scholarly journals On complete objects in the category of T0 closure spaces

2003 ◽  
Vol 4 (1) ◽  
pp. 25 ◽  
Author(s):  
D. Deses ◽  
Eraldo Giuli ◽  
E. Lowen-Colebunders
Keyword(s):  
General Theory ◽  
Complete Lattices ◽  
Closure Spaces

<p>In this paper we present an example in the setting of closure spaces that fits in the general theory on “complete objects” as developed by G. C. L. Brümmer and E. Giuli. For V the class of epimorphic embeddings in the construct Cl<sub>0</sub> of T<sub>0</sub> closure spaces we prove that the class of V-injective objects is the unique firmly V-reflective subconstruct of Cl0. We present an internal characterization of the Vinjective objects as “complete” ones and it turns out that this notion of completeness, when applied to the topological setting is much stronger than sobriety. An external characterization of completeness is obtained making use of the well known natural correspondence of closures with complete lattices. We prove that the construct of complete T<sub>0</sub> closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins.</p>

The Paradoxes of Allais and Ellsberg

1986 ◽  
Vol 2 (1) ◽  
pp. 23-53 ◽  
Author(s):  
Isaac Levi
Keyword(s):  
General Theory ◽  
Rational Choice ◽  
Expected Utility ◽  
John Dewey ◽  
Expected Value ◽  
Human Knowledge ◽  
Numerical Indicator ◽  
Charles Peirce ◽  
Oriented Approach

In The Enterprise of Knowledge (Levi, 1980a), I proposed a general theory of rational choice which I intended as a characterization of a prescriptive theory of ideal rationality. A cardinal tenet of this theory is that assessments of expected value or expected utility in the Bayesian sense may not be representable by a numerical indicator or indeed induce an ordering of feasible options in a context of deliberation. My reasons for taking this position are related to my commitment to the inquiry-oriented approach to human knowledge and valuation favored by the American pragmatists, Charles Peirce and John Dewey. A feature of any acceptable view of inquiry ought to be that during an inquiry points under dispute ought to be kept in suspense pending resolution through inquiry.

scholarly journals The Abel map for surface singularities III: Elliptic germs

2021 ◽  
Author(s):  
János Nagy ◽  
András Némethi
Keyword(s):  
General Theory ◽  
Present Note ◽  
Affine Subspace ◽  
Normal Surface ◽  
Rational Homology ◽  
Surface Singularities ◽  
Abel Map ◽  
Appl Math ◽  
Rational Homology Sphere

AbstractThe present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy and Némethi in Adv Math 371:20, 2020; Nagy and Némethi in Pure Appl Math Q 16(4):1123–1146, 2020). Besides the general theory, by the study of specific families we wish to show the power of this new method. Indeed, using the general theory of Abel maps applied for elliptic singularities in this note we are able to prove several key properties for elliptic singularities (e.g. the statements of the next paragraph), which by ‘old’ techniques were not reachable. If $$({\widetilde{X}},E)\rightarrow (X,o)$$ ( X ~ , E ) → ( X , o ) is the resolution of a complex normal surface singularity and $$c_1:{\mathrm{Pic}}({\widetilde{X}})\rightarrow H^2({\widetilde{X}},{\mathbb {Z}})$$ c 1 : Pic ( X ~ ) → H 2 ( X ~ , Z ) is the Chern class map, then $${\mathrm{Pic}}^{l'}({\widetilde{X}}):= c_1^{-1}(l')$$ Pic l ′ ( X ~ ) : = c 1 - 1 ( l ′ ) has a (Brill–Noether type) stratification $$W_{l', k}:= \{{\mathcal {L}}\in {\mathrm{Pic}}^{l'}({\widetilde{X}})\,:\, h^1({\mathcal {L}})=k\}$$ W l ′ , k : = { L ∈ Pic l ′ ( X ~ ) : h 1 ( L ) = k } . In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. E.g., we show that the closure of any $$W(l',k)$$ W ( l ′ , k ) is an affine subspace. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent.

RADICAL CLOSURES IN COMPLETE LATTICES. GENERAL THEORY

1976 ◽  
Vol 29 (3) ◽  
pp. 303-317
Author(s):  
V A Andrunakievič ◽  
Ju M Rjabuhin ◽  
K K Kračilov
Keyword(s):  
General Theory ◽  
Complete Lattices

Slip and Fall Characterization of Floors

2004 ◽  
Author(s):  
Peter J. Poczynok ◽  
Ralph L. Barnett
Keyword(s):  
General Theory ◽  
Contact Point ◽  
Force Plate ◽  
Frictional Resistance ◽  
Value Theory ◽  
Reliability Theory ◽  
Duty Cycles ◽  
Time Period ◽  
Five Disciplines

During ambulation, every maneuver causes the feet to impose a tangential loading at each contact with the floor. If the frictional resistance at the contact point is less than the associated tangential loading, slipping occurs and sometimes falling. There are five disciplines, some recently developed, that enable one to develop the general theory for predicting the number of walkers who will slip within a given time period on a statistically homogeneous and isotropic floor. These include force-plate studies, floor duty cycles, tribometry, extreme value theory of slipperiness, and floor reliability theory. When used with some additional bookkeeping notions, the general theory will be extended to real floors traversed by walkers with multiple ambulation styles and wearing a variety of footwear. In contrast, conventional slip and fall theory does not account for floor usage, different footwear and various ambulation styles, nor can it be used to determine the number of walkers who actually slip on a given floor.

