Rectangularity and paramonotonicity of maximally monotone operators

Maximally monotone operators play a key role in modern optimization and variational analysis. Two useful subclasses are rectangular (also known as star monotone) and paramonotone operators, which were introduced by Brezis and Haraux, and by Censor, Iusem and Zenios, respectively. The former class has useful range properties while the latter class is of importance for interior point methods and duality theory. Both notions are automatic for subdifferential operators and known to coincide for certain matrices; however, more precise relationships between rectangularity and paramonotonicity were not known. Our aim is to provide new results and examples concerning these notions. It is shown that rectangularity and paramonotonicity are actually independent. Moreover, for linear relations, rectangularity implies paramonotonicity but the converse implication requires additional assumptions. We also consider continuous linear monotone operators, and we point out that in Hilbert space both notions are automatic for certain displacement mappings

A Note on One Less Known Class of Generated Residual Implications

This paper builds on our contribution [Havlena and Hlinena, 2016] which studied modelling of the conjunction in human language. We have discussed three different ways of constructing conjunction. We have dealt with generated t-norms, generated means and Choquet integral. In this paper we construct the residual operators based on the above conjunctions. The only operator based on a t-norm is an implication. We show that this implication belongs to the class of generated implications I^g_N which was introduced in [Smutna, 1999] and studied in [Biba and Hlinena, 2012]. We study its properties. More, we investigate this class of generated implications. Some important properties, including relations between some classes of implications, are given

Equivalence after extension for compact operators on Banach spaces

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators that are equivalent after extension, we prove the necessity of similar relationships between the $s$-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary $s$-functions. We investigate equivalence after extension for operators on $\ell^{p}$-spaces. We show that two operators that act on different $\ell^{p}$-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension, however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces $X$ and $Y$, we investigate consequences of an operator on $X$ being equivalent after extension to a compact operator on $Y$. We show that, in this case, a closed finite codimensional subspace of $Y$ must embed into $X$, and that certain general Banach space properties must transfer from $X$ to $Y$. We also show that no operator on $X$ can be equivalent after extension to an operator on $Y$, if $X$ and $Y$ are essentially incomparable Banach spaces

Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, the satisfaction of which is parametrized by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided.Comment: 36 page

A Rule-Based Approach to Analyzing Database Schema Objects with Datalog

Database schema elements such as tables, views, triggers and functions are typically defined with many interrelationships. In order to support database users in understanding a given schema, a rule-based approach for analyzing the respective dependencies is proposed using Datalog expressions. We show that many interesting properties of schema elements can be systematically determined this way. The expressiveness of the proposed analysis is exemplarily shown with the problem of computing induced functional dependencies for derived relations. The propagation of functional dependencies plays an important role in data integration and query optimization but represents an undecidable problem in general. And yet, our rule-based analysis covers all relational operators as well as linear recursive expressions in a systematic way showing the depth of analysis possible by our proposal. The analysis of functional dependencies is well-integrated in a uniform approach to analyzing dependencies between schema elements in general.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854