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43 Lui s A. Caffarelli an d Xavier Cabre, Full y nonlinear elliptic equations, 199 5 42 Victo r duillemin and Shlomo Sternberg, Variation s on a theme by Kepler, 199 0 41 Alfre d Tarski and Steven Gtvant, A formalization o f theory without variables, 198 7 40 R . H. Bing, Th e geometric topology of 3-manifolds , 198 3 39 N . Jacobson, Structur e and representations of Jordan algebras, 196 8 38 O . Ore, Theor y of graphs, 196 2 37 N . Jacobson, Structur e of rings, 195 6 36 W . H. Gottschalk an d G. A. Hedlund, Topologica l dynamics , 195 5 35 A , C Schaeffe r an d D. C. Spencer, Coefficien t region s for Schlicht functions , 195 0 34 J . L, Walsh, Th e location of critical points of analytic and harmonic functions, 195 0 33 J . F. Ritt, Differentia l algebra , 195 0 32 R . L. Wilder, Topolog y of manifolds, 194 9 31 E . Hille and R. S. Phillips, Functiona l analysis and , 195 7 30 T . Radd, Lengt h and area, 194 8 29 A . Weil, Foundation s of algebraic geometry, 194 6 28 G . T. Whyburn, Analyti c topology, 194 2 27 S . Lefschetz, Algebrai c topology, 194 2 26 N . Levinson, Ga p and density theorems, 194 0 25 Garret t Birkhoff, Lattic e theory, 194 0 24 A . A. Albert, Structur e of algebras, 193 9 23 G . Szego, Orthogona l polynomials, 193 9 22 C , N. Moore, Summabl e series and convergence factors, 193 8 21 J . M. Thomas, Differentia l systems , 193 7 20 J . L , Walsh, Interpolatio n an d approximation b y rationa l s i n th e complex domain, 193 5 19 R . E. A. C Pale y and N. Wiener, Fourie r transforms i n the complex domain, 193 4 18 M Morse , Th e calculus of variations in the large, 193 4 17 J . M. Wedderburn, Lecture s on matrices, 193 4 16 G . A. Bliss, Algebrai c functions, 193 3 15 M . H. Stone, Linea r transformations i n Hilbert space and their applications to analysis, 1932 14 J . F. Ritt, Differentia l equation s from the algebraic standpoint, 193 2 13 R . L. Moore, Foundation s of point , 193 2 12 S . Lefschetz, Topology , 193 0 11 D . Jackson, Th e theory of approximation, 193 0 10 A . B. Coble, Algebrai c geometry and theta functions, 192 9 9 G . D. Birkhoff, Dynamica l systems, 192 7 8 L . P. Eisenhart, Non-Riemannia n geometry, 192 2 7 E . T. Bell, Algebrai c arithmetic, 192 7 6 G , C Evans , Th e logarithmic potential, discontinuous Dirichlet and Neumann problems, 1927 5.1 G . C . Evans, .Functionals an d thei r applications; selecte d topics , includin g integra l equations, 191 8 12 O - Veblen, Analysi s situs, 192 2 4 L . E. Dickson, O n invariants and the theory of number s W. F. Osgood, Topic s in the theory of functions of several complex variables, 191 4 {Continued in the back of this publication) This page intentionally left blank http://dx.doi.org/10.1090/coll/025 America n Mathematica l Societ y Colloquiu m Publication s Volum e 25

Lattic e Theor y

Garret t Birkhof f

America n Mathematica l Societ y Providence , Rhod e Islan d 1991 Mathematics Subject Classification. Primar y 06-02 .

Library o f Congres s Car d Numbe r 66-2370 7 International Standar d Boo k Numbe r 0-8218-1025- 1 International Standar d Seria l Numbe r 0065-925 8

Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s granted t o quot e brie f passage s fro m thi s publicatio n i n reviews, provided th e customary acknowledgmen t o f the sourc e i s given . Republication, systematic copying, or multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be made b y e-mai l t o reprint-permissionfimath.ams.org .

© Copyrigh t 1940 , 1948 , 1967 by the America n Mathematica l Society . Al l rights reserved . Second (Revised ) Edition , 194 8 Third Edition , sevent h printin g wit h correction s 1993 . Third Edition , eight h printin g wit h correction s 1995 . Printed i n the Unite d State s o f America . The America n Mathematica l Societ y retain s al l rights except thos e granted t o the United State s Government . @ Th e pape r use d i n this boo k i s acid-free an d fall s within th e guideline s established t o ensur e permanence an d durability .

