References. Springer International Publishing AG 2017 S. Givant, Introduction to Relation Algebras, DOI /

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1 References 1. Abian, A.: Boolean rings with isomorphisms preserving suprema and infima. London Mathematical Society 3 (19713), pp Andréka, H. and Givant, S.: Functionally dense relation algebras. Algebra Universalis 68 (2013), pp Andréka, H., Givant, S., Jipsen, P., and Németi, I.: On Tarski s axiomatic foundations of the calculus of relations. Journal of Symbolic Logic, toappear. 4. Andréka, H., Givant, S., Mikulás, S., Németi, I., and Simon, A.: Notions of density that imply representability in algebraic logic. Annals of Pure and Applied Logic 91 (1998), pp Andréka, H., Givant, S., and Németi, I.: The lattice of varieties of representable relation algebras. Journal of Symbolic Logic 59 (1994), pp Andréka, H. and Maddux, R. D.: Representations for small relation algebras. Notre Dame Journal of Formal Logic 35 (1994), pp Arrow, K. J.: A difficulty in the concept of social welfare. Journal of Political Economy 58 (1950), pp Arrow, K. J.: Social choice and individual values. Third edition. Yale University Press, New Haven CN, 2012, xvi pp. 9. van Benthem, J. A. F. K.: Language in action. Categories, lambdas and dynamic logic. Studies in Logic and the Foundations of Mathematics, vol. 130, North Holland Publishing Company, Amsterdam, 1991, x pp. 10. Bird, R. and de Moor, O.: Algebra of programming. International Series in Computer Science, Prentice Hall, Upper Saddle River, NJ, 1997, 295 pp. Springer International Publishing AG 2017 S. Givant, Introduction to Relation Algebras, DOI /

2 542 References 11. Birkhoff, G.: Lattice theory. First edition. American Mathematical Society Colloquium Publications, vol. 25, American Mathematical Society, Providence, RI, 1940, 155 pp. 12. Birkhoff, G.: On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society 31 (1944), pp Birkhoff, G.: Subdirect unions in universal algebra. Bulletin of the American Mathematical Society 50 (1944), pp Birkhoff, G.: Sobre los grupos de automorfismos. Revista de la Unión Matemática Argentina 110 (1946), pp Boole, G.: The mathematical analysis of logic Being an essay towards a calculus of deductive reasoning. MacMillan, Barclay, and MacMillan, Cambridge, 1847, 82 pp. 16. Boole, G.: An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities. MacMillan and Company, Cambridge, and Walton and Maberly, London, 1854, v + iv pp. 17. Boyd, J.: The algebra of group kinship. Journal of Mathematical Psychology 6 (1969), pp Brams, S. J. and Fishburn, P. C.: Approval voting. In: K. Arrow, A. Sen, and K. Suzumura (eds.) Handbook of social choice and welfare, Elsevier Science, Amsterdam, 2002, pp Brink, C., Kahl, W., and Schmidt, G. (eds): Relational methods in computer science. Advances in Computing, Springer-Verlag, Vienna, 1997, xv pp. 20. Burris, S. and Sankappnavar, H. P.: A course in universal algebra. Graduate Texts in Mathematics, Springer-Verlag, New York, 1981, xvi pp. 21. Chang, C. C., Jónsson, B., and Tarski, A.: Refinement properties for relational structures. Fundamenta Mathematicae 55 (1964), pp Chin, L. H.: Distributive and modular laws in relation algebras. Doctoral dissertation, University of California at Berkeley, Berkeley CA, 1948, 62 pp. 23. Chin, L. H. and Tarski, A.: Distributive and modular laws in the arithmetic of relation algebras. University of California Publications in Mathematics, New Series 1 (1951), pp Comer, S.: Multivalued loops and their connection with algebraic logic. Unpublished manuscript, 1979, 173 pp.

3 References Comer, S.: Multi-valued algebras and their graphical representation. Unpublished manuscript, 1986, 103 pp. 26. De Morgan, A.: On the syllogism, no. IV, and on the logic of relations. Transactions of the Cambridge Philosophical Society 10 (1864), pp Dirichlet, J. P. G. L.: Vorlesungen über Zahlentheorie. Third edition, edited and with supplements by R. Dedekind, F. Vieweg und Sohn Verlag, Braunschweig, Düntsch, I.: Relation algebras and their application in temporal and spatial reasoning. Artificial Intelligence Review 23 (2005), pp Düntsch, I., Schmidt, G., and Winter, M.: A necessary relation algebra for mereotopology. Studia Logica 69 (2001), pp Frias, M. F.: Fork algebras in algebra, logic and computer science. Advances in Logic, vol. 2, World Scientific Publishing Company, River Edge, NJ, 2002, xi pp. 31. Frobenius, G. and Stickelberger, L.: Über Gruppen von vertauschbaren Elementen. Journal für die reine und angewandte Mathematik 86 (1879), pp Galvin, F.: Horn sentences. Doctoral dissertation, University of Minnesota, Minneapolis and St. Paul MN, (1965), iii + 47 pp. 33. Galvin, F.: Horn sentences. Annals of Mathematical Logic 1 (1970), pp Givant, S.: The structure of relation algebras generated by relativizations. Contemporary Mathematics, vol. 156, American Mathematical Society, Providence RI, 1994, xvi pp. 35. Givant, S.: The calculus of relaltions as a foundation for mathematics. Journal of Automated Reasoning 37 (2006), pp Givant, S.: Duality theories for Boolean algebras with operators. Springer Monographs in Mathematics, Springer-Verlag, New York, 2014, xiv pp. 37. Givant, S. and Andréka, H.: Simple relation algebras. Springer-Verlag, to appear. 38. Givant, S. and Halmos, P.: Introduction to Boolean algebras. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2009, xiv pp. 39. Grätzer, G.: Universal algebra. Second edition. Springer-Verlag, New York, 2008, xix pp.

