A Brief History of Algebra with a Focus on the Distributive Law and Semiring Theory
aa r X i v : . [ m a t h . HO ] J u l A Brief History of Algebra with a Focus on theDistributive Law and Semiring Theory
Peyman NasehpourDepartment of Engineering ScienceGolpayegan University of TechnologyGolpayegan, Isfahan [email protected], [email protected] 1, 2018
Abstract
In this note, we investigate the history of algebra briefly. We particularly focuson the history of rings, semirings, and the distributive law. “ I am sure that no subject loses more than mathematics by any attempt todissociate it from its history. ”- J. W. L. Glaisher (1848–1928), English mathematician“
One can conceive history as an argument without end. ”- Pieter Catharinus Arie Geijl (1887–1966), Dutch historian The word “algebra” is derived from the Arabic word Al-Jabr, and this comes from thetreatise written in 820 by the medieval Persian mathematician [44], Muhammad ibnMusa al-Khwarizmi, entitled, in Arabic “
Kit¯ab al-mukhtas.ar f¯i h. isab al-ˇgabr wa-’l-muq¯abala ”, which can be translated as “The Compendious Book on Calculation byCompletion and Balancing” [48]. This is a translation of Geyl’s sentence in his book with the title “Napoleon: Voor en Tegen in de FranseGeschiedschrijving”, published in 1946. The original sentence in Dutch [21] is: “Men kan de geschiedschri-jving opvatten als een discussie zonder eind. ” Key words and phrases . History of Algebra, Distributive Law, Semiring Theory2010
Mathematics Subject Classification . 01A05, 97A30
A Brief History of Algebra
One may consider the history of algebra to have two main stages, i.e., “classical al-gebra” that mostly was devoted to solving the (polynomial) equations and “abstractalgebra”, also called “modern algebra” that is all about the study of algebraic struc-tures, something that today algebraists do.
Classical algebra can be divided into three sub-stages:1. In the very early stages of algebra, concepts of algebra were geometric, proposedby the Babylonians apparently and developed by the Greeks and later revivedby Persian mathematician Omar Khayyam (1048–1131) since the main purposeof Khayyam in this matter was to solve the cubic equations with intersectingconics [15, p. 64].2. The main step of passing from this substage to the substage of equation-solvingwas taken by Persian mathematician al-Khwarizmi (c.780–c.850) in his treatiseAl-Jabr that it gives a detailed account of solving polynomials up to the seconddegree.3. While the mathematical notion of a function was implicit in trigonometric tables,apparently the idea of a function was proposed by the Persian mathematicianSharaf al-Din al-Tusi (died 1213/4), though his approach was not very explicit,perhaps because of this point that dealing with functions without symbols isvery difficult. Anyhow algebra did not decisively move to the dynamic functionsubstage until the German mathematician Gottfried Leibniz (1646–1716) [33].One may also view the history of the development of classical algebra in anotherperspective namely passing from rhetorical substage to full symbolic substage. Ac-cording to this view, the history of algebra is divided into three substages through thedevelopment of symbolism: 1. Rhetorical algebra, 2. Syncopated algebra, and 3. Sym-bolic algebra [46].
1. In rhetorical algebra, all equations are written in full sentences, but in symbolicalgebra, full symbolism is used, the method that we do today. So far as weknow, rhetorical algebra was first developed by the ancient Babylonians and wasdeveloped up to 16 th century.2. Syncopated algebra used some symbolism but not as full as symbolic algebra.It is said that syncopated algebraic expression first appeared in the book Arith-metica by the ancient Greek mathematician Diophantus (born sometime betweenAD 201 and 215 and passed away sometime between AD 285 and 299) and wascontinued in the book Brahma Sphuta Siddhanta by the ancient Indian mathe-matician Brahmagupta (598–c.670 CE). Also, see
The Evolution of Algebra , Science,
2. The full symbolism can be seen in the works of the French mathematician, Ren´eDescartes (1596–1650), though early steps of symbolic algebra was taken byMoroccan mathematician Ibn al-Bann¯a al-Marak¯ush¯i (1256–c.1321) and An-dalusian mathematician Ab¯u al-Hasan al-Qalas¯ad¯i (1412–1486) [41, p. 162].