12 1 1 1 6 1 5 1 4 1 3 1 2 1 1 PREFACE T O TH E THIR D EDITIO N

The purpos e o f thi s editio n is threefold : t o mak e th e deepe r idea s o f lattic e theory accessibl e t o mathematician s generally , t o portra y it s structure , an d t o indicate som e o f it s mos t interestin g applications . A s i n previou s editions , a n attempt i s made to includ e current developments , includin g variou s unpublishe d ideas o f my own ; however , unlik e previou s editions , this editio n contain s onl y a very incomplete bibliography . I am summarizing elsewheref m y ideas about the role played by theory in mathematics generally. I shall therefore discuss below mainly its logical structure, which I hav e attempted to reflect i n my table o f contents. The beauty o f lattice theory derive s i n part fro m th e extreme simplicit y o f it s basic concepts: (partial) ordering, least upper and greatest lower bounds. I n thi s respect, it closely resembles theory. Thes e ideas are developed in Chapters I-V below , where it i s shown that thei r apparent simplicity conceals many subtle variations includin g fo r example , th e propertie s o f modularity , semimodularity , pseudo-complements and orthocomplements . At thi s level , lattice-theoreti c concept s pervad e th e whol e o f moder n algebra, though many textbooks on algebra fai l to make this apparent. Thu s lattices and groups provide two of the most basic tools of " universal algebra ", and in particular the structur e o f algebrai c system s i s usuall y mos t clearl y reveale d throug h th e analysis o f appropriat e lattices . Chapter s V I an d VI I tr y t o develo p thes e remarks, and to include enough technical applications to the theory o f groups and loops with operators to make them convincing . A differen t aspec t o f lattic e theor y concern s th e foundation s o f se t theor y (including genera l topology ) an d rea l analysis. Her e the us e o f variou s (partial ) orderings t o justif y transfinit e induction s an d othe r limitin g processe s involve s some of the most sophisticated constructions of all mathematics, some of which are even questionable ! Chapter s VIII-XI I describ e thes e processe s fro m a lattice - theoretic standpoint . Finally, many o f the deepest and mos t interesting applications o f lattice theor y concern (partially ) ordered mathematical structure s having also a binary additio n or multiplication: lattice-ordered groups , monoids , vector spaces, rings, and s (like the real field). Chapter s XIII-XVII describ e the properties of such systems,

t G . Birkhoff , What can lattices do for you?, a n articl e i n Trends in Lattice Theory, Jame s C . Abbot, ed. , Van Nostrand, Princeton , N.J. , 1967 . v VI PREFACE and als o thos e o f positiv e linea r operator s o n partiall y ordere d vecto r spaces . The theory o f such systems, indeed, constitute s th e most rapidly developin g par t of lattice theory at th e present time . The labor o f writing this boo k has bee n enormous, even though I hav e made no attempt at completeness. I wis h to express my deep appreciation t o those many colleagues an d student s wh o hav e criticize d part s o f my manuscrip t i n variou s stages o f preparation . I n particular , I ow e a ver y rea l deb t t o th e following : Kirby Baker , Orri n Frink , Georg e Gratzer , C . Grandjot , Alfre d Hales , Pau l Halmos, Samue l H . Holland , M . F. Janowitz, Roge r Lyndon , Donal d MacLaren , Richard S . Pierce, George Raney, Arlan Ramsay, Gian-Carlo Rota, Walter Taylor, and Ala n G . Waterman . My thanks are also due to the National Scienc e Foundation fo r partial suppor t of research in this area and of the preparation o f a preliminary edition of notes, and to th e Argonn e Nationa l Laborator y an d th e Ran d Corporatio n fo r suppor t o f research int o aspects o f lattice theory o f interest t o members o f their staffs . Finally, I wis h t o than k Laur a Schlesinge r an d Lorrain e Dohert y fo r thei r skillful typin g o f the entire manuscript . TABLE O F CONTENT S

CHAPTER I TYPES O F LATTICE S ... .

