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

Abstract

A Boolean-algebraic framework for maximal-degree U-k-seminets is presented, unifying combinatorial and algebraic properties. This work extends Aczel’s quasigroup theory and Belousov’s k-net constructions by introducing 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 nonequipotent sets, and
(3) inequalities governing µ and t ((t+2)/2 < µ ≤ t).
Theorems are validated via tabulated solutions for m = t−d, demonstrating scalable applications in finite geometry and network design. The framework bridges partial quasigroups and block designs, offering algorithmic tools for seminets with maximal degree constraints.