The transition of algebra from the “classical” to the “modern” form occurred in aboutthe middle of 19 th century when mathematicians noticed that classical tools are notenough to solve their problems. During the time of classical algebra in the Renais-sance, Italian mathematicians Scipione del Ferro (1465–1526) and Niccol`o Tartaglia(1499/1500–1557) found the solution of the equations of degree 3 [29, p. 10] and an-other Italian mathematician Lodovico Ferrari (1522–1565) solved equations of degree4, but, then, it was the Italian mathematician and philosopher Paolo Ruffini (1765–1822) and later the Norwegian mathematician Niels Henrik Abel (1802–1829) whoused abstract algebra techniques to show that equations of degree 5 and of higher thandegree 5 are not always solvable using radicals (known as Abel-Ruffini theorem [47, § Mathematical Analysis of Logic (1847), was mostprobably the first who formulated an example of a non-numerical algebra, a formalsystem, which can be investigated without explicit resource to their intended interpre-tations [18, p. 1]. It is also good to mention that the first statement of the moderndefinition of an abstract group was given by the German mathematician Walther FranzAnton von Dyck (1856–1934) [56].
An algebraic structure on a set (called underlying set or carrier set) is essentially acollection of finitary operations on it [14, p. 41, 48]. Since ring-like structures havetwo binary operations, often called addition and multiplication, with multiplication dis-tributing over addition and the algebraic structure “semiring” is one of them, we con-tinue this note by discussing the history of the distributive law in mathematics briefly.The distributive law, in mathematics, is the law relating the operations of multipli-cation and addition, stated symbolically, a ( b + c ) = ab + ac . Ancient Greeks wereaware of this law. The first six books of Elements presented the rules and techniquesof plane geometry. Book I included theorems about congruent triangles, construc-tions using a ruler and compass, and the proof of the Pythagorean theorem about the3engths of the sides of a right triangle. Book II presented geometric versions of thedistributive law a ( b + c + d ) = ab + ac + ad and formulas about squares, such as ( a + b ) = a + 2 ab + b and a − b = ( a + b )( a − b ) [6, p. 32].Apart from arithmetic, the distributive law had been noticed, years before the birthof abstract algebra, by the inventors of symbolic methods in the calculus. While the firstuse of the name distributive operation is generally credited to the French mathemati-cian Franc¸ois-Joseph Servois (1768–1847) [45], the Scottish mathematician DuncanFarquharson Gregory (1813–1844), who wrote a paper in 1839 entitled On the RealNature of Symbolical Algebra , brought out clearly the commutative and distributivelaws [11, p. 331].Boole in his book entitled
Mathematical Analysis of Logic (1847) mentioned thislaw by giving the name distributive to it and since he was, most probably, the firstwho formulated an example of a non-numerical algebra, one may consider him the firstmathematician who used this law in abstract algebra [4]. Some years later, Cayley, in apaper entitled
A Memoir on the Theory of Matrices (1858), showed that multiplicationof matrices is associative and distributes over their finite addition [13].The distributive law appeared naturally in ring-like structures as well. Though thefirst axiomatic definition of a ring was given by the German mathematician AbrahamHalevi (Adolf) Fraenkel (1891–1965) and he did mention the phrase “the distributivelaw” (in German “die distributiven Gesetze”), but his axioms were stricter than thosein the modern definition [20, p. 11]. Actually, the German mathematician EmmyNoether (1882–1935) who proposed the first axiomatic modern definition of (commu-tative) rings in her paper entitled “Idealtheorie in Ringbereichen”, did also mention thephrase “Dem distributiven Gesetz” among others such as “Dem assoziativen Gesetz”and “Dem kommutativen Gesetz” [42, p. 29].
