No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Counting and Computing Join-Endomorphisms in Lattices
Abstract
Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. We study the cardinality of the set of all join-endomorphisms of a given finite lattice L. In particular, we show that when L is , the discrete order of n elements extended with top and bottom, where is the Laguerre polynomial of degree n. We also study the following problem: Given a lattice L of size n and a set of size m, find the greatest lower bound Open image in new window. The join-endomorphism Open image in new window has meaningful interpretations in epistemic logic, distributed systems, and Aumann structures. We show that this problem can be solved with worst-case time complexity in for powerset lattices, for lattices of sets, and for arbitrary lattices. The complexity is expressed in terms of the basic binary lattice operations performed by the algorithm.
... This approach works since for any set S of join-endomorphisms ( S(C) S)(c) = C {f (c) | f ∈ S}. The problem, however, is that the number of join-endomorphisms over a distributive lattice can be non-polynomial in the size of the lattice [19]. ...
... In [19] the authors investigate the cardinality of the set E(L) of all join-endomorphisms of a given lattice L. (A join-endomorphism is a self-map that preserves finite joins, hence it is a space function without the continuity requirement.) The authors also provide efficient algorithms to compute the meet of a given set of join-endomorphisms. ...
Spatial constraint systems (scs) are semantic structures for reasoning about spatial and epistemic information in concurrent systems. We develop the theory of scs to reason about the distributed information of potentially infinite groups. We characterize the notion of distributed information of a group of agents as the infimum of the set of join-preserving functions that represent the spaces of the agents in the group. We provide an alternative characterization of this notion as the greatest family of join-preserving functions that satisfy certain basic properties. For completely distributive lattices, we establish that distributed information of a group is the greatest information below all possible combinations of information in the spaces of the agents in the group that derive a given piece of information. We show compositionality results for these characterizations and conditions under which information that can be obtained by an infinite group can also be obtained by a finite group. Finally, we provide an application on mathematical morphology where dilations, one of its fundamental operations, define an scs on a powerset lattice. We show that distributed information represents a particular dilation in such scs.
... using equation (16). Then, for f , g cotight, we have ...
It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.
... Counting structures of interest has become a rising theme of investigation in lattice theory and related areas of research. For example, [1] counts various kinds of doubly idempotent semirings, [16] deals with join-endomorphisms in lattices, the subject of [3] are topological spaces and [5] generates and counts a certain kind of bisemilattices. However, there is no work concerning the number of closure operators in general lattices or orders. ...
This paper investigates the interplay between isolated sublattices and closure operators. Isolated sublattices are a special kind of sublattices which can serve to diminish the number of elements of a lattice by means of a quotient. At the same time, there are simple formulae for the relationship between the number of closure operators in the original lattice and the quotient lattice induced by isolated sublattices. This connection can be used to derive an algorithm for counting closure operators, provided the lattice contains suitable isolated sublattices.
Let L be a distributive lattice and E(L) be the set of join endomorphisms of L. We consider the problem of finding f⊓E(L)g given L and f,g∈E(L) as inputs. (1) We show that it can be solved in time O(n) where n=|L|. The previous upper bound was O(n2). (2) We characterize the standard notion of distributed knowledge of a group as the greatest lower bound of the join-endomorphisms representing the knowledge of each member of the group. (3) We show that deciding whether an agent has the distributed knowledge of two other agents can be computed in time O(n2) where n is the size of the underlying set of states. (4) For the special case of S5 knowledge, we show that it can be decided in time O(nαn) where αn is the inverse of the Ackermann function.
