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Optional intervals event, sequential operation and their applications in physics, computer science and applied mathematics

Zhongyuan. Li, Yanlei. Gong, Lei. Yu, Yue. Cao, Bo. Yin

TL;DR

The paper develops a formal framework that treats events through Optional Intervals Events (OIE) and introduces two binary, time-coupled operators: $\oplus$ (Complete Sequential Addition) for parallel-start scenarios and $\otimes$ (Complete Sequential Multiplication) for ordered sequences. It proves that each operation yields a semigroup structure, constructs a quotient semigroup under a commutativity relation, and provides Cayley-diagram visualizations to reveal a novel algebraic object. The framework is then mapped to physical and computational contexts, offering axiomatic accounts of simultaneous starts, scheduling, and modeling beyond straightforward observation. Practical implications span physics, computer science, and applied mathematics, with discussion of implementation rules, feasibility constraints, and potential extensions including probabilistic and semiring considerations. The work thus proposes a robust, subjectively-planned, algebraic approach to timing and sequencing of events that unifies concurrent and sequential notions within a single formal system.

Abstract

In this paper, we apply algebraic theories such as set theory and group theory to analyze the execution interval and sequence of events. We propose a triple, the "Optional Intervals Event", as an abstract entity mapping to events, and define two sequential operations, "Complete Sequential Addition" for concurrent events and "Complete Sequential Multiplication" for ordered sequential events. We prove that complete sequential addition and complete sequential multiplication form semigroups, and further construct a quotient semigroup for complete sequential addition semigroup that satisfies the commutative law. Then we draw Cayley tables to visualize algebraic structures, and this additionally reveals a new mathematical object. With these new concepts, we provide a rigorous formalization of "starting simultaneously" thereby resolving measurement errors by using quotient semigroup of complete sequential addition semigroup, and explain why both one-by-one races and head-to-head races can determine the first place. Finally, we present other implications derived from our work. This work enables more rigorous event scheduling, physical simulations, and computational modeling beyond observational limitations.

Optional intervals event, sequential operation and their applications in physics, computer science and applied mathematics

TL;DR

The paper develops a formal framework that treats events through Optional Intervals Events (OIE) and introduces two binary, time-coupled operators: (Complete Sequential Addition) for parallel-start scenarios and (Complete Sequential Multiplication) for ordered sequences. It proves that each operation yields a semigroup structure, constructs a quotient semigroup under a commutativity relation, and provides Cayley-diagram visualizations to reveal a novel algebraic object. The framework is then mapped to physical and computational contexts, offering axiomatic accounts of simultaneous starts, scheduling, and modeling beyond straightforward observation. Practical implications span physics, computer science, and applied mathematics, with discussion of implementation rules, feasibility constraints, and potential extensions including probabilistic and semiring considerations. The work thus proposes a robust, subjectively-planned, algebraic approach to timing and sequencing of events that unifies concurrent and sequential notions within a single formal system.

Abstract

In this paper, we apply algebraic theories such as set theory and group theory to analyze the execution interval and sequence of events. We propose a triple, the "Optional Intervals Event", as an abstract entity mapping to events, and define two sequential operations, "Complete Sequential Addition" for concurrent events and "Complete Sequential Multiplication" for ordered sequential events. We prove that complete sequential addition and complete sequential multiplication form semigroups, and further construct a quotient semigroup for complete sequential addition semigroup that satisfies the commutative law. Then we draw Cayley tables to visualize algebraic structures, and this additionally reveals a new mathematical object. With these new concepts, we provide a rigorous formalization of "starting simultaneously" thereby resolving measurement errors by using quotient semigroup of complete sequential addition semigroup, and explain why both one-by-one races and head-to-head races can determine the first place. Finally, we present other implications derived from our work. This work enables more rigorous event scheduling, physical simulations, and computational modeling beyond observational limitations.

Paper Structure

This paper contains 54 sections, 339 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Analysis of the properties of event
  3. Execution status of event
  4. The starting timestamp and ending timestamp of an event
  5. Atomic events and composite events
  6. Error event
  7. Optional intervals event and analysis
  8. Optional intervals event
  9. Set of optional intervals events
  10. Error optional intervals event
  11. Two sequential operations on the elements of a finite set of optional intervals events
  12. The set that is naturally isomorphic to the Cartesian product of the sets of interval 2-tuples of each member in a finite set of optional intervals events.
  13. Infeasible and feasible subset of the set that is naturally isomorphic to the Cartesian product of all members of the $\mathrm{2Tuple}\mathcal{SS}$ instance of a finite $\mathrm{OIE}\mathcal{S}$ instance in an index order
  14. Set of infeasible interval 2-tuples of members of a finite $\mathrm{OIE}\mathcal{S}$ instance under an index order
  15. Compatibility of the set of tuples of infeasible interval 2-tuples of $\mathrm{oie}\mathcal{S}$ instances
  16. ...and 39 more sections

Figures (3)

  • Figure 1: 5-dimensional Full CSA Diagram
  • Figure 2: 6-dimensional Full CSA Diagram
  • Figure 3: 6-dimensional Full CSA diagram with 0

Theorems & Definitions (32)

  • Definition 1: Starting timestamp of an event & helper function
  • Definition 2: Ending timestamp of an event & helper function
  • Definition 3: The interval 2-tuple of an event & helper function
  • Definition 4: Composite event's starting timestamp and ending timestamp
  • Definition 5: Error event
  • Definition 6: Functions for finding the minimum 1st item and the maximum 2nd item of a $\mathrm{2Tuple}\mathcal{S}$ Instance
  • Definition 7: Optional intervals event
  • Definition 8: Atomic $\mathrm{OIE}$
  • Definition 9: Composite $\mathrm{OIE}$
  • Definition 10: Member variable of $\mathrm{OIE}$: meta $\mathrm{OIE}$ instance tuple & helper function
  • ...and 22 more