scholarly journals On simpleness of semirings and complete semirings

2014 ◽  
Vol 13 (06) ◽  
pp. 1450015 ◽  
Author(s):  
Yefim Katsov ◽  
Tran Giang Nam ◽  
Jens Zumbrägel
Keyword(s):  
Commutative Monoids ◽  
Complete Lattices ◽  
Infinite Element ◽  
Morita Invariant ◽  
Simple Congruence ◽  
Invariant Properties ◽  
The Relationship ◽  
Finite Distributive Lattices ◽  
The Ideal

In this paper, we investigate various classes of semirings and complete semirings regarding the property of being ideal-simple, congruence-simple, or both. Among other results, we describe (complete) simple, i.e. simultaneously ideal- and congruence-simple, endomorphism semirings of (complete) idempotent commutative monoids; we show that the concepts of simpleness, congruence-simpleness, and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; we also describe congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we obtain the following further results: The ideal-simpleness, congruence-simpleness, and simpleness of semirings are Morita invariant properties; a complete description of simple semirings containing the infinite element; the "Double Centralizer Property" representation theorem for simple semirings; a complete description of simple semirings containing a projective minimal one-sided ideal; a characterization of ideal-simple semirings having either an infinite element or a projective minimal one-sided ideal; settling a conjecture and a problem as published by Katsov in 2004 for the classes of simple semirings containing either an infinite element or a projective minimal left (right) ideal, showing, respectively, that semirings of those classes are not perfect and that the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.

scholarly journals A characterization of complete lattices

1955 ◽  
Vol 5 (2) ◽  
pp. 311-319 ◽  
Author(s):  
Anne C. Davis
Keyword(s):  
Complete Lattices

scholarly journals Free mu-lattices

2000 ◽  
Vol 7 (28) ◽  
Author(s):  
Luigi Santocanale
Keyword(s):  
Ordered Set ◽  
Order Relation ◽  
Equivalence Classes ◽  
Complete Lattices ◽  
Models Of Computation ◽  
Fix Point

A mu-lattice is a lattice with the property that every unary <br />polynomial has both a least and a greatest fix-point. In this paper<br />we define the quasivariety of mu-lattices and, for a given partially<br />ordered set P, we construct a mu-lattice JP whose elements are<br />equivalence classes of games in a preordered class J (P). We prove<br />that the mu-lattice JP is free over the ordered set P and that the<br />order relation of JP is decidable if the order relation of P is <br />decidable. By means of this characterization of free mu-lattices we<br />infer that the class of complete lattices generates the quasivariety<br />of mu-lattices.<br />Keywords: mu-lattices, free mu-lattices, free lattices, bicompletion<br />of categories, models of computation, least and greatest fix-points,<br />mu-calculus, Rabin chain games.

The theory of semi-functors

1993 ◽  
Vol 3 (1) ◽  
pp. 93-128 ◽  
Author(s):  
Raymond Hoofman
Keyword(s):  
General Theory ◽  
Natural Transformation ◽  
Lambda Calculus ◽  
Adjoint Functor ◽  
Typed Lambda Calculus ◽  
The World ◽  
Natural Transformations ◽  
Theoretical Characterization

The notion ofsemi-functorwas introduced in Hayashi (1985) in order to make possible a category-theoretical characterization of models of the non-extensional typed lambda calculus. Motivated by the further use of semi-functors in Martini (1987), Jacobs (1991) and Hoofman (1992a), (1992b) and (1992c), we consider the general theory of semi-functors in this paper. It turns out that the notion ofsemi natural transformationplays an important part in this theory, and that various categorical notions involving semi-functors can be viewed as 2-categorical notions in the 2-category of categories, semi-functors and semi natural transformations. In particular, we find that the notion ofnormal semi-adjunctionas defined in Hayashi (1985) is the canonical generalization of the notion of adjunction to the world of semi-functors. Further topics covered in this paper are the relation between semi-functors and splittings, the Karoubi envelope construction, semi-comonads, and a semi-adjoint functor theorem.

Extensions of Closure Spaces

1977 ◽  
Vol 29 (6) ◽  
pp. 1277-1286 ◽  
Author(s):  
K. C. Chattopadhyay ◽  
W. J. Thron
Keyword(s):  
General Theory ◽  
Extension Theory ◽  
Topological Spaces ◽  
Completely Regular ◽  
Closure Spaces ◽  
Basic Concepts

Extension theory has been intensively studied for completely regular spaces and is fairly well developed for T0-topological spaces. (See, for example, [1] and [5]). However, except for definitions of some of the basic concepts in [4] and results on embedding of closure spaces in cubes in [2] and [7], ours is the first study of the general theory of extensions of G0-closure spaces. (Definitions will be given following these introductory paragraphs).

scholarly journals Even every Join-extension solves a Universal problem

1976 ◽  
Vol 21 (2) ◽  
pp. 220-223 ◽  
Author(s):  
Isidore Fleischer
Keyword(s):  
Complete Lattices

AbstractGeneralization to arbitrary join-extensions of a poset of a recent characterization of those which are complete lattices.

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