Spatial constraint systems (scs) are semantic structures for reasoning about spatial and epistemic information in concurrent systems. We develop the theory of scs to reason about the distributed information of potentially infinite groups. We characterize the notion of distributed information of a group of agents as the infimum of the set of join-preserving functions that represent the spaces of the agents in the group. We provide an alternative characterization of this notion as the greatest family of join-preserving functions that satisfy certain basic properties. For completely distributive lattices, we establish that the distributed information of c amongst a group is the greatest lower bound of all possible combinations of information in the spaces of the agents in the group that derive c. We show compositionality results for these characterizations and conditions under which information that can be obtained by an infinite group can also be obtained by a finite group. Finally, we provide an application on mathematical morphology where dilations, one of its fundamental operations, define an scs on a powerset lattice. We show that distributed information represents a particular dilation in such scs.
The importance of complete lattices in mathemati-cal morphology is well-known. There are some as-pects of the subject that can, however, be clarified by using complete lattices equipped with the ad-ditional structure of a binary operation subject to conditions making them quantales. This extended abstract shows how quantales, and more generally quantale modules, can be used to capture a num-ber of constructions for dilations in a uniform way. The treatment is partly an exposition of material that is not technically novel, but which provides a valuable conceptual perspective on mathematical morphology.
We introduce spatial and epistemic process calculi for reasoning about spatial information and knowledge distributed among the agents of a system. We introduce domain-theoretical structures to represent spatial and epistemic information. We provide operational and denotational techniques for reasoning about the potentially infinite behaviour of spatial and epistemic processes. We also give compact representations of infinite objects that can be used by processes to simulate announcements of common knowledge and global information.
Modular lattices, introduced by R. Dedekind, are an important subvariety of
lattices that includes all distributive lattices. Heitzig and Reinhold
developed an algorithm to enumerate, up to isomorphism, all finite lattices up
to size 18. Here we adapt and improve this algorithm to construct and count
modular lattices up to size 23, semimodular lattices up to size 22, and
lattices of size 19. We also show that is a lower bound for the
number of nonisomorphic modular lattices of size n.
Preface Acknowledgements Foreword Introduction 1. A primer on ordered sets and lattices 2. Order theory of domains 3. The Scott topology 4. The Lawson Topology 5. Morphisms and functors 6. Spectral theory of continuous lattices 7. Compact posets and semilattices 8. Topological algebra and lattice theory: Applications Bibliography List of symbols List of categories Index.
We show how the ntcc calculus, a model of temporal concurrent constraint programming with the capability of modeling asynchronous and non-deterministic timed behavior, can be used for modeling real musical processes. In particular, we show how the expressiveness of ntcc allows to implement complex interactions among musical processes handling different kinds of partial information. The ntcc calculus integrates two dimensions of soft computing: a horizontal dimension dealing with partial information and a vertical one in which non determinism comes into play. This integration is an improvement over constraint satisfaction and concurrent constraint programming models, allowing a more natural representation of a variety of musical processes. We use the nondeterminism facility of ntcc to build weaker representations of musical processes that greatly simplifies the formal expression and analysis of its properties. We argue that this modeling strategy provides a runnable specification for music problems that eases the task of formally reasoning about them. We show how the linear temporal logic associated with ntcc gives a very expressive setting for formally proving the existence of interesting musical properties of a process. We give examples of musical specifications in ntcc and use the linear temporal logic for proving properties of a realistic musical problem.
Spatial constraint systems (scs) are semantic structures for reasoning about spatial and epistemic information in concurrent systems. We develop the theory of scs to reason about the distributed information of potentially infinite groups. We characterize the notion of distributed information of a group of agents as the infimum of the set of join-preserving functions that represent the spaces of the agents in the group. We provide an alternative characterization of this notion as the greatest family of join-preserving functions that satisfy certain basic properties. For completely distributive lattices, we establish that the distributed information of c amongst a group is the greatest lower bound of all possible combinations of information in the spaces of the agents in the group that derive c. We show compositionality results for these characterizations and conditions under which information that can be obtained by an infinite group can also be obtained by a finite group. Finally, we provide an application on mathematical morphology where dilations, one of its fundamental operations, define an scs on a powerset lattice. We show that distributed information represents a particular dilation in such scs.