The most familiar examples for semirings in classical algebra are the semiring of non-negative integers or the semiring of nonnegative real numbers. The first examplesof semirings in modern algebra appeared in the works of the German mathematicianRichard Dedekind (1831–1916) [16], when he worked on the algebra of the ideals ofrings [25]. In point of fact, it was Dedekind who proposed the concept of ideals in hisearlier works on number theory, as a generalization of the concept of “ideal numbers”developed by the German mathematician Ernst Kummer (1810–1893) [37]. Otherssuch as the English mathematician Francis Sowerby Macaulay (1862–1937) and theGerman mathematicians Emanuel Lasker (1868–1941), Emmy Noether (1882–1935),Wolfgang Krull (1899–1971) and Paul Lorenzen (1915–1994) also studied the algebraof ideals of rings [35,38–40,42]. Semirings appeared implicitly in the works of the Ger-man mathematician David Hilbert (1862–1943) and the American mathematician Ed-ward Vermilye Huntington (1874–1952) in connection with the axiomatization of thenatural and nonnegative rational numbers [25, 31, 32]. But, then, it was the Americanmathematician Harry Schultz Vandiver (1882–1973) who used the term “semi-ring” inhis 1934 paper entitled “Note on a simple type of algebra in which cancellation lawof addition does not hold” for introducing an algebraic structure with two operations4f addition and multiplication such that multiplication distributes on addition, whilecancellation law of addition does not hold [49]. The foundations of algebraic theoryfor semirings were laid by Samuel Bourne and others in the l950’s [1, p. 6]. Forexample, the concept of ideals for semirings was introduced by Samuel Bourne [5].In the years between 1939 and 1956, Vandiver published at least six more papers onsemirings [50–55], but it seems he was not successful to draw the attention of mathe-maticians to consider semirings as an independent algebraic structure that is worth tobe developed [27]. In fact, semirings, as most of the other concepts in mathematics,were not developed as an exercise for generalization, only for the sake of generaliza-tion! Actually, in the late 1960s, semirings were considered as a more serious topicby researchers when real applications were found for them. The Polish-born Americanmathematician Samuel Eilenberg (1913–1998) and a couple of other mathematiciansstarted developing formal languages and automata theory systematically [19], whichthey have strong connections to semirings. Since then many mathematicians and com-puter scientists have broadened the theory of semirings and related structures [23].Definitely, the reference books on semirings and other ring-like algebraic structuresincluding the books [1, 3, 17, 24–26, 30, 36, 43] have helped to the popularity of theserather new algebraic structures. Today many journals specializing in algebra have edi-tors who are responsible for semirings. Semirings not only have significant applicationsin different fields such as automata theory in theoretical computer science, (combinato-rial) optimization theory, and generalized fuzzy computation, but are fairly interestinggeneralizations of two broadly studied algebraic structures, i.e., rings and bounded dis-tributive lattices [23,25]. The number of publications in the field of semiring theory, thebeauty of the semirings and their broad applications in different areas of science shouldconvince us that today semiring theory is an established one and its development, evenin pure mathematics, is valuable and important.
In order to have a bit more of “an argument without end”, the reader may like to referthe books [6–10, 57] on the general history of mathematics and the books [2, 34, 48] onthe history of algebra.
Acknowledgements
The author’s main interests in algebra are in commutative algebra and semiring theory.It is a pleasure to thank both Professor Dara Moazzami and Professor Winfried Brunsto help and encourage the author in order to work on these fields of algebra. Theauthor is supported in part by the Department of Engineering Science at GolpayeganUniversity of Technology and his special thanks go to the Department for providing allthe necessary facilities available to him for successfully conducting this research.5 eferences [1] J. Ahsan, J. N. Mordeson and M. Shabir,
Fuzzy Semirings with Applications toAutomata Theory , Springer, Berlin, 2012.[2] H.-W. Alten, A. Djafari Naini, B. Eick, M. Folkerts, H. Schlosser, K.-H. Schlote,H. Wesem¨uller-Kock and H. Wußing, , Springer Spektrum, Berlin, 2014.[3] S. Bistarelli,
Semirings for Soft Constraint Solving and Programming , Springer-Verlag, Berlin, 2004.[4] G. Boole,
The Mathematical Analysis of Logic, Being an Essay towards a Calcu-lus of Deductive Reasoning , Cambrige: Macmilan, Barclay,& Macmilan, Lon-don: George Bell, 1847.[5] S. Bourne,
The Jacobson radical of a semiring , Proc. Nat. Acad. Sci. (1951),163–170.[6] M. J. Bradley, The Birth of Mathematics: Ancient Times to 1300
Vol. 1. ChelseaHouse Infobase Publishing, New York, 2006.[7] M. J. Bradley,
The Age of Genius: 1300 to 1800 , Vol. 2. Chelsea House InfobasePublishing, New York, 2006.[8] M. J. Bradley,
The Foundations of Mathematics: 1800 to 1900 , Vol. 3. ChelseaHouse Infobase Publishing, New York, 2006.[9] M. J. Bradley,