CHAPTER I I POSTULATES FO R LATTICE S .2 0

CHAPTER II I STRUCTURE AN D REPRESENTATIO N THEOR Y .... 5 5

CHAPTER I V GEOMETRIC LATTICE S 8 0

CHAPTER V COMPLETE LATTICE S Il l

CHAPTER V I UNIVERSAL ALGEBR A 13 2

CHAPTER VI I APPLICATIONS T O ALGEBR A 15 9

CHAPTER VII I TRANSFINITE INDUCTIO N 18 0

CHAPTER I X APPLICATIONS T O GENERA L TOPOLOG Y 21 1

CHAPTER X METRIC AN D TOPOLOGICA L LATTICE S 23 0

CHAPTER X I BOREL ALGEBRA S AN D VO N NEUMAN N LATTICE S . . .25 4

CHAPTER XI I APPLICATIONS T O LOGI C AN D PROBABILIT Y .27 7 vii Vlii CONTENT S

CHAPTER XII I LATTICE-ORDERED GROUP S 28 7

CHAPTER XI V LATTICE-ORDERED MONOID S 31 9

CHAPTER X V VECTOR LATTICE S . 34 7

CHAPTER XV I POSITIVE LINEA R OPERATOR S ...... 38 0

CHAPTER XVI I LATTICE-ORDERED S 39 7

BIBLIOGRAPHY . . . 41 1

ADDITIONAL REFERENCES .41 5

INDEX 41 7 BIBLIOGRAPHY

I. BOOKS . A n attempt is made to identify eac h book by a mnemonic letter sequence. Fo r example, LT1 and LT2 refer to the first two editions of the present book . Othe r books especi- ally relevan t include : [A-H] P . Alexandrot T and H . Hopf , Topologie, Springer , 1935 . [Ba] S . Banach , Theorie des operations linearei, Warsaw , 1833 . [B-M] G . Birkhof f an d S . Ma c Lane , A survey of modern algebra, 3 d ed., Macmillan , 1965 . [Bo] G . Boole , An investigation into the laws of thought, London , 185 4 (reprinte d b y Ope n Court Publishing Co. , Chicago, 1940) . [Ca] C . Caratheodory, Mass und Integral und ihre Algebraisierung, Birkhauser , 1956 . Also , Measure and Integration, Chelsea , 1963 . [CG] J . von Neumann , Continuous geometry, Princeton , 1960 . [Da] M . M . Day , Normed linear spaces, 2d ed. . Springer , 1962 . (Contain s materia l o n Banac h lattices.) [D-S] N . Dunfor d an d J. Schwartz , Theory of linear operators. Par t I, New York , 1958 . [Fu] L . Fuchs, Partially ordered algebraic systems, Pergamo n Press , 1963 . {GA] E . H . Moore , Introduction to a form of general analysis, AM S Cotloq . Pubi . no . 2 , New Haven , 1910 . [Hal] P . R . Halmos , Lectures on Boolean algebras, Va n Nostrand , 1963 . [Hau] F . HausdorfT , Orundziige der Mengenlehre, Leipzig , 1914 . [KaVP] L . V . Kantorovich , B . C . Vulich an d A . G . Pinsker , Functional analysis in paHially ordered spaces (i n Russian), Mosco w - Leningrad, 1950 . [Kel] J . L . Kelley, General topology, Va n Nostrand, 1955 . [KG] F . Maeda , Kontinuierliche Oeometrien, Springer, 1958. [K-N] J . L . Kelley, I . Namioka and others, Linear topological spaces. Va n Nostrand, Prince - ton, 1963 . [Li] L . R . Lieber , Lattice theory, Galoi s Institut e o f Mathematic s an d Art , Brooklyn , N.Y., 1959 . [MA] B . L . van de r Waerden , Moderne Algebra, 2 vols., Springer , 1930 . [Pe] A . L . Peressini , Ordered topological vector spaces. Harpe r an d Row , 1967 . [Sch] H . H . Schaefer , Topological vector spaces, Macmillan, 1966 . (Chapte r V i s o n vecto r lattices; Appendix on th e spectr a o f positiv e linea r operators. ) [Sch] E . Schroder , Algebra der Logik, 3 vols., Leipzig , 1890-5 . [Sik] R . Sikorski , Boolean Algebras, 2 d ed., Springer , 1964 . [Sk] L . A . Skornjakov , Complemented modular lattices and regular rings, Gos , Izd.-Mat . Lit., Moscow , 1961 . [SuJ M . Suzuki, Structure of a group and the structure of its lattice of subgroups. Springer , 1956. [Symp] Lattice theory, Proc . Sympos. Pur e Math . vol . 2 , AMS , Providence, R.I. , 1961 . [Sz] G . Szasz, Introduction to lattice theory, 3 d ed., Academic Pres s and Akademia i Kiado , 1963. [Ta] A . Tareki, Cardinal algebras, Oxfor d Univ . Press , 1949 . [Tr] M . L . Dubreil-Jacotin , L . Lesieur , an d R . Croisot , Lecons sur la theorie des treillis, Gauthier-Villars, Paris , 1953 . [UA] P . M . Conn . Universal algebra, Harper an d Row , 1965 . [Vu] B Z . Vulikh , Introduction to the theory of partially ordered spaces, Noordhoff . [W-R] A . N . Whitehea d an d B . Russell , , 3 vols. , Cambridg e Univ . Press, 1925 , 1927 . 411 412 BIBLIOGRAPHY

