scholarly journals On rings of sets

1968 ◽  
Vol 8 (4) ◽  
pp. 723-730 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Sets ◽  
The Past ◽  
Complete Ring ◽  
Irreducible Elements ◽  
Necessary And Sufficient

In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].

scholarly journals On a topology between Tα and Tγα

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3209-3221
Author(s):  
Dimitrije Andrijevic
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closure Operators ◽  
New Class ◽  
Necessary And Sufficient

Using the topology T in a topological space (X,T), a new class of generalized open sets called ?-preopen sets, is introduced and studied. This class generates a new topology Tg which is larger than T? and smaller than T??. By means of the corresponding interior and closure operators, among other results, necessary and sufficient conditions are given for Tg to coincide with T? , T? or T??.

scholarly journals A characterization of point semiuniformities

1996 ◽  
Vol 19 (2) ◽  
pp. 311-316
Author(s):  
Jennifer P. Montgomery
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Groups ◽  
Necessary And Sufficient

The concept of a uniformity was developed by A. Well and there have been several generalizations. This paper defines a point semiuniformity and gives necessary and sufficient conditions for a topological space to be point semiuniformizable. In addition, just as uniformities are associated with topological groups, a point semiuniformity is naturally associated with a semicontinuous group. This paper shows that a point semiuniformity associated with a semicontinuous group is a uniformity if and only if the group is a topological group.

An Internal Solution to the Problem of Linearization of a Convexity Space

1976 ◽  
Vol 19 (4) ◽  
pp. 487-494 ◽  
Author(s):  
D. A. Szafron ◽  
J. H. Weston
Keyword(s):  
Topological Space ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Linear Structure ◽  
Linear Topological Space ◽  
Necessary And Sufficient Conditions ◽  
Convexity Space ◽  
Internal Solution ◽  
Convexity Structure ◽  
Necessary And Sufficient

Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.

James, Intentionality, and Analysis

2020 ◽  
Author(s):  
Henry Jackman
Keyword(s):  
William James ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Core ◽  
The Past ◽  
Knowing That ◽  
Traditional Account ◽  
The World ◽  
Necessary And Sufficient

William James was always gripped by the problem of intentionality (or “knowing”), that is, of how our thoughts come to be about the world. Nevertheless, coming up with a sympathetic reading of James’s account requires appreciating that James’s approach to analyzing a phenomenon is very different from that which most contemporary philosophers have found natural. In particular, rather than trying to give necessary and sufficient conditions for a thought’s being about an object, James presented an account of intentionality that focused on certain core cases (particularly those where we actually see or handle the objects of our thoughts), and explained the extension of our “knowing” talk to other cases (objects and events in the past, unobservables, etc.) in terms of various pragmatically relevant relations that can be found between those cases and the “core.” Once this account of intentionality is in place, a number of features of James’s approach to truth come in to clearer focus, and can seem less problematic than they would if one presupposed a more traditional account of intentionality and analysis.

Some properties of retract lattices of monounary algebras

2012 ◽  
Vol 62 (2) ◽  
Author(s):  
Danica Jakubíková-Studenovská ◽  
Jozef Pócs
Keyword(s):  
Sufficient Conditions ◽  
Boolean Lattice ◽  
Necessary And Sufficient Conditions ◽  
Monounary Algebra ◽  
Irreducible Elements ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ∅(A, f) is complemented. In this case R ∅(A, f) forms a boolean lattice.

scholarly journals Near-ring-semigroups of continuous selfmaps

1988 ◽  
Vol 37 (2) ◽  
pp. 277-291 ◽  
Author(s):  
K.D. Magill
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analogous Problem ◽  
Multiplicative Semigroup ◽  
Necessary And Sufficient

We find necessary and sufficient conditions on a topological space X so that S(X), the semigroup of all continuous selfmaps of X, is isomorphic to the multiplicative semigroup of a near-ring. The analogous problem is also considered for the semigroup of all continuous selfmaps which fix some point of X.

scholarly journals The largest proper congruence onS(X)

1984 ◽  
Vol 7 (4) ◽  
pp. 663-666 ◽  
Author(s):  
K. D. Magill
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

S(X)denotes the semigroup of all continuous selfmaps of the topological spaceX. In this paper, we find, for many spacesX, necessary and sufficient conditions for a certain type of congruence to be the largest proper congruence onS(X).

Locke on Being Self to My Self

2021 ◽  
pp. 118-144
Author(s):  
Ruth Boeker
Keyword(s):  
Personal Identity ◽  
Final Section ◽  
John Locke ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Present Moment ◽  
Intuitive Knowledge ◽  
The Past ◽  
Necessary And Sufficient ◽  
Over Time

John Locke accepts that every perception gives me immediate and intuitive knowledge of my own existence. However, this knowledge is limited to the present moment when I have the perception. If I want to understand the necessary and sufficient conditions of my continued existence over time, Locke argues that it is important to clarify what “I” refers to. According to Locke, persons are thinking intelligent beings who can consider themselves as extended into the past and future and who are concerned for their happiness and accountable for their actions. I show that the concept of self that he develops in the context of his discussion of persons and personal identity is richer and more complex than the I-concept that he invokes in his version of the cogito. In the final section I turn to the reception of Locke’s view by some of his early critics and defenders, including Elizabeth Berkeley Burnet, an anonymous author, and Catharine Trotter Cockburn.

scholarly journals Some Properties of Interior and Closure in General Topology

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 624
Author(s):  
Soon-Mo Jung ◽  
Doyun Nam
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Open Set ◽  
Necessary And Sufficient ◽  
Open Subspace

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

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