4 544 References 40. Halmos, P.: Lectures on Boolean algebras. D.Van Nostrand Company, Princeton, NJ, 1963, vi pp. 41. Henkin, L., Monk, J. D., and Tarski, A.: Cylindric algebras. PartI. Studies in Logic and the Foundations of Mathematics, vol. 64, North- Holland Publishing Company, Amsterdam, 1971, vi pp. 42. Henkin, L., Monk, J. D., Tarski, A., Andréka, H., and Németi, I.: Cylindric set algebras. Lecture Notes in Mathematics, vol. 883, Springer- Verlag, Berlin Heidelberg New York, 1981, viii pp. 43. Hirsch, R. and Hodkinson, I.: Complete representations in algebraic logic. Journal of Symbolic Logic 62 (1997), pp Hirsch, R. and Hodkinson, I.: Relation algebras by games. Studies in Logic and the Foundations of Mathematics, vol. 147, Elsevier Science, North-Holland Publishing Company, Amsterdam, 2002, 712 pp. 45. Hirsch, R., Hodkinson, I., and Maddux, R.: Provability with finitely many variables. Bulletin of Symbolic Logic 8 (2002), pp Jipsen, P.: Computer-aided investigations of relation algebras. Doctoral dissertation, Vanderbilt University, Nashville TN, 1992, iii + 82 pp. 47. Jipsen, P.: Discriminator varieties of Boolean algebras with residuated operators. In: C. Rauszer (ed.), Algebraic methods in logic and in computer science, Banach Center Publications, vol. 28 (1993), Institute of Mathematics, Polish Academy of Sciences, pp Jipsen, P. and Lukács, E.: Minimal relation algebras. Algebra Universalis 32 (1994), pp Jónsson, B.: Representation of modular lattices and of relation algebras. Transactions of the American Mathematical Society 92 (1959), pp Jónsson, B.: Varieties of relation algebras. Algebra Universalis 15 (1982), pp Jónsson, B.: Relation algebras and Schröder categories. Discrete Mathematics 70 (1988), pp Jónsson, B. and Tarski, A.: Direct decompositions of finite algebraic systems. Notre Dame Mathematical Lectures 5, North State Press, Notre Dame, IN, 1947, vi + 64 pp. 53. Jónsson, B. and Tarski, A.: Representation problems for relation algebras. Bulletin of the American Mathematical Society 54 (1948), pp. 80 and 1192, Abstract Jónsson, B. and Tarski, A.: Boolean algebras with operators. Part I. American Journal of Mathematics 73 (1951), pp

5 References Jónsson, B. and Tarski, A.: Boolean algebras with operators. Part II. American Journal of Mathematics 74 (1952), pp Kahl, W. and Schmidt, G.:Exploring (finite) relation algebras using tools written in Haskell. Report Number , Institute for Software Technology, Department of Computing Science, Federal Armed Forces University Munich, Munich, 2000, 158 pp. 57. Kamel, H.: Relational algebra. Doctoral dissertation, University of Pennsylvania, Philadelphia PA, 1952, viii pp. 58. Kamel, H.: Relational algebra and uniform spaces. Journal of the London Mathematical Society 29 (1954), pp Köthe, G.: Abstrakte Theorie nichkommutative Ringe mit einer Anwendung auf die Darstellungstheorie kontinuierlicher Gruppen. Mathematische Annalen 103 (1930), pp Kramer, R. L.: Relativized relation algebras. In: H. Andréka, J. D. Monk, and I. Németi (eds.), Algebraic logic, Colloquia Mathematica Societatis János Bolyai, vol. 54, North-Holland Publishing Company, Amsterdam, 1991, pp Lambek, J.: Relations old and new. In: Orlowska, E. and Szalas, A.: Relational methods for computer science applications, Studies in Fuzziness and Soft Computing, Physica Verlag Rudolf Liebing KG, Vienna, 2001, pp Langford, C. H.: Theorems on deducibility. Annals of Mathematics, Second Series 28 (1927), pp Lewis, C. I.: A survey of symbolic logic. University of California Press, Berkeley, 1918, vi pp. 64. Lovász, L.: Operations with structures. Acta Mathematica Academiae Scientiarum Hungaricae 18 (1967), pp Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Mathematische Annalen 76 (1915), pp Löwenheim, L.: Einkleidung der Mathematik in den Schöderschen Relativkalkül. Journal of