Convergence functions and their related topologies

Kent, D. C.

doi:10.4064/fm-54-2-125-133

Cited by 77 publications

(43 citation statements)

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“…Let X be a nonempty set and F(X) the collection of all filters on X, a subset F of F(X) × X is called a filter-theoretical convergence relation [16] if (1) ([x], x) ∈ F for all x ∈ X, where [x] is the principle ultrafilter generated by the singleton {x};…”

Section: M-scott Convergencementioning

confidence: 99%

M-Scott Convergence and M-Scott Topology on Posets

Yao¹

2011

Honam Mathematical Journal

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Section: M-scott Convergencementioning

confidence: 99%

M-Scott Convergence and M-Scott Topology on Posets

Yao¹

2011

Honam Mathematical Journal

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“…This pseudo-difference map was used in [12] to determine the group of lattice automorphisms of the lattice of convergence structures where a slightly different definition was used for a convergence structure corresponding to the definition used by Kent [8]. THEOREM …”

Section: S^2 G R(α ) Since ^X π ^2 ί #(#) and Similarly Gf X G S(x) mentioning

confidence: 99%

“…Then C(X) with this order becomes a complete, distributive lattice [7,8] in which for any {q 3 -1 j eJ}, a subset of C(X), (Vqj)(x) = Γiq 3 (x)(jeJ) and (Λffy)(α?) = Ugy(&)(ie J).…”

mentioning

confidence: 99%

Complementation in the lattice of convergence structures

Riecke¹

1977

Pacific J. Math.

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“…All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9]. Let L be any lattice, partially ordered by an order relation <.…”

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confidence: 99%

Order-Topological lattices

Erné

1980

Glasgow Math. J.

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1. Introduction and basic concepts. The observation that convergence of real sequences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the i present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9]. Let L be any lattice, partially ordered by an order relation <. Furthermore, x = y v z means x = V {y> z}, and x = y A z means x = /\{y, z}. For the sake of convenience, we set

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Convergence functions and their related topologies

Cited by 77 publications

References 0 publications

M-Scott Convergence and M-Scott Topology on Posets

M-Scott Convergence and M-Scott Topology on Posets

Complementation in the lattice of convergence structures

Order-Topological lattices

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