Optional Intervals Event and Two n-ary Finitary Operations: An Algebraic Framework for Unifying Parallel-Serial Execution and Axiomatizing Simultaneity from an Epistemological Perspective

Abstract

This paper proposes an algebraic framework for analyzing event execution intervals and sequences, introducing "Optional Intervals Event (OIE)" as a 4-tuple abstraction (C, F, I, A) that serves as a pre-execution planning tool for real-world events. The OIE establishes a mapping to real-world events and stores all feasible execution intervals together with dependency relationships among sub-events. Based on this abstraction, we define two n-ary finitary operations: (i) "Complete Sequence Addition", which models concurrent events with a certain degree of equal opportunity within a shared time domain; and (ii) "Complete Sequence Multiplication", which models strictly ordered sequential events. We analyze the algebraic properties of these operations, including closure, non-commutativity, permutational equivalence, and orbit spaces. We prove that, for any non-degenerate finite OIE set, Complete Sequence Addition yields a single-orbit space due to permutational equivalence, whereas Complete Sequence Multiplication may yield multiple orbits. This orbital divergence rigorously captures the fundamental symmetry gap between concurrent and sequential execution. In computer science, this framework establishes an axiomatic algebraic system that formally unifies parallel and serial execution as n-ary finitary operations. It enables constraint-aware pre-execution planning and characterizes concurrent symmetry via orbitspace analysis and permutational equivalence. We also discuss applications to probability theory and physics, including the distinction between process symmetry and outcome symmetry and a novel axiomatization of simultaneity from an epistemological perspective.