Boolean-Algebraic Framework for Maximal-Degree U-k-Seminets: Theoretical Foundations and Computational Advances

Abstract

A Boolean-algebraic framework for maximal-degree U-k-seminets is presented, unifying combinatorial and algebraic properties. This work ex 
tends Aczel’s quasigroup theory and Belousov’s k-net constructions by in traducing a computational framework for U-k-seminets of maximal 
degree µ. Key results include: (1) explicit bounds for µ in terms of set cardinality t and t-order d (µ = t−d+2), (2) existence conditions for no 
equipotent sets, and (3) inequalities governing µ and t ((t+2)/2 < µ ≤ t). Theorems are validated via tabulated solutions for m = t − d, demonstrating scal able applications in finite geometry and network design. The framework bridges partial quasigroups and block designs, offering algorithmic tools for seminets with maximal degree constraints.