scholarly journals Near-rings of quotients of endomorphism near-rings

1975 ◽  
Vol 19 (4) ◽  
pp. 345-352 ◽  
Author(s):  
Michael Holcombe
Keyword(s):  
Additive Group ◽  
Finite Products ◽  
Definition Of ◽  
Group Object ◽  
Rings Of Quotients ◽  
Natural Way

Let be a category with finite products and a final object and let X be any group object in . The set of -morphisms, (X, X) is, in a natural way, a near-ring which we call the endomorphism near-ring of X in Such nearrings have previously been studied in the case where is the category of pointed sets and mappings, (6). Generally speaking, if Γ is an additive group and S is a semigroup of endomorphisms of Γ then a near-ring can be generated naturally by taking all zero preserving mappings of Γ into itself which commute with S (see 1). This type of near-ring is again an endomorphism near-ring, only the category is the category of S-acts and S-morphisms (see (4) for definition of S-act, etc.).

scholarly journals Boundaries in digital planes

1990 ◽  
Vol 3 (1) ◽  
pp. 27-55 ◽  
Author(s):  
Efim Khalimsky ◽  
Ralph Kopperman ◽  
Paul R. Meyer
Keyword(s):  
Jordan Curve ◽  
Ordered Set ◽  
Continuous Functions ◽  
Jordan Curve Theorem ◽  
Connected Sets ◽  
Digital Plane ◽  
Parameter Interval ◽  
Totally Ordered Set ◽  
Definition Of ◽  
Natural Way

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.In [KKM] we introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends our previous work by considering some questions associated with image processing:How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? Our construction gives a unified answer to these questions.The crucial step in making our approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits us to define path, arc, and curve as certain continuous functions on such a parameter interval.

scholarly journals On Near-Rings of Quotients

1979 ◽  
Vol 22 (2) ◽  
pp. 77-86 ◽  
Author(s):  
A. Oswald
Keyword(s):  
Finite Rank ◽  
Minimum Condition ◽  
Suitable Definition ◽  
Definition Of ◽  
Rings Of Quotients

In (2), Holcombe investigated near-rings of zero-preserving mappings of a group Γ which commute with the elements of a semigroup S of endomorphisms of Γ and examined the question: under what conditions do near-rings of this type have near-rings of right quotients which are 2-primitive with minimum condition on right ideals? In the first part of this paper (§2) we investigate further properties of near-rings of this type. The second part of the paper (§3) deals with those near-rings which have semisimple near-rings of right quotients. Our results here are analogous to those of Goldie (1); in particular, with a suitable definition of finite rank we prove that a near-ring which has a semisimple near-ring of right quotients has finite rank

scholarly journals CONTACT STRUCTURES, σ-CONFOLIATIONS AND CONTAMINATIONS IN 3-MANIFOLDS

2009 ◽  
Vol 11 (02) ◽  
pp. 201-264 ◽  
Author(s):  
ULRICH OERTEL ◽  
JACEK ŚWIATKOWSKI
Keyword(s):  
Contact Structure ◽  
Contact Structures ◽  
Tight Contact ◽  
Branched Surface ◽  
Definition Of ◽  
Branched Surfaces ◽  
Natural Way

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called σ-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the σ-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact structure up to isotopy. After defining tightness for all pure contaminations in a natural way, generalizing the definition of tightness for contact structures, we obtain some conditions on (the embedding of) a branched surface in a 3-manifold sufficient to guarantee that any pure contamination carried by the branched surface is tight. We also find conditions sufficient to prove that a branched surface carries only overtwisted (non-tight) contact structures. Our long-term goal in developing these methods is twofold: Not only do we want to study tight contact structures and pure contaminations, but we also wish to use them as tools for studying 3-manifold topology.

scholarly journals Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

2018 ◽  
Vol 15 (08) ◽  
pp. 1830003 ◽  
Author(s):  
Víctor Manuel Jiménez ◽  
Manuel de León ◽  
Marcelo Epstein
Keyword(s):  
Second Order ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Text Structures ◽  
Cosserat Medium ◽  
Homogeneity Property ◽  
Definition Of ◽  
Cosserat Media ◽  
Natural Way

A Lie groupoid, called second-order non-holonomic material Lie groupoid, is associated in a natural way to any Cosserat medium. This groupoid is used to give a new definition of homogeneity which does not depend on a material archetype. The corresponding Lie algebroid, called second-order non-holonomic material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously obtained ones in terms of non-holonomic second-order [Formula: see text]-structures.

scholarly journals Cartesian differential categories revisited

2015 ◽  
Vol 27 (1) ◽  
pp. 70-91 ◽  
Author(s):  
G. S. H. CRUTTWELL
Keyword(s):  
General Version ◽  
Finite Products ◽  
Definition Of ◽  
Differential Categories

We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over categories with finite products, so that every category with finite products has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.

scholarly journals On Aggregation and Computation on Domain Values

1992 ◽  
Vol 21 (414) ◽  
Author(s):  
Kim Skak Larsen
Keyword(s):  
Query Languages ◽  
Relational Algebra ◽  
Way Of Thinking ◽  
Definition Of ◽  
Almost All ◽  
Natural Way

