algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

and

network theory In mathematics, computer science, and network science, network theory is a part of graph theory. It defines networks as Graph (discrete mathematics), graphs where the vertices or edges possess attributes. Network theory analyses these networks ...

, a Wang algebra is a

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

A

, over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

or (more generally) a commutative unital ring, in which

A

has two additional properties:
(Rule i) For all elements ''x'' of

A

, ''x'' + ''x'' = 0 (universal additive nilpotency of degree 1).
(Rule ii) For all elements ''x'' of

A

, ''x'x'' = 0 (universal multiplicative nilpotency of degree 1).

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. From 1935 to 1940, several Chinese electrical engineering researchers published papers on the method. The original Wang algebra is the

Grassman algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...

over the finite field mod 2. At the 57th annual meeting of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, held on December 27–29, 1950,

Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott function ...

and

Richard Duffin Richard James Duffin (1909 – October 29, 1996) was an American physicist, known for his contributions to electrical transmission theory and to the development of geometric programming and other areas within operations research. Education and ...

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. In 1969

Wai-Kai Chen Wai-Kai Chen (; born December 23, 1936, in Nanjing) is a Chinese-American professor emeritus of electrical engineering and computer science. Biography Wai-Kai Chen's youth was troubled by the Sino-Japanese War of 1937–1945 followed by the civil ...

used the Wang algebra formulation to give a unification of several different techniques for generating the trees of a graph. 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. According to Guo Jinhai, professor in the Institute for the History of Natural Sciences of the

Chinese Academy of Sciences The Chinese Academy of Sciences (CAS; ) is the national academy for natural sciences and the highest consultancy for science and technology of the People's Republic of China. It is the world's largest research organization, with 106 research i ...

, 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.

Wang Algebra and the Spanning Trees of a Graph

;The Wang Rules for Finding all Spanning Trees of a Graph G :#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

Commutative algebra Electrical engineering Network theory Ring theory {{Engineering-stub, *Electrical