II. PAPER S G. Birkhoff , [1 ] On the combination of subalgebras, Proc . Cambridg e Philos . Soc . 2 9 (1933) , 441-64; [2] Combinatorial relations in projective geometries, Ann . o f Math . 3 6 (1935) , 743-8 ; [3] On the structure of abstract algebras, Proc . Cambridg e Philos . Soc . 3 1 (1935) , 433-54 ; [4] Moore-Smith convergence in general topology, Ann. of Math. 38 (1937), 39-56; [5] Generalized arithmetic, Duk e Math . J . 9 (1942) , 283-302 ; [6 ] Lattice-ordered groups, Ann . o f Math . 4 3 (1942), 298-331 ; [7 ] Uniformly semi-primitive multiplicative processes, Trans . AM S 10 4 (1962), 37-51 . G. Birkhof f an d O . Frink, [II Representations of lattices by sets, Trans. AMS 6 4 (1948), 299-316. G. Birkhof f an d J . vo n Neumann , [1 ] On the logic of quantum mechanics, Ann . o f Math . 3 7 (1936), 823-43. G. Birkhof f an d R . S . Pierce, [1] Lattice-ordered rings, An . Acad . Brasil . Ci . 28 (1956) , 41-69. J. Certaine , [1 ] Lattice-ordered groupoids and some related probletns, Harvar d Doctora l Thesi s (unpublished), 1943 . M. M . Day, [1] Arithmetic of ordered systems, Trans . AM S 5 8 (1945) , 1-43 . R. Dedekind, [1] Vber Zerlegungen von Zahlen durch ihre grossten gemeinsamen Teiler, Festschrif t Techn. Hoch . Branuschwei g (1897) , and Ges . Werke , Vol . 2 , 103-48 ; [2] Vber die von drei Moduln erzeugte Dualgruppe, Math . Ann . 5 3 (1900) , 371-403 , an d Ges . Werke , Vol , 2 , 236-71. R. P . Dilworth , [I] N on-commutative residuated lattices, Trans . AM S 4 6 (1939) , 426-44 ; [2 ] Lattices with unique complements Trans. AM S 5 7 (1945) , 123-54 . R. P . Dilwort h an d Pete r Crawley , [1] Decomposition theory for lattices without chaiyi coyidition, Trans. AM S 9 6 (1960) , 1-22 . C. J. Everet t an d S . Ulam, [1 ] On ordered groups, Trans . AM S 57 (1945) , 208-16 . H. Freudcnthal , [1 ] Teilweisc geordnete Moduln, Proc . Akad . Wet . Amsterda m 3 9 (1936) , 641-51. (). Frink , [I] Topology in lattices, Trans . AM S 5 1 (1942) , 569-82 ; [2 ] Complemented modular lattices and jjrojertive spaces of infinite dimension, ibid. , 6 0 (1946) , 452-67 . H. Gordon , [1 ] Relative uniform convergence, Math . Ann . 15 3 (1964), 418-27. H. Hahn , [I] Cbe r di e nichtarchimedisch e Grossen.systeme , S.-B . Akad . Wiss . Wie n Il a 11 6 (1907), 601-55. A. Hales , [1 ] On the non-existence of free complete Boolean algebras, Fundamenta l Math . 5 4 (1964), 45-66 . J. Hashimot o [1 ] Ideal theory for lattices, Math . Japonic a 2 (1952) , 149-86 . M. Henrikso n an d J . R . Isbell , [1 ] Lattice-ordered rings and function rings, Pacifi c J . Math . 12 (1962), 533-65. 0. Holder , [1 ] Die Axiome der Quantitat und die Lehre com Mass, Leipzi g Ber. , Math.-Phys . CI. 5 3 (1901) , 1-64 . S. S . Holland, Jr., [1] .4 Radon-Sikodym theorem for dimension lattices. Trans . AMS 10 8 (1963), 66-87. E. V . Huntington, j 1] Sets of independent postulates for the algebra of logic, Trans. AMS 5 (1904), 288-309. B. Jonsson , fl] Representations of complemented modular lattices, Trans . AM S 9 7 (1960) , 64-94 . B. Jonsso n an d A . Tarski, . [1] Direct decompositions of finite algebraic systems, Notr e Dam e Mathematical Lectures , No . 5 (1947) , 6 4 pp. R. V . Kadison, [I] A representation theorem for commutative topological algebra, Mem . AM S #7 , (1951), 3 9 pp. S. Kakutani , [1 ] Concrete representation of abstract {L)-spaces and the mean ergodic theorem, Ann. of Math. 4 2 (1941), 523-37; [2] Concrete representation of {M)-spaces, ibid. , 994-1024, L. Kantorovich, [I] Lineare halbgeordnete Raume, Mat . Sb . 2 (44 ) (1937) , 121-68 . 1. Kaplansky , [1 ] Any orthocomplemented complete modular lattice is a continuous geometry, Ann. o f Math. 6 1 (1955) , 524-41 . BIBLIOGRAPHY 413

