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{{Short description|Algebraic structure in network theory}}

In [[algebra]] and [[network theory]], a '''Wang algebra''' is a [[commutative algebra]] <math>A</math>, over a [[Field (mathematics)|field]] or (more generally) a [[Commutative ring|commutative unital ring]], in which <math>A</math> has two additional properties:<br>(Rule i) For all elements ''x'' of  <math>A</math>, ''x'' + ''x'' = 0 (universal additive [[Nilpotent|nilpotency]] of degree 1).<br>(Rule ii) For all elements ''x'' of <math>A</math>, ''x''{{char|'''⋅'''}}''x'' = 0 (universal multiplicative nilpotency of degree 1).<ref name=Duffin>{{cite journal|title=An analysis of the Wang algebra of networks|journal=Trans. Amer. Math. Soc.|volume=93|pages=114–131|mr=0109161|doi=10.1090/s0002-9947-1959-0109161-6|year=1959|last1=Duffin|first1=R. J.|author-link=Richard Duffin|doi-access=free}}</ref><ref>{{cite book|author=Chen, Wai-Kai|chapter-url=https://books.google.com/books?id=kAcBm96mUvsC&pg=PA332| title=Applied Graph Theory | isbn=9780444601933 | date=2 December 2012 |publisher=North-Holland|chapter=5.4 The Wang-algebra formulation|pages=332–352}} [https://books.google.com/books?id=kAcBm96mUvsC&pg=PA333 p. 333], [https://books.google.com/books?id=kAcBm96mUvsC&pg=PA334 p. 334]</ref>

==History and applications==
Rules (i) and (ii) were originally published by K. T. Wang (Wang Ki-Tung, 王 季同) in 1934 as part of a method for analyzing electrical networks.<ref>{{cite journal|author=K. T. Wang|title=On a new method of analysis of electrical networks|journal=Memoir 2|publisher=National Research Institute of Engineering, Academia Sinica|year=1934}}</ref> From 1935 to 1940, several Chinese electrical engineering researchers published papers on the method. The original Wang algebra is the [[Exterior algebra|Grassman algebra]] over the [[Finite field|finite field mod 2]].<ref name=Duffin/> At the 57th annual meeting of the [[American Mathematical Society]], held on December 27–29, 1950, [[Raoul Bott]] and [[Richard Duffin]] introduced the concept of a Wang algebra in their abstract (number 144''t'') ''The Wang algebra of networks''. They gave an interpretation of the Wang algebra as a particular type of Grassman algebra mod 2.<ref>{{cite journal|author=Whyburn, W. M.|author-link=William Whyburn|title=The annual meeting of the society|journal=Bulletin of the American Mathematical Society|volume=57|issue=2|date=March 1951|pages=109–152|doi=10.1090/S0002-9904-1951-09479-3|mr=1565283|s2cid=120638163 |doi-access=free}} (See p. 136.)</ref> In 1969 [[Wai-Kai Chen]] used the Wang algebra formulation to give a unification of several different techniques for generating the [[Tree (graph theory)|trees of a graph]].<ref>{{cite journal|doi=10.1080/00207216908900016|title=Unified theory on the generation of trees of a graph Part I. The Wang algebra formulation |year=1969 |last1=Chen |first1=Wai-Kai |journal=International Journal of Electronics |volume=27 |issue=2 |pages=101–117 }}</ref> The Wang algebra formulation has been used to systematically generate King-Altman directed graph patterns. Such patterns are useful in deriving rate equations in the theory of enzyme kinetics.<ref>{{cite journal|doi=10.1186/1471-2105-10-238|title=Generating rate equations for complex enzyme systems by a computer-assisted systematic method |year=2009 |last1=Qi |first1=Feng |last2=Dash |first2=Ranjan K. |last3=Han |first3=Yu |last4=Beard |first4=Daniel A. |journal=BMC Bioinformatics |volume=10 |page=238 |pmid=19653903 |pmc=2729780 |doi-access=free }}</ref>

According to Guo Jinhai, professor in the Institute for the History of Natural Sciences of the [[Chinese Academy of Sciences]], Wang Ki Tung's pioneering method of analyzing electrical networks significantly promoted electrical engineering not only in China but in the rest of the world; the Wang algebra formulation is useful in electrical networks for solving problems involving topological methods, graph theory, and Hamiltonian cycles.<ref>{{cite journal|author=郭金海 (Guo Jinhai)|title=王季同的电网络分析新方法及其学术影响 (Wang Ki-Tung's New Method for the Analysis of Electric Network and Its Scientific Influence)|journal=The Chinese Journal for the History of Science and Technology, No. 4|year=2003|publisher=Institute for the History of Natural Sciences, Chinese Academy of Sciences|pages=33–40|url=http://english.ihns.cas.cn/institute/OS/journal/cjhst/201011/t20101109_61156.html}}</ref>

==Wang Algebra and the Spanning Trees of a Graph==
;The Wang Rules for Finding all Spanning Trees of a Graph G<ref name="Kauffman">{{cite web|author=Kauffman, Louis H.|author-link=Louis Kauffman|title=Wang Algebra and the Spanning Trees of a Graph|website=Mathematics Department, University of Chicago Illinois|url=http://homepages.math.uic.edu/~kauffman/WangAlgebra.pdf}}</ref>
:#For each node write the sum of all the edge-labels that meet that node.
:#Leave out one node and take the product of the sums of labels for all the remaining nodes.
:#Expand the product in 2. using the Wang algebra.
:#The terms in the sum of the expansion obtained in 3. are in 1-1 correspondence with the spanning trees in the graph.

==References==
<references/>

[[Category:Algebra]]
[[Category:Electrical engineering]]
[[Category:Network theory]]


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