Orthomodular lattices in causal structure of space-time

Document Type : Original Paper

Authors

Department of pure mathematics and computer science of Hakim Sabzevari University

Abstract

Rough set theory provides a convenient framework to study and compare algebraic operators in many mathematical structures. In this paper, we establish a connection between the causal structure of space-time in Einstein's theory of relativity and the theory of Rough sets based on coverage, and by means of this, we define the covering approximation operators for the causal structure and show that some of these operators are the same basic and common operators in the causal structure of space-time such as I^± ، J^±، D، ⊥, and some operators like ⊥' and ⊥'⊥' and different operators in the causal structure. Recently, causal logic on space-times has been introduced by a complete orthomodular lattice consisting of all constants of the clouser operator ⊥⊥. Here, through the orthogonal operator ⊥' , we introduce another complete lattice and determine the elements of this lattice in causal space-times. Also, we provide a necessary and sufficient condition for this lattice to be orthomodular in global hyperbolic space-times. Finally, we show that these two lattices are isomorphic in the case of two-dimensional Minkowski space-time, but this is not necessarily true in the general case.

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History

  • Receive Date: 12 November 2023
  • Revise Date: 20 June 2024
  • Accept Date: 26 July 2024

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