J. L . Kelley, [1] Convergence in topology, Duk e Math . J. 1 7 (1950) , 277-83 . M. G. Krein and M . A. Rutman, [1] Linear operators leaving invariant a cone in a Banach space, Uspehi Mat . Nau k 3 (1948) , 3-95 . (AM S Translations No. 26. ) L. Lesieur , [I] Sur les demigroupes reticules satisfaisant a une condition de chaine, Bull . Soc . Math. France 8 3 (1955) , 161-93 . L. Loomis, [1] On the representation of a-complete Boolean algebras, Bull . AMS 53 (1947), 757-60; [2] The lattice theoretic background of the dimension theory, Mem . AM S #18 , A 1955 . S. Ma c Lane , [1 ] A lattice formulation for transcendence degrees and p-bases, Duk e Math. J. 4 (1938), 455-68 . H. MacNeille , [I] Partially ordered sets, Trans . AM S 4 2 (1937) , 416-60 . F. Maeda , [1 ] Matroid lattices of infinite length, J . Sci . Hiroshim a Univ . Ser . A-1 5 (1952) , 177-82. K. Menger , F . Alt , an d O . Schreiber , [1 ] New foundations of projective and affine geometry, Ann. of Math . 3 7 (1936) , 456-82. E. H . Moore , [1] The Ne w Have n Mathematica l Colloquium , Introduction to a form of general analysis, AM S Colloq . Publ . vol . 2 , New Haven, 1910 . I. Namioka , fl] Partially ordered linear topological spaces, Mem . AM S #2 4 (1957) . J. vo n Neumann, [1 ] Continuous geometries an d Examples of continuous geometries, Proc . Nat. Acad. Sci . U.S.A . 2 2 (1936) , 707-13 . M. H. A . Newman , [1 ] A characterization of Boolean lattices and rings, J . Londo n Math . Soc . 16 (1941), 256-72. T. Ogasawara , [1 ] Theory of vector lattices, I , J . Sci . Hiroshim a Univ . 1 2 (1942) , 37-100 ; [2 ] Theory of vector lattices, II , ibid . 1 3 (1944), 41-161. (Se e Math . Revs . 1 0 (1949), 545-6. ) O. Ore , [1] On the foundations of , I, Ann . of Math. 3 6 (1935), 406-37; [2] On the foundations of abstract algebra, II, ibid. , 3 7 (1936) , 265-92 . C. S. Peirce , [1] On the algebra of logic, Amer. J. Math . 3 (1880), 15-57 . L. Rieger , [1 ] On the free R.-comjilete Boolean algebras. . ., Fund . Math . 3 8 (1951) , 35-52 . F. Riesz , [1 ] Sur la theorie generate des operations lineaires, Ann . o f Math . 4 1 (1940) , 174-206 . G.-C. Rota , [1 ] On the foundations of combinatorial theory. 1. Theory of Mobius functions, Zeits, i. Wahrscheinlichkeitzreehnun g 2 (1964), 340-68 . Th. Skolem , [I] Om konstitutionen av den identiske kalkuls grupper, Thir d Scand , Math. Congr . (1913), 149-63 . M. H. Stone, [1] The theory of rejxresentations for Boolean algebras, Trans. AMS 40 (1936), 37-111; [2] Applications of the theory of Boolean rings to general topology, ibid. , 4 1 (1937) , 375-481 ; [3] Topological representations of distributive lattices and Broutverian logics, Cas . Mat. Fys. 6 7 (1937), 1-25 ; [4 ] A general theory of spectra, Prbc . Nat . Acad . Sci . U.S.A . 2 7 (1941) , 83-7 . A. Tarski, [1] Sur les classes closes par rapport a certaincs operations elementaires, Fund . Math. 1 6 (1929), 181-305 ; [2 ] Zur Grundlagung der Booleschcn Algebra. I , ibid. , 2 4 (1935) , 177-98 ; [3] Grundzuge des Systememkalkuts, ibid. , 2 5 (1936) , 503-26 an d 2 6 (1936) , 283-301. M. Ward an d R . P . Dilworth , [1] Residuated lattices, Trans . AM S 4 5 (1939) , 335-54 . L. Weisner, [1] Abstract theory of inversion of finite series, Trans . AMS 38 (1935), 474-484. H. Whitney , [1 ] The abstract properties of linear dependence, Amer . J. 5 7 (1935) , 507-33 . P. Whitman , [I ] Free lattices, Ann . o f Math . 4 2 (1941) , 325-9 ; [2 ] Free lattices. II , ibid . 4 3 (1942), 104-15 . K. Yosid a an d S . Kakutani , [1 ] Operator-theoretical treatment of Markoff ys process and mean ergodic theorem, Ann. o f Math . 4 2 (1941) , 188-228 . This page intentionally left blank ADDITIONAL REFERENCES

Since the third edition [LT3J of Lattice Theory was published in 1967, containing two chapters on the role of lattices in universal algebra, there has been enormous progress in both areas. The journal Algebra Universalis, founded in 197 1 by George Gratzer, has maintained a high quality of contributions to them ever since. Sinc e 1984 , a second journal Order, founded by Ivan Rival shortly after the publication of the volume [Rivj which he edited, ha s provided a second, more applications oriented vehicle of publication with combinatorial emphasis. In addition to these journals, many books have been published explaining new developments concerned wit h th e theories of ordere d set s and lattices , an d their man y applications. Abou t fifteen of th e mos t importan t o f thes e book s ar e liste d below . The y supplemen t th e man y footnotes of [LT3I, which were intended to document major developments prior to around 1965 in 400 pages.