Modern Mathematics: 1900 to 1950 , Vol. 4. Chelsea House In-fobase Publishing, New York, 2006.[10] M. J. Bradley,
Mathematics Frontiers: 1950 to the Present , Vol. 5. Chelsea HouseInfobase Publishing, New York, 2006.[11] F. Cajori,
A History of Mathematics , The Macmillan Company, London, 1909.[12] A. Cayley,
VII. On the theory of groups, as depending on the symbolic equa-tion θ n = 1 , The London, Edinburgh, and Dublin Philosophical Magazine andJournal of Science (42) (1854), 40–47.[13] A. Cayley, A Memoir on the Theory of Matrices , Philos. Trans. R. Soc. Lond., (1858), 17–37.[14] P. M. Cohn,
Universal Algebra , D. Reidel Publishing Company, Dordrecht, 1981.[15] R. Cooke,
Classical Algebra, Its Nature, Origins, and Uses , John Wiley andSons, lnc. Publication, Hoboken, 2008.[16] R. Dedekind, ¨Uber die Theorie der ganzen algebraiscen Zahlen , SupplementXI to P.G. Lejeune Dirichlet: Vorlesung ¨uber Zahlentheorie 4 Aufl., Druck undVerlag, Braunschweig, 1894. 617] M. Droste, W. Kuich and H. Vogler,
Handbook of Weighted Automata , EATCSMonographs in Theoretical Computer Science, Springer, Berlin, 2009.[18] J. M. Dunn and G. M. Hardegree,
Algebraic methods in philosophical logic ,Oxford University Press, New York, 2001.[19] S. Eilenberg,
Automata, Languages, and Machines , Vol. A., Academic Press,New York, 1974.[20] A. A. Fraenkel, ¨Uber die Teiler der Null und die Zerlegung von Ringen , disserta-tion, Marburg University, 1914.[21] P. C. A. Geyl,
Napoleon: Voor en Tegen in de Franse Geschiedschrijving , Oost-hoek’s Uitgevers Mij. N.V., Utrecht, 1946.[22] W. J. Gilbert and S. A. Vanstone,
Classical Algebra , 3rd end., Waterloo Mathe-matics Foundation, Waterloo, 1993.[23] K. Głazek,
A guide to the literature on semirings and their applications in math-ematics and information sciences , Kluwer, Dordrecht, 2002.[24] J. S. Golan,
Power Algebras over Semirings, with Applications in Mathematicsand Computer Science , Kluwer, Dordrecht, 1999.[25] J. S. Golan,
Semirings and Their Applications , Kluwer, Dordrecht, 1999.[26] J. S. Golan,
Semirings and Affine Equations over Them: Theory and Applica-tions , Kluwer, Dordrecht, 2003.[27] J. S. Golan,
Some recent applications of semiring theory , International Confer-ence on Algebra in Memory of Kostia Beider at National Cheng Kung University,Tainan, 2005.[28] M. Gondran, M. Minoux,
Graphs, Dioids and Semirings , Springer, New York,2008.[29] H. Grant, I. Kleiner,
Turning Points in the History of Mathematics , Birkh¨aser,New York, 2015.[30] U. Hebisch and H. J. Weinert,
Semirings, Algebraic Theory and Applications inComputer Science , World Scientific, Singapore, 1998.[31] D. Hilbert, ¨Uber den Zahlbegriff , Jber. Deutsch. Math.-Verein. (1899), 180–184.[32] E. V. Huntington, Complete sets of postulates for the theories of positive integraland positive rational numbers , Trans. Amer. Math. Soc. (1902), 280–284.[33] V. J. Katz and B. Barton, Stages in the history of algebra with implications forteaching , Educational Studies in Mathematics (2007), 185–201.[34] I. Kleiner, A History of Abstract Algebra , Birkh¨auser, Boston, 2007.735] W. Krull,
Axiomatische Begr¨undung der Algemeinen Idealtheorie , Sitz. phys.med. Soc. Erlangen (1924), 47–63.[36] W. Kuich and A. Salomaa, Semirings, Automata, Languages , Springer-Verlag,Berlin, 1986.[37] E. E. Kummer, ¨Uber die Zerlegung der aus Wurzeln der Einheit gebildeten com-plexen Zahlen in ihre Primfactoren , Jour. f¨ur Math. (Crelle) (1847) 327–367.[38] E. Lasker, Zur Theorie der Moduln und Ideale , Math. Ann. (1905), 19–116.[39] P. Lorenzen, Abstrakte Begr¨undung der multiplikativen Idealtheorie , Math. Z., (1939), 533–553.[40] F. S. Macaulay, Algebraic Theory of Modular Systems , Cambridge UniversityPress, Cambridge, 1916.[41] C. B. Boyer and U. C. Merzbach,
A History of Mathematics , 3rd edn, John Wileyand Sons Inc., Hoboken, 2011.[42] E. Noether,
Idealtheorie in Ringbereichen , Mathematische Annalen (1–2)(1921), 24–66.[43] G. Pilz, Near-Rings, The Theory and Its Aplications , Revised edition, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1983.[44] G. Saliba,
Science and medicine , Iranian Studies, (3–4) (1998), 681–690.[45] F. J. Servois, Essai sur un nouveau d’exposition des principes du calculdiff´erentiel , Ann. Math. Pures Appl. (1814), 93–140.[46] L. Stalling, A brief history of algebraic notation , School Science and Mathemat-ics, (5), 230–235.[47] J.-P. Tignol,
Galois’ Theory of Algebraic Equations , 2nd edn., World Scientific,Singapore, 2016.[48] B. L. van der Waerden,
A History of Algebra: From al-Khw¯arizmi to EmmyNoether , Springer, Berlin, 1985.[49] H. S. Vandiver,