The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words such that ) has a canonical monoid structure inherited from the bijection with the set of join-continuous map s from the chain to itself. We explicitly describe this monoid structure and, relying on a general characterization of idempotent join-continuous map s from a complete lattice to itself, we characterize idempotent paths as upper zigzag paths. We argue that these paths are counted by the odd Fibonacci numbers. Our method yields a geometric/combinatorial proof of counting results, due to Howie and to Laradji and Umar, for idempotents in monoids of monotone endomap s on finite chains.
This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening relation algebras is not finitely axiomatizable. These algebras play a role similar to representable relation algebras, and we identify a finitely-based variety of cyclic involutive GBI-algebras that includes all weakening relation algebras. We also show that algebras of down-closed sets of partially-ordered groupoids are bounded cyclic involutive GBI-algebras.
The notion of constraint system (cs) is central to declarative formalisms from concurrency theory such as process calculi for concurrent constraint programming (ccp). Constraint systems are often represented as lattices: their elements, called constraints, represent partial information and their order corresponds to entailment. Recently a notion of n-agent spatial cs was introduced to represent information in concurrent constraint programs for spatially distributed multi-agent systems. From a computational point of view a spatial constraint system can be used to specify partial information holding in a given agent's space (local information). From an epistemic point of view a spatial cs can be used to specify information that a given agent considers true (beliefs). Spatial constraint systems, however, do not provide a mechanism for specifying the mobility of information/processes from one space to another. Information mobility is a fundamental aspect of concurrent systems.
Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 2
2n/3 order preserving maps (and 2
2
in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.
The basic operations of mathematical morphology, dilation and erosion, were introduced by Matheron and Serra. They were initially defined as Minkowski addition and subtraction on subsets of the Euclidean space, using translations, unions and intersections. Following Sternberg, they were generalized to the set of grey-level images with the help of umbras. Recently Serra and Matheron have generalized morphological operations to complete lattices, that is, sets in which the operations of supremum and infimum are well-defined. This generalization has proven useful by extending the scope of mathematical morphology to other structures. In this paper we show that it is also necessary for a mathematically coherent application of morphological operators to grey-level images. Indeed • In the continoous case, the definition of dilations on umbras is not exactly the same as for ordinary Euclidean sets; here the union must be replaced by a supremum operation similar to the one in the complete lattice of closed sets. Moreover, dilations and erosions can be defined directly with lattice-theoretic methods, without recourse to umbras. • In the digital case, when the set of grey-levels is bounded, the problem of grey-level overflow can be dealt with correctly only by taking into account the complete lattice structure of the set of grey-level images. Otherwise the properties of morphological operators are lost.
From a well-known decomposition theorem, we propose a tree representation for distributive and simplicial lattices. We show how this representation (called ideal tree) can be efficiently computed (linear time in the size of the lattice given by any graph whose transitive closure is the lattice) and compared with respect to time and space complexity. As far as time complexity is concerned, we simply consider the time needed for computations of basic lattice operations such as meet or join and reachability (x ⩽ y). Therefore an ideal tree can be considered as a good data structure for a distributive lattice, since for a lattice L = (X,E) it uses O(¦X¦) space and allows computations of reachability, meet and join operations in O(¦M(L)¦), where M(L) denotes the suborder of the meet irreducible elements in L. Furthermore, optimal bit-vector encoding for distributive lattices can be easily derived from this data structure. Relationships with encoding proposed by Aït-Kaci et al. [3], Caseau [5] are also discussed. Intensive use of this ideal tree allow us to achieve best running time algorithms for most of the applications in which distributive lattices are involved; as for example, constructing the lattice of ideals or generating ideals for a given partial order. Therefore this data structure can be used in many areas such as scheduling theory, in which several algorithms are based on a dynamic programming approach of the lattice of ideals of the precedence ordering; or distributed programming, in which some of the debugging tools rely on the calculation of the lattice of ideals of the causality ordering of the events.
This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.