property (T) and rigidity for actions on Banach spaces

We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when 1. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement

Eigenvarieties and invariant norms: Towards p-adic Langlands for U(n)

We give a proof of the Breuil-Schneider conjecture in a large number of cases, which complement the indecomposable case, which we dealt with earlier in [Sor]. In some sense, only the Steinberg representation lies at the intersection of the two approaches. In this paper, we view the conjecture from a broader global perspective. If $U_{/F}$ is any definite unitary group, which is an inner form of \GL(n) over \K, we point out how the eigenvariety \X(K^p) parametrizes a global $p$-adic Langlands correspondence between certain $n$-dimensional $p$-adic semisimple representations $\rho$ of \Gal(\bar{\Q}|\K) (or what amounts to the same, pseudo-representations) and certain Banach-Hecke modules $\mathcal{B}$ with an admissible unitary action of U(F\otimes \Q_p), when $p$ splits. We express the locally regular-algebraic vectors of $\mathcal{B}$ in terms of the Breuil-Schneider representation of $\rho$. Upon completion, this produces a candidate for the $p$-adic local Langlands correspondence in this context. As an application, we give a weak form of local-global compatibility in the crystalline case, showing that the Banach space representations $B_{\xi,\zeta}$ of Schneider-Teitelbaum [ScTe] fit the picture as predicted. There is a compatible global mod $p$ (semisimple) Langlands correspondence parametrized by \X(K^p). We introduce a natural notion of refined Serre weights, and link them to the existence of crystalline lifts of prescribed Hodge type and Frobenius eigenvalues. At the end, we give a rough candidate for a local mod $p$ correspondence, formulate a local-global compatibility conjecture, and explain how it implies the conjectural Ihara lemma in [CHT].Comment: Comments and suggestions are very welcom

Quantale Modules and their Operators, with Applications

The central topic of this work is the categories of modules over unital quantales. The main categorical properties are established and a special class of operators, called Q-module transforms, is defined. Such operators - that turn out to be precisely the homomorphisms between free objects in those categories - find concrete applications in two different branches of image processing, namely fuzzy image compression and mathematical morphology

On the universal module of pp-adic spherical Hecke algebras

Let $\widetilde{G}$ be a split connected reductive group with connected center $Z$ over a local non-Archimedean field $F$ of residue characteristic $p$, let $\widetilde{K}$ be a hyperspecial maximal compact open subgroup in $\widetilde{G}$. Let $R$ be a commutative ring, let $V$ be a finitely generated $R$-free $R[\widetilde{K}]$-module. For an $R$-algebra $B$ and a character $\chi:{\mathfrak H}_V(\widetilde{G},\widetilde{K})\to B$ of the spherical Hecke algebra ${\mathfrak H}_V(\widetilde{G},\widetilde{K})={\rm End}_{R[\widetilde{G}]}{\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we consider the specialization $$M_{\chi}(V)={\rm ind}_{\widetilde{K}}^{\widetilde{G}}V\otimes_{{\mathfrak H}_V(\widetilde{G},\widetilde{K}),\chi}B$$ of the universal ${\mathfrak H}_V(\widetilde{G},\widetilde{K})$-module ${\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$. For large classes of $R$ (including ${\mathcal O}_F$ and $\overline{\mathbb F}_p$), $V$, $B$ and $\chi$, arguing geometrically on the Bruhat Tits building we give a sufficient criterion for $M_{\chi}(V)$ to be $B$-free and to admit a $\widetilde{G}$-equivariant resolution by a Koszul complex built from finitely many copies of ${\rm ind}_{\widetilde{K}Z}^{\widetilde{G}}(V)$. This criterion is the exactness of certain fairly small and explicit ${\mathfrak N}$-equivariant $R$-module complexes, where ${\mathfrak N}$ is the group of ${\mathcal O}_F$-valued points of the unipotent radical of a Borel subgroup in $\widetilde{G}$. We verify it if $F={\mathbb Q}_p$ and if $V$ is an irreducible $\overline{\mathbb F}_p[\widetilde{K}]$-representation with highest weight in the (closed) bottom $p$-alcove, or a lift of it to ${\mathcal O}_F$. We use this to construct $p$-adic integral structures in certain locally algebraic representations of $\widetilde{G}$