[Abb] Jame s C Abbott , (ed) , Trends in Lattice Theory, Va n Nostrand, 1970 . Fou r readable essays on different topics , explaining their status as of tha t date in non-technical lan- guage. [Bi5] G.-C . Rota and J. S. Oliveira (eds.), Selected Papers in Algebra and Topology by Garrett Birkhoff, Birkhauser, 1987 . In 600 pages, this volume makes available many papers with my comments on them. [B-W] K . A. Baker and R. Wille (eds.), Lattice Theory and its Applications, Haldermann Verlag, 1995. Contains 1 5 articles by distinguished experts on lattice theory and its applications. [C-D] Pete r Crawley and Robert Dilworth, Algebraic Theory of Lattices, Prentice-Hall, 1973 . A self-contained surve y of man y properties of 'compactl y generated ' lattices b y those who discovered them; cf. [Dil] . [Die] Kennet h Bogart , R . Freese , an d J . Kun g (eds.) , The Dilworth Theorems, Birkhauser , 1990. This volume reprints selected papers and contains biographical information about a leading contributor. [D-P] B . A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridg e Univ. Press, 1990 , provides a very readabl e introductio n to man y topics in the spirit of the journal Order. [FL] Ralp h Freese , J . Jezek , an d J . B . Nation, Free Lattices, Am. Math . Society , 1995 . A definitive boo k by leading authorities on the topic identified b y its title. [Fuchs] Laszl o Fuchs , Teilweise geordneten algebraische Strukturen, Akad . Kiado , Budapest , 1966. A scholarl y revie w explainin g man y result s abou t partiall y ordere d algebrai c structures not touched on in [LT3]. [GLT] Georg e Gratzer , General Lattice Theory, Birkhauser, 1978 . A n incisiv e an d origina l explanation of man y results discovered after 1965 , by a leading contributor. [L-Z] W . A. J. Luxembur g and A . C . Zaanen , Riesz Spaces, North-Holland , 1971 . A schol- arly and thoroug h discussion of vecto r lattices, called Ries z spaces i n deference to N. Bourbaki.

415 416 ADDITIONAL REFERENCES

[Male] A . I. Mai' cev, Algebraic Systems, Springer, 1973 . English translation of a fundamental book o n genera l algebra , bein g writte n (i n Russian ) b y Mai'ce v (1909-67 ) shortl y before his untimely death. [MMT] Ralp h N. McKenzie, Geo. F. McNulty and Walter Taylor, Algebras, Lattices, Varieties, (vol. I) , Wadsworth & Brooks/Cole, 1987 . A highly origina l ye t leisurely introductio n to a planned fou r volume treatise covering all these subjects in depth. [M-M] F . Maeda-S . Maeda , Theory of Symmetric Lattices, Springer , 1970 . A highly origina l discussion of matroi d lattices and related topics. [Riv] Iva n Riva l (ed. ) Ordered Sets, Reidel, 1982 . Contains 23 papers on posets and lattices by leadin g experts , wit h applications to compute r an d socia l sciences , discussion s of open questions, and a list of over 100 0 references. [Sch] H . H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974. A highly orginal volume, comparable with [L-Z], but emphasizing the L-space s and JW-space s of Kaku- tani, and assuming some familiarity with the treatise Linear Operators by Dunford and Schwartz. Leon Henkin et al (eds.), Proceedings of the Tarski Symposium, Am. Math. Soc., 1974. Contains a rich collection of papers concerned with Tarski's '.' Published as Proc. XXV Symp. Pure Math, by Am. Math. Society. George Gratzer, Universal Algebra, 2 d ed., Springer, 1979 . With valuable, supplements by Bjarn i Jonsson an d Walter Taylor, this is the definitive monograp h on the subject, with encyclopaedic coverage. INDEX

Absolute (value), 295 Chain, 2 . 8 Characteristic function, 117 ACC, 180 —subalgebra, 135 Alexander's Theorem, 224 ci-groupoid, 327 Algebra, 132 Closed, 112 Algebraic lattice, 187 —/-ideal, 307,315,352 Algebraically independent, 144 —sublattice, 112 Almost/-ring, 406 Closure , 111,123 Antireflexive, 1,2 0 —property, 7, 111 Antisymmetric, 1 c/-, 327 Antitone, 3 Coftnal , 214 Archimedean, 290,322,398 Collineation, 104 Arnold's Theorem, 221 Commutation lattice, 332 Artin equivalence, 336 Commutative law, 8 Ascending Chain Condition. 180 Compact , 186 Associative law, 8 Compact space, 222 Atomic lattice, 196 ff , 16 , 3,24,135 Complemented lattice, 16 Averaging operator, 408 Complete lattice, 6,111 of Choice, 205 —/-group, 312 Completely distributive, 119 Banach lattice, 366 Completion by cuts, 126 , 23 Conditional completion, 127 Basis, 221 Conditionally complete, 114 Betweenness, 20,36,42 Congruence , 26,136 Bidirectedset,251 Conrad's Theorem, 311 Binary geometric lattice, 107 law, 8 , 18,44 Continuous geometry, 238 generalized—algebra, 48 , 209 —lattice, 17,43 Convergent, 213 —polynomial, 61 Convex, 7,11 —ring, 47 Coprime, 329 —subalgebra, 18 Crypto-, 154 Boole's isomorphism, 277 Borel algebra, 254 DCC, 180 Bounded variation, 239 Dedekind /-monoid, 338 Breadth, 98 Dense-in-itself, 200,242 Brouwerian lattice, 45,128,138 Dependent, 86 —logic, 281 Desarguesian, 104 Cardinal power, 55 Descending Chain Condition, 180 —product, 55 Diagram, 4 —sum, 55 Dilworth Theorems, 71,97,103,146 Carrier, 310 Dimension function, 271 Center, 67,175 —ortholattice, 274 417 418 INDEX

