Partial orders, maxels and Mobius functions
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This more advanced lecture connects the Boole-Mobius transform between Boolean functions and Boole polynumbers, which is a key tool in understanding circuit analysis from the point of view of the Algebra of Boole. We include a brief discussion of Mobius functions on partially ordered sets (posets). We also examine the inverse of the Boole-Mobius transform in terms of matrix inversion, and point out that this is better understood in terms of the modern theory of maxels for matrix arithmetic. For more on maxels, see the MathFoundations lectures from MF166-172. We are here providing a simplified introduction to the Incidence Algebra of a poset. We also introduce a very novel bar notation to describe negative numbers -- a long overdue improvement on fundamental notation for arithmetic!! This video will be part of the Boole's Logic and Circuit Analysis online course which is being developed by Norman. The course will become available hopefully next year, to Patrons of Norman's channel, and will provide prospective electrical engineers with a thorough grounding in the application of Boole's original ideas to modern circuit analysis. You can become a Patron, and support Norman's efforts at simplifying and rejuvenating mathematics education and research, at}:https://www.patreon.com/njwildberger: