Wang algebra
{{Short description|Algebraic structure in network theory}}
In algebra and network theory, a Wang algebra is a commutative algebra , over a field or (more generally) a commutative unital ring, in which has two additional properties:
(Rule i) For all elements x of , x + x = 0 (universal additive nilpotency of degree 1).
(Rule ii) For all elements x of , x{{char|⋅}}x = 0 (universal multiplicative nilpotency of degree 1).{{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}}{{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]
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.{{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}} From 1935 to 1940, several Chinese electrical engineering researchers published papers on the method. The original Wang algebra is the Grassman algebra over the finite field mod 2. 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 144t) The Wang algebra of networks. They gave an interpretation of the Wang algebra as a particular type of Grassman algebra mod 2.{{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.) In 1969 Wai-Kai Chen used the Wang algebra formulation to give a unification of several different techniques for generating the trees of a graph.{{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 }} 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.{{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 }}
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.{{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 |issue=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}}
Wang Algebra and the Spanning Trees of a Graph
;The Wang Rules for Finding all Spanning Trees of a Graph G{{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}}
:#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.