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On soft bitopological ordered spaces. (English) Zbl 1540.54008

Summary: This paper introduces soft bitopological ordered spaces, combining soft topological spaces with partial order relations. The authors extensively investigate increasing, decreasing, and balancing pairwise open and closed soft sets, analyzing their properties. They prove that the collection of increasing (decreasing) open soft sets forms an increasing (decreasing) soft topology. The paper thoroughly examines increasing and decreasing pairwise soft closure and interior operators. Notably, it introduces \(bi\)-ordered soft separation axioms, denoted as \(PST_i\) (\(PST_i^\bullet\), \(PST_i^\ast\), \(PST_i^{\ast\ast})\)-ordered spaces, \(i = 0, 1, 2\), showcasing their interrelationships through examples. It explores separation axiom distinctions in bitopological ordered spaces, referencing relevant literature. The paper investigates new types of regularity and normality in soft bitopological ordered spaces and their connections to other properties. Importantly, it establishes the equivalence of three properties for a soft bitopological ordered space satisfying the conditions of being \(TP^\ast\)-soft regularly ordered: \(PST_2\)-ordered, \(PST_1\)-ordered, and \(PST_0\)-ordered. It introduces the concept of a \(bi\)-ordered subspace and explores its hereditary property. The authors define soft bitopological ordered properties using ordered embedding soft homeomorphism maps and verify their applicability for different types of \(PST_i\)-ordered spaces, \(i = 0, 1, 2\). Finally, the paper identifies the properties of being a \(TP^\ast\); (\(PP^\ast\))-soft \(T_3\)-ordered space and a \(TP\)-soft \(T_4\)-ordered space as a soft bitopological ordered property.

MSC:

54A40 Fuzzy topology
54E55 Bitopologies
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
06F30 Ordered topological structures
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[1] Proof. Case: The proof of the first, second, and third statement in this theorem can be easily derived from Definition 4.1.
[2] If a soft set ω Π is contained in another soft set λ Π and Iint s 12 (ω Π ) is the largest IP O-soft set contained within ω Π , then Iint s 12 (
[3] For the intersection of soft sets ω Π and λ Π , Iint s 12 [ω Π ⊓λ Π ] is contained within both Iint s 12 (ω Π )
[4] and Iint s 12 (λ Π ), which are the largest IP O-soft sets contained within ω Π and λ Π , respec-tively.
[5] Obvious. Theorem 4.4. For any SBT OS (Υ, η 1 , η 2 , Π, ≲), and any soft sets ω Π and λ Π in P (Υ) Π , the following statements hold:
[6] Proof. It is stated that the proof is similar to that of a previous theorem therefore has not been submitted.
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