quantales finite observations and strong bisimulation abstract it has often been claimed that bisimulation makes distinctions that cannot be observed in practice abramsky and vickers proposed an algebraic framework based on quantales for describing observations on concurrent processes without hidden transitions and used it in order to provide an observational explanation of several process equivalences ranging from hoare trace equivalence to ready simulation we follow their approach and argue that strong bisimulation can be explained in the same way at least in the case of image computable labelled transition systems

global invariants for measured foliations new exotic invariants for measured foliations are constructed which we call the u classes of a pair f u the dependence of the s classes on the geometry of the foliation gis examined and the dynamics of a foliation is shown to determine the u classes in many cases we use the u classes to study the classifying space bfsl of foliations with a transverse invariant volume form and we show the homotopy groups of bfsl are uncountably generated starting in degrees q 3 new invariants for groups of volume preserving diffeomorphisms also arise from the t classes these invariants are nontrivial and related to the geometric aspects of the group action relations between the tl classes and the secondary classes of a foliation are exhibited 0 introduction in this paper we construct new exotic characteristic classes for foliations which have a transverse invariant measure these invariants which we call the t classes are similar to the secondary classes and take values in the real cohomology h m of the foliated manifold m theorem 1 6 the values of the m classes are strongly related to the geometry of the foliation proposition 3 4 the u classes have applications to the study of foliations with a transverse invariant volume form theorems 5 1 and 5 2 to the computations of the homology groups of the discrete group of volume preserving diffeomorphisms of rw theorem 7 4 and to the problem of relating the secondary classes of a foliation with its global geometry 15 201 the a classes arose from an attempt to obtain general invariants of a foliation which are intermediate between the leaf and secondary classes they are a special case of a family of universal invariants associated to a c2 foliation the weil operators in codimension one the introduction of the weil operator is a decisive ingredient in duminy s elegant proof of the sullivan conjecture about the godbillon vey class 9 the properties of the weil operators in higher codimensions are given in 15 where they are used to study the residuable secondary classes we briefly describe the construction of the weil operators and their relations with the l classes and other foliation invariants in 2 for a foliation with all leaves compact we note that the weil operators are identically zero proposition 2 6 we denote by f a c2 foliation of codimension q on a paracompact manifold m and i denotes a transverse invariant measure for sf recall that ti assigns to each received by the editors june 28 1982 and in revised form november 9 1982 1980 mathematics subject classification primary 57r30 57r20 58f1 1 secondary 58h10 28d15

submanifolds with splitting tangent sequence a theorem of van de ven states that a projective submanifold of complex projective space whose holomorphic tangent bundle sequence splits holomorphically is necessarily a linear subspace note that the sequence always splits differentiably but in general not holomorphically we are interested in generalizations to the case when the ambient space is a homogeneous manifold different from projective space quadrics grassmannians or abelian manifolds for example split submanifolds are closely related to totally geodesic submanifolds

representations of fuzzy topologies we make use of the point space concept 2 to give some representations of fuzzy topologies by classical topologies we show that fuzzy convergence and fuzzy continuity with respect given fuzzy topologies are reducible to the convergence and the continuity with respect to suitable associated topologies

a note on the hypergraph functor the purpose of this note is to show that in the case of spatial locales l the hypergraph functor has a right adjoint functor

category theoretic aspects of chain valued frames part ii applications to lattice valued topology this paper is part ii of a two part series dealing with category theoretic aspects of chain valued frames using the categorical properties established for l frm in part i for l a complete chain this paper constructs upper free functor l and lower free functor r the functor l is used to create a class of non generated l frames factor the lpt spectrum functor through the s spectrum functor resolve the relationship between l sobriety and i l sobriety and give insight into the construction of universal l topological spaces and the functor r is used to create a class of non generated l frames factor the w l functor through a new spectrum functor s and give insight into the construction of co universal l topological spaces these facts universal spaces are anti stratified co universal spaces are stratified l frm produces both kinds of spaces via l and r l frm via its dual l loc is adjunctive with l top comprise a coherent argument that l top must accommodate both kinds of spaces resolving the philosophical debate over the place of the constant maps condition with respect to the axioms of l topologies

category theoretic aspects of chain valued frames part i categorical and presheaf theoretic foundations this paper is part i of a two part series dealing with category theoretic aspects of chain valued frames a new categorical motivation for lattice valued frames is given from presheaves and then under the assumption that l be a complete chain it is established that standard category theoretic properties completeness cocompleteness factorization structures hold for l frm the category of l frames and l frame morphisms properties which are a platform for the upper free functor l and lower free functor r used in part ii to give a broad range of applications of chain valued frames to lattice valued topology