<p>Query languages often allow a limited amount of anthmetic and string operations on domain values, and sometimes sets of values can be dealt with through aggregation and sometimes even set comparisons. We address the question of how these facilities can be added to a relational language in a natural way. Our discussions lead us to reconsider the definition of the standard operators, and we introduce a new way of thinking about relational algebra computations.</p><p>We define a language FC, which has an iteration mechanism as its basis. A tuple language is used to carry out almost all computations. We prove equivalence results relating FC to relational algebra under various circumstances.</p>

scholarly journals Cotorsion radicals and projective modules

1971 ◽  
Vol 5 (2) ◽  
pp. 241-253 ◽  
Author(s):  
John A. Beachy
Keyword(s):  
Projective Module ◽  
Projective Modules ◽  
Left Module ◽  
Perfect Ring ◽  
Artinian Rings ◽  
Trace Ideal ◽  
Dual Notion ◽  
Rings Of Quotients ◽  
Torsion Radical ◽  
Natural Way

We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.Torsion radicals in categories of modules have been studied extensively in connection with quotient categories and rings of quotients. (See [8], [12] and [13].) In this paper we consider the dual notion, which we have called a cotorsion radical. We show that the cotorsion radicals of the category RM correspond to the idempotent ideals of R. Thus they also correspond to TTF classes in the sense of Jans [9].It is well-known that the trace ideal of a projective module is idempotent. We show that this is in fact a consequence of the natural way in which every projective module determines a cotorsion radical. As an application of these techniques we study a question raised by Endo [7], to characterize rings with the property that every finitely generated, projective and faithful left module is completely faithful. We prove that for a left perfect ring this is equivalent to being an S-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.

A new definition of conjugacy for semigroups

2018 ◽  
Vol 17 (02) ◽  
pp. 1850032 ◽  
Author(s):  
Janusz Konieczny
Keyword(s):  
Group Theory ◽  
Inverse Semigroup ◽  
Directed Graphs ◽  
Conjugacy Classes ◽  
Inverse Semigroups ◽  
Transformation Semigroups ◽  
Arbitrary Semigroup ◽  
Basic Transformation ◽  
Definition Of ◽  
Natural Way

The conjugacy relation plays an important role in group theory. If [Formula: see text] and [Formula: see text] are elements of a group [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] for some [Formula: see text]. The group conjugacy extends to inverse semigroups in a natural way: for [Formula: see text] and [Formula: see text] in an inverse semigroup [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] and [Formula: see text] for some [Formula: see text]. In this paper, we define a conjugacy for an arbitrary semigroup [Formula: see text] that reduces to the inverse semigroup conjugacy if [Formula: see text] is an inverse semigroup. (None of the existing notions of conjugacy for semigroups has this property.) We compare our new notion of conjugacy with existing definitions, characterize the conjugacy in basic transformation semigroups and their ideals using the representation of transformations as directed graphs, and determine the number of conjugacy classes in these semigroups.

scholarly journals QUANTUM MECHANICS AT PLANCK'S SCALE AND DENSITY MATRIX

2003 ◽  
Vol 12 (07) ◽  
pp. 1265-1278 ◽  
Author(s):  
A. E. SHALYT-MARGOLIN ◽  
J. G. SUAREZ
Keyword(s):  
Quantum Mechanics ◽  
Density Matrix ◽  
Semiclassical Approximation ◽  
Black Hole Entropy ◽  
Uncertainty Relations ◽  
Statistical Entropy ◽  
Fundamental Length ◽  
Definition Of ◽  
Liouville’S Equation ◽  
Natural Way

In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies in the fact that here the density matrix are subjected to deformation, whereas in the previous approaches only commutators are deformed. The density matrix obtained by deforming the quantum-mechanical one is named the density pro-matrix throughout this paper. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation. The proposed approach allows a description of the dynamics. In particular, the explicit form of the deformed Liouville's equation and the deformed Shrödinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that the results obtained here allow one to deduce in a simple and natural way the Bekenstein–Hawking's formula for black hole entropy in semiclassical approximation.

scholarly journals Some remarks on certain classes of semilattices

1982 ◽  
Vol 5 (1) ◽  
pp. 21-30 ◽  
Author(s):  
P. V. Ramana Murty ◽  
M. Krishna Murty
Keyword(s):  
Necessary And Sufficient Condition ◽  
Dense Element ◽  
Pseudocomplemented Semilattice ◽  
Distributive Semilattice ◽  
Interesting Corollary ◽  
Definition Of ◽  
Almost All ◽  
Necessary And Sufficient ◽  
Distributive Semilattices ◽  
Natural Way

In this paper the concept of a∗-semilattice is introduced as a generalization to distributive∗-lattice first introduced by Speed [1]. It is shown that almost all the results of Speed can be extended to a more eneral class of distributive∗-semilattices. In pseudocomplemented semilattices and distributive semilattices the set of annihilators of an element is an ideal in the sense of Grätzer [2]. But it is not so in general and thus we are led to the definition of a weakly distributive semilattice. In§2we actually obtain the interesting corollary that a modular∗-semilattice is weakly distributive if and only if its dense filter is neutral. In§3the concept of a sectionally pseudocomplemented semilattice is introduced in a natural way. It is proved that given a sectionally pseudocomplemented semilattice there is a smallest quotient of it which is a sectionally Boolean algebra. Further as a corollary to one of the theorems it is obtained that a sectionally pseudocomplemented semilattice with a dense element becomes a∗-semilattice. Finally a necessary and sufficient condition for a∗-semilattice to be a pseudocomplemented semilattice is obtained.

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