Sierpinski object for affine systems
Introduction
The notion of Sierpinski space (see, e.g., [36]) plays a significant role in general topology. In particular, one can show the following three important properties (see, e.g., [1], [22]):
- (1)
A topological space is iff it can be embedded into some power of .
- (2)
The injective objects in the category of topological spaces are precisely the retracts of powers of .
- (3)
A topological space is sober iff it can be embedded as a front-closed subspace into some power of .
Some of the above-mentioned results have already been extended to lattice-valued topology (see, e.g., [19], [33], [34]). In particular, there already exists a convenient characterization of the category of fuzzy topological spaces in terms of the Sierpinski object of E.G. Manes [33].
In [35], S. Vickers introduced the concept of topological system as a common framework for both point-set and point-free topologies. He showed that the category of topological spaces is isomorphic to a full (regular mono)-coreflective subcategory of the category of topological systems, which gave rise to the so-called spatialization procedure for topological systems (from systems to spaces and back). Inspired by the notion of S. Vickers, R. Noor and A.K. Srivastava [23] have recently presented the concept of Sierpinski object in the category of topological systems, providing topological system analogues of items (1), (2) above.
Motivated by the notion of lattice-valued topological system of [7], [30] and the results of [23], in this paper, we show lattice-valued system analogues of the above three items (fuzzifying, therefore, some of the achievements of [23]). To better incorporate various lattice-valued settings available in the literature, we use the affine context of Y. Diers [10], [11], [12] and build our systems over an arbitrary variety of algebras (see, e.g., [9] for the similar approach). Choosing a particular variety gives a particular lattice-valued setting (for example, variety of frames [17] provides the setting of lattice-valued topological systems of [7]).
Section snippets
Affine spaces and systems
This section recalls from [9] the notions of affine system and space, and also their related spatialization procedure. To better encompass various many-valued frameworks, we employ a particular instance of the setting of affine sets of Y. Diers [10], [11], [12], which is based in varieties of algebras.
Definition 1 Let be a family of cardinal numbers, which is indexed by a (possibly, proper or empty) class Λ. An Ω-algebra is a pair , which comprises a set A and a family of maps
Sierpinski object for affine systems
Motivated by the ideas of R. Noor and A.K. Srivastava [23], in this section, we introduce an affine system analogue of the Sierpinski space. The respective analogue is based in the concept of Sierpinski object in a concrete category of E.G. Manes [20], [21]. Restated in the modern language of concrete categories of, e.g., [1], the concept in question can be defined as follows (cf. [23, Definition 2.7]).
Definition 10 Given a concrete category C, a C-object S is called a Sierpinski object provided that for
Properties of the Sierpinski affine system
In this section, we are going to show affine system analogues of the three properties of the Sierpinski space, mentioned in the introductory section.
Sierpinski space versus Sierpinski system
In this section, we compare the Sierpinski affine space and the Sierpinski affine system. We notice first that there already exists a lattice-valued analogue of the Sierpinski space [33], [34]. Moreover, its affine version has already been studied in, e.g., [29], [31], which motivates our next definition.
Definition 42 Sierpinski (L-)affine space is the pair , where stands for the subalgebra of , which is generated by the identity map . □
According to [29, Theorem 3.2] (cf. also [31,
Conclusion
This paper makes another step in our effort to bring the theory of lattice-valued topology under the setting of affine sets of Y. Diers [10], [11], [12]. In particular, we have considered an affine setting for topological systems of S. Vickers [35] and introduced an affine system analogue of the well-known Sierpinski space. Our study was motivated by the paper of R. Noor and A.K. Srivastava [23], who presented the Sierpinski topological system and studied its basis properties. With the help of
Acknowledgements
The authors would like to express their sincere gratitude to the anonymous referees of this paper for their helpful remarks and suggestions, e.g., Example 12, Example 43 are due to one of them.
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2023, Fuzzy Sets and SystemsCitation Excerpt :Additionally, throughout the paper, we will shorten “with respect to” to “w.r.t.” and “if and only if” to “iff”. This section recalls from, e.g., [7] the notion of affine topological space. To better encompass numerous lattice-valued topological frameworks, we will rely on a particular instance of the setting of affine sets of Y. Diers [8–10] based in varieties of algebras (in the categorically-algebraic sense as shown below).
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2021, Fuzzy Sets and SystemsCitation Excerpt :From now on, we will use the definition of a Sierpinski object as stated in Definition 26. For convenience of the reader, we provide an example of Sierpinski object in the category of affine spaces [8,32]. The next theorem shows an important relationship between Sierpinski objects in the categories of affine spaces and composite affine spaces, which is the main result of this paper.
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2021, Fuzzy Sets and SystemsCitation Excerpt :Now we are mentioning some more problems for future research. J. T. Denniston et al. [5] studied the category AfSys(L) of L-affine systems (where L is a fixed member of a fixed variety of algebras A) and introduced the Sierpinski (L-) affine system. They proved that the Sierpinski (L-) affine system is a Sierpinski object in the category AfSys(L).
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This research was supported by the bilateral project “New Perspectives on Residuated Posets” of the Austrian Science Fund (FWF) (project No. I 1923-N25) and the Czech Science Foundation (GAČR) (project No. 15-34697L).