Reflectors and coreflectors in the category of fuzzy topological spaces

Research in recent years has revealed that the construct of fuzzy topological spaces behaves quite differently from that of topological spaces with respect to certain categorical properties. In this paper we discuss some of these aspects. Since the topological construct L-FTS contains nontrivial both initially and finally closed full subconstructs, and each such construct gives rise to a natural autonomous theory of fuzzy topology, it can be said to some extent that fuzzy topology should consist of a system of closely related topology theories, including the classical topology theory as a special case, with each applying to one such subconstruct. Therefore in the first part of this paper the theory of sobriety is established for each finally and initially closed full subconstruct of L-FTS to illustrate this idea. The second topic of this paper is the relationship between the construct of stratified L-fuzzy topological spaces and several other familiar constructs in fuzzy topology, for example, the constructs ofSostak fuzzy topological spaces and L-fuzzifying topological spaces. © 2000 Academic PressKey Words: fuzzy topology; sobriety; frame; topological construct; co-tower extension; total fuzzy topology. PRELIMINARIESAfter a development period of more than 30 years, fuzzy topology has become rather diverse in its topics as well as its methods. The recent book [17] is a comprehensive treatment of the general topology and lattice theoretic aspects of fuzzy topology. In this paper we deal with some categorical properties of the construct of fuzzy topological spaces.

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“…In the case L = 0 1 , the concrete reflectivity and co-reflectivity of FNS in FTS were proved in [32].…”

Section: An Operatormentioning

confidence: 94%

Some Categorical Aspects of Fuzzy Topology

Liu

Zhang

Luo

2000

Journal of Mathematical Analysis and Applications

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“…The full subconstruct with objects the fuzzy neigbourhood spaces [9], [20] is denoted by FNS, and the full subconstruct with objects the maximal fuzzy topological spaces [19] is denoted by MAX. We recall that If r C Ix, then the fuzzy topology generated by I?…”

Section: Preliminariesmentioning

confidence: 99%

Stability under Homeomorphisms in FTS

Löwen

Wuyts

1997

Mathematische Nachrichten

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“…Theorem 2.13 [16]. If (X, Ω) is a fuzzy topological space, then (X, Ω) is a fuzzy neighborhood space if and only if for all θ ∈ Ω and for all Then µ is called N-topo nilpotent if and only if for all ν ∈ Σ R (0), and for all > 0, there exists n 0 ∈ Z + such that for all n ≥ n 0 ,…”

Section: Corollary 27 Let (R + ·T(σ R )) Be a Fuzzy Neighborhoodmentioning

confidence: 99%

N‐topo nilpotency in fuzzy neighborhood rings

Ahsanullah

Al-Thukair

2004

International Journal of Mathematics and Mathematical Sciences

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We introduce the notion of N-topo nilpotent fuzzy set in a fuzzy neighborhood ring and develop some fundamental results. Here we show that a fuzzy neighborhood ring is locally inversely bounded if and only if for all 0 < α < 1, the α-level topological rings are locally inversely bounded. This leads us to prove a characterization theorem which says that if a fuzzy neighborhood ring on a division ring is Wuyts-Lowen WNT 2 and locally inversely bounded, then the fuzzy neighborhood ring is a fuzzy neighborhood division ring. We also present another characterization theorem which says that a fuzzy neighborhood ring on a division ring is a fuzzy neighborhood division ring if the fuzzy neighborhood ring contains an N-topo nilpotent fuzzy neighborhood of zero.2000 Mathematics Subject Classification: 54A40, 54H13.1. Introduction. This paper is a continuation into the investigation of the compatibility of the Lowen fuzzy neighborhood topologies with algebraic structures. In the present text, we introduce and study the notion of N-topo nilpotent fuzzy set (fuzzy neighborhood topologically nilpotent fuzzy set) in a fuzzy neighborhood ring. We prove that this notion is a good extension of its classical counterpart. We also prove that the notion of bounded fuzzy set introduced in [5] is a good extension. In [2], we introduced the concept of locally inversely bounded fuzzy neighborhood ring; here we show that a fuzzy neighborhood ring is locally inversely bounded if and only if its level topological rings are locally inversely bounded. This leads us to establish two characterization theorems, which in the first place, show that if a fuzzy neighborhood ring on a division ring is Wuyts-Lowen WNT 2 and locally inversely bounded, then it is a fuzzy neighborhood division ring; and secondly, a fuzzy neighborhood ring on a division ring is a fuzzy neighborhood division ring if the fuzzy neighborhood ring contains an Ntopo nilpotent fuzzy set of fuzzy neighborhood of zero. In this regard, some pleasant properties related to N-topo nilpotency are achieved. Finally, we study the notion of N-topo quasi-regularity in connection with N-topo nilpotency in fuzzy neighborhood rings.

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Reflectors and coreflectors in the category of fuzzy topological spaces

Cited by 35 publications

References 5 publications

Some Categorical Aspects of Fuzzy Topology

Some Categorical Aspects of Fuzzy Topology

Stability under Homeomorphisms in FTS

N‐topo nilpotency in fuzzy neighborhood rings

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