Contact reduction and groupoid actions

We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of on , and we show that the quotients of fibers by suitable Lie subgroups are either contact or locally conformal symplectic manifolds with structures induced by the one on .

We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.

     

Kites and residuated lattices

We investigate a construction of an integral residuated lattice starting from an integral residuated lattice and two sets with an injective mapping from one set into the second one. The resulting algebra has a shape of a Chinese cascade kite, therefore, we call this algebra simply a kite. We describe subdirectly irreducible kites and we classify them. We show that the variety of integral residuated lattices generated by kites is generated by all finite-dimensional kites. In particular, we describe some homomorphisms among kites.     

Continuous-trace groupoid crossed products

Let be a second countable, locally compact groupoid with Haar system, and let be a bundle of -algebras defined over the unit space of on which acts continuously. We determine conditions under which the associated crossed product is a continuous trace -algebra.

     

Cancellative residuated lattices

Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups). Although cancellative monoids are defined by quasi-equations, the class of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of .We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of to the lattice of subvarieties of . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzies characterization of categorically equivalent varieties.     

Induced residuated maps

The purpose of this paper is to introduce a class of mappings from a lattice L, whose elements are residuated maps, into itself. The main results of this paper identify certain injective residuated mappings of L and order automorphisms of a sublattice of L with mappings from this class.     

Regularity of residuated mappings

Communicated by Boris M. Schein     

Injectives in residuated algebras

Injectives in several classes of structures associated with logic are characterized. Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BL-algebras, NM-algebras and bounded hoops.     

Semiconic idempotent residuated structures

An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form avariety SCIL, which is not locally finite, but it is proved that SCIL has the finite embeddability property (FEP). More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains.     

On distributive residuated groupoids

In this note we extend the well-known fact (see [1], p. 45) that an implicative (or Brouwerian) lattice is distributive to a large class of residuated groupoids.     

强Ockham代数与剩余格

首先讨论了Ockham代数与剩余格的关系,引入了强Ockham代数的概念,并讨论了它的基本性质．然后,将著名的风蕴涵和风算子推广到Ockham代数上,证明了添加广义R0蕴涵和广义风算子后的Ockham代数L成为剩余格的充要条件是L为强Ockham代数．最后给出若干重要例子,以此来说明强Ockham代数的条件是独立的．     

Singly generated quasivarieties and residuated structures

A quasivariety $\mathsf{K}$ of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A . It is structurally complete if and only if the free ℵ₀-generated algebra in $\mathsf{K}$ can serve as A . A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of $\mathsf{K}$ all satisfy the same existential positive sentences. We prove that if $\mathsf{K}$ is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if $\mathsf{K}$ is relatively semisimple then it is generated by one $\mathsf{K}$-simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics.     

Commutative idempotent residuated lattices

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct.     

Constructing simple residuated lattices

A method of constructing residuated lattices is presented. As an application, examples of simple, integral, cancellative, distributive residuated lattices are given that are not linearly ordered. This settles a problem raised in [5] and [2].     

On calculating residuated approximations

For an order-preserving map f : L → Q between two complete lattices L and Q, there exists a largest residuated map ρ _f under f, which is called the residuated approximation of f. Andreka, Greechie, and Strecker introduced the notion of the shadow σ _f of f Iterations of the shadow are called the umbral mappings. The umbral mappings form a decreasing net that converges to the residuated approximation ρ _f of f. The umbral number u _f of f is the smallest ordinal number α such that the equation \({\sigma^{(\alpha)}_{f} = \rho_{f}}\) holds. In order to speed up the computation of the umbral number u _f of f and find some relation between the structure of L and u _f , we present the concept of the order skeleton of a lattice \({L, \tilde{L} = L/\sim}\), determined by a certain congruence relation ~ on L where each equivalence class [x] is the maximal autonomous chain containing x. If [x] is finite for each \({x \in L}\), then \({L_{o} := \{ \Lambda [x]\,|\, x \in L \}}\) is a join-subcomplete sub-semilattice of L isomorphic to the order skeleton \({\tilde{L}}\) of L; for every order-preserving mapping f : L → Q from such a lattice L to a complete lattice Q, we define f _o : L _o → Q by \({f_{o} := f|_{{L}_{o}}}\) and prove that \({u_{f} = u_{{f}_{o}}}\). For a lattice L with no infinite chains, the order skeleton \({\tilde{L}}\) of L is distributive if and only if the shadow σ _f of f is residuated for every complete lattice Q and every mapping f : L → Q. Related topics are discussed.     

The groupoid of prevarieties of lattices

Siberian Mathematical Journal -     

Longitudinal smoothness of the holonomy groupoid

Iakovos Androulidakis and Georges Skandalis have defined a holonomy groupoid for any singular foliation. This groupoid, whose topology is usually quite bad, is the starting point for the study of longitudinal pseudodifferential calculus on such foliation and its associated index theory. These studies can be highly simplified under the assumption of the holonomy groupoid being longitudinally smooth. In this note, we rephrase the period bounding lemma that asserts that a vector field on a compact manifold admits a strictly positive lower bound for its periodic orbits in order to prove that the holonomy groupoid is always longitudinally smooth.