Directed group, 287,289. 312 Graded lattice. 16,4 0 -set, 186.21 3 —poset, 5, 40 Direct product, 8,139 Graph, 98 Disjoint, 68.295 Gratzer-Schmidt Theorem. 18 8 Disjunctive implication, 149 Greatest, 2, 4 —normal form, 61 —lower bound, 6 Distributive element, 69 Hahn's ordinal power. 208 —inequality, 9 Hahn's Theorems, 310, 399 —lattice, 1 2 Hausdorff's Axiom, 192 — triple, 37 Hausdorff space. 213 —valuation, 234 Height. 5 Distributivity, 11,117,29 2 Holder Theorems, 300. 322 Divisibility monoid, 320 Homogeneous continua, 201 Dual, 3 Dual ideal, 25 Ideal. 25 Dual space, 358 — completion. 113 , 127 Duality Principle, 3 Idempotent, 8. Ill Identical implication, 14 8 Endomorphism, 24, 135 Identity, 14 8 Epimorphism, 24, 134 Implication algebra, 46 Equasigroup, 16 0 Incidence function, 10 1 Equationally definable, 15 3 — matrix, 10 1 Ergodic Theorems. 392 ff Inclusion, 1 Existential identity, 14 9 Incomparable, 2 Extensive, 11 1 indecomposable. 334 Family of algebras, 15 0 Independent, 86 Field of sets, 1 2 —/-ideals, 307 Filter of sets, 25 Inf. 6 Finitary closure operation, 18 5 Infinitary operation , 11 7 First category, 258 Infinite distributivity, 117 , 314 Frechet k-spaee, 220 I nject i v e ca tegory, 15 6 Free algebra. 143 Integral, 324 — Boolean algebra, 61 —po-groupoid. 324 — Borel algebra, 257 Integrally closed. 290, 356 — complemented lattice, 14 6 Interval, 7 — distributive lattice, 34, 59 — topology, 250 — lattice. 14 5 Intrinsic neighborhood topology, 241 — modular lattice. 63 ff, 147 Involution, 3 —semi lattice. 30 Isomorphism, 3. 32, 134 — vector lattice, 354 ff Isotonc. 2, 9. 24 — word algebra, 1 4 i Iterated limits. 215 /-ring, 403 Iwasawa Theorem, 317 Freely generated. 14 3 Janowitz Theorem, 268 Frobenius/-monoid. 341 /-closed, 56 Fully characteristic, 135 . 152 o/-normal polynomial, 60 Funayama-Nakayama Theorem, 13 8 Join, 6 Function lattice, 351 Functor, 15 5 J oin-inaccessible. 18 6 Join-irreducible. 31, 58 Galois connection, 12 4 Join-morphism, 24 Generators (set of), 13 4 Join semilattice. 22 Geometric lattice, 80 Jordan decomposition, 239, 293 gib.. 6 Jordan-Dedekind Chain Condition, 5 , 41, 81. Glivenko, Theorem o(, 130 164.177 INDEX 419

Jordan-Holder Theorem, 16 4 Moore family, 111 . 133 Morphism, 24,134 kakutani Theorem, 375 (M) -space, 376 Kaplansky Theorems, 227, 273 Net, 312 Konig's Theorem, 98 Neutral element, 69, 77, 174 Krein Representation Theorem, 358 Newman algebra, 49 Ruratowski Theorem, 257 Noetherian /-monoid, 336 Kurosh-Ore Theorem, 75,90.166 —poset, 18 0 Normal loop, 161 Lattice, 6 — space, 213 Lattice-ordered group, 287 Nowhere dense set, 216 -loop, 29 7 Null system, 10 9 Latticoid, 22. 23, 32 Least, 2, 4 Operator, 161 —upper bound, 6 Order. 4 Length. 5 —convergence. 244, 362 Lexicographic order, 19 9 Order-dense, 200 /-group, 287 Ordered group, 209 /-ideal, 304, 349 Ordinal, 180 , 202 /-loop, 297 — power, 207 /-monoid, 319 — polynomial, 202 /-morphism, 304 — product, 199 Loomis-Sikorski Theorem, 255 —sum, 19 8 Loop, 16 1 Ore's Theorem, 16 8 Lower bound, 6 Oriented graph, 20 —semimoduiar, 15 Ortholattice, 52 /-ring. 397 Orthomodular lattice, 53 (L) -space, 373 Outer measure, 262 l.u.b., 6 , 1 Partition lattice, 15 , 95 Malcev Theorem, 16 3 Permutable relations, 95, 16 1 Maximal, 4, 329, 334 —subgroups, 17 2 —ideal, 28 Perspective, 74 Af-closed, 56 Perspectivity.91,265 Measure, 260 Planar ternary ring, 10 6 —algebra, 260 Point lattice, 80 Median (law), 32, 35 Polarity, 12 2 Meet. 6 Polymorphism, 15 3 Meet-continuous lattice, 18 7 Polynomial (lattice). 29 Meet-morphism, 24 po-group, 287 Meet-semifattice, 22 po-groupoid, 319 Metric lattice, 231 po-ioop, 297 .Metric space, 211 po-monoid, 319 Metric transitivity, 394 po-ring. 397 Minimal, 4 po-semigroup, 319 —Boolean polynomial, 62 Poset, 1 Modular identity, 1 2 Positive cone, 288 — inequality, 9 — linear operator. 381 ff — lattice, 13,3 6 Primary element, 336 —pair, 82, 95 Prime element, 329, 334 Module, 15 9 — ideal, 28 Moebius function. 10 2 -interval, 235 Monomorphism, 24, 134 Primitive matrix, 386 420 INDEX