fuzzy points and attachment in this paper a binary relation between l sets called attachment is defined by means of a suitable family of completely coprime filters in l examples are given using several kinds of lattice ordered algebras and a class of attachments is determined whose elements generalize pu liu s quasi coincidence relation mapping any l set on x to the set of the attached l points one gets a frame map from the l powerset l x to the powerset p s x of the set s x of l points on x this allows one to define functors from the category l top of l topological spaces either to the category top of topological spaces or to the category topsys of topological systems a comparison with the hypergraph functor is also included

a comparison theorem if x is complete these maps are isomorphisms but in general they are not this ibllows from the following standard example due to serre 4 p 232 one takes a certain ip 1 bundle x over a non singular elliptic curve and removes a curve y the result is a non singular algebraic surface u x y on which every regular function is constant h the associated complex space ua of u is a stein naanifold biholomorphic to x where denotes tl 0 4 in particular we see that in this example the comparison map o h u gv h gv is not an isomorphism the point is that in serre s example the degree of the normal bundle nr x of y in x is zero this suggests the following problem let x be an algebraic variety over c ycx a closed subvariety defined by the ideal j find sufficient conditions on the normal sheaf nr x of y in x such that for every coherent algebraic sheaf f on x y the comparison maps h x y van

necessity of non stratified and anti stratified spaces in lattice valued topology this paper surveys the necessity of non stratified spaces from these viewpoints the characteristic functor the l spectrum and l soberification functors the upper free functor associated with l valued frames and two functorial embeddings associated with topological systems from semantic domains interestingly additional arguments also emerge en route for stratified spaces so that this paper ultimately argues for the necessity of both stratified and non stratified spaces in fixed basis topology the schemum of l top s and variable basis topology e g loc top

flat bundles and residues for foliations in this note we compute the residues for a vector field preserving a foliation we assume the vector field disappears only to first order along its singular set and that its singular set consists of closed separated leaves under these conditions the residue is completely determined by the characteristic classes of the flat bundle v o which is the normal bundle v of the foliation restricted to the singular set and the action of the vector field on v o for a foliation of even codimension q whose normal bundle has euler class zero we interpret the pontrjagin ring of the normal bundle in dimension 2q as an obstruction to the existence of such a vector field we show that if the euler class of the normal bundle of a foliation is zero and it admits such a vector field then many of its secondary characteristic classes must be also be zero this is one of the few results relating the geometry of a foliation to its characteristic classes we will make extensive use of the material in 6 it will be very helpful to the reader to be familiar with this paper

ampleness and connectedness in complex g p this paper determines the ampleness of the tangent bundle of the complex homogeneous space g p by calculating the maximal fibre dimension of the desingularization of a nilpotent subvariety of the lie algebra of g

hypergraph functor and attachment using an arbitrary variety of algebras the paper introduces a fuzzified version of the notion of attachment in a complete lattice of guido to provide a common framework for the concept of hypergraph functor considered by different authors in the literature the new notion also gives rise to a category of variable basis topological spaces which is a proper supercategory of the respective category of rodabaugh

relationship of algebraic theories to powersets over objects in set and set c this paper deals with a particular question when do powersets in lattice valued mathematics form algebraic theories or monads in clone form our approach in this and related papers is to consider powersets over objects in the ground categories set and setxc from the standpoint of algebraic theories in clone form c is a particular subcategory of the dual of the category of semi quantales for both fixed basis powersets over objects of set and variable basis powersets over objects of setxc necessary and sufficient conditions are found under which the family of all such powersets over a ground object forms an algebraic theory in clone form of standard construction in such results a distinguished role emerges for unital quantales

sobriety and spatiality in varieties of algebras the paper considers a generalization of the classical papert papert isbell adjunction between the categories of topological spaces and locales to an arbitrary variety of algebras and illustrates the obtained results by the category of algebras over a given unital commutative quantale

subvarieties with partially ample normal bundle we show that local complete intersection subvarieties of smooth projective varieties which have partially ample normal bundle possess the g2 property this generalizes results of hartshorne and b descu schneider

fuzzy algebras as a framework for fuzzy topology the paper introduces a variety based version of the notion of the l m fuzzy topological space of kubiak and sostak and embeds the respective category into a suitable modification of the category of topological systems of vickers the new concepts provide a common framework for different approaches to fuzzy topology and topological systems existing in the literature paving the way for studying the problem of interweaving algebra and topology in mathematics which was raised by denniston melton and rodabaugh in their recent research on variable basis topological systems over the category of locales