Explore advanced concepts in Boolean algebra and circuit analysis in this 34-minute lecture. Delve into the Boole-Mobius transform, connecting Boolean functions and Boole polynumbers. Examine partial orders, maxels, and Mobius functions on partially ordered sets (posets). Learn about the inverse of the Boole-Mobius transform through matrix inversion and the modern theory of maxels. Discover a novel bar notation for negative numbers, revolutionizing fundamental arithmetic notation. Gain insights into the Incidence Algebra of a poset and its applications in electrical engineering. Cover topics including linear orders, partial orders on natural numbers, divisibility relations, and maxel algebra of posets.