Principal ideal, 25 Stone Representation Theorem, 226 Probability measure , 260 Strictly characteristic. 13 5 — algebra, 260 — meet-irreducible, 19 4 Projective category, 15 6 Strong unit, 300, 307, 357 Projective intervals, 14 , 232 Structure lattice, 13 8 ~ geometry , 80, 92 Subalgebra, 13 3 —pseudo-metric, 38 2 Subdirect product, 14 0 Projectivity, 10 4 —decomposition theorem , 19 3 Pseudo-complement, 45, 12 5 Subdirectly irreducible , 14 0 Pseudo-metric, 231 Sublattice, 7 Subnet, 214 , 16 0 Substitution Property, 13 6 Quasi-metric lattice, 231 Sup, 6 Quasi-ordered set, 20 Symmetric partition lattice, 15 , 95 Quasi-ordering, 20 Tarski Theorems, 115 , 119, 345 Radical, 336 fi -lattice, 217 Reflexive, 1 7 -space, T\ -space, 117 , 212 Regular measure, 263 0 Topological lattice, 248 —open set, 216 Topological space , 116 , 212 —/-ring, 402 Transftnite induction , 203 — ring, 18 9 Transition operator, 390 — topological space, 213 Transitive 1 , 20 Relation algebra, 343 Transitivity o f perspectiviiy, 75. 269 Relative uniform convergence, 369 Relatively complemented , 16 , 70 Translation, 13 7 Remak's Principle, 16 7 Transpose, 1 4 Residual, 325 Transposition principle, 1 3 Residuated, 325 TychonoiT Theorem. 224 Restricted direct product, 289 UWB-lattice, 37 1 Reynolds' identity, 40 8 Uniformly monoton e norm , 371 Ring of sets, 12,5 6 — positive, 384 — semiprimitive, 387, 399 a-lattice, 254 Unique factorization, 68 Sasaki-Kujiwara Theorem, 94 Uniquely complemented, 12 1 Schwan's Lemma, 73 Unity. 376 Sectionally complemented, 28, 70 Universal algebra, 13 2 Semi-ideal, 12 0 — bounds, 2 Semilattice, 9, 21 Upper bound, 6 Semimodular, 15 , 39 — semimodular, 1 5 Separable, 261 Sheffer's stroke symbol, 45 Valuation, 230 Simbireva Theorems. 303, 305 Vector lattice, 347 Similar algebras, 13 4 Von Neumann lattice, 267 Simple algebra, 13 8 Wallman Theorem, 225 Simple lattice, 71 Weak unit, 308 Skew lattice, 23 Weight, 12 9 Souslin Problem, 202 Weil-ordered, 180 , 202 Species of algebra, 14 1 Whitmans Theorem, 17 1 Sperner's Lemma, 99 Width, 98 Standard element, 69 Word algebra, 14 1 —ideal, 27 Star convergence, 245 — problem, 14 6 Steinitz-MacLane Kxchang e Property, 19 7 Zero, 319 Stone lattice, 13 0 Zorn s Property. 19 1 Other Titles in This Series (Continued from the front of this publication)

3.1 G . A. Bliss, Fundamenta l existenc e theorems, 191 3 3.2 E . Kasner, Differential-geometri c aspect s of dynamics, 191 3 2 E . H. Moore, Introductio n t o a form o f general analysi s M. Mason, Selecte d topic s i n th e theory o f boundary valu e problem s o f differentia l equations E. J. Wiiczyriski, Projectiv e differential geometry , 191 0 1 H . S. White, Linea r systems of curves on algebraic surface s F. S. Woods, Form s on noneuclidean spac e E. B. Va n Vleck, Selecte d topic s in th e theor y o f divergen t serie s and o f continue d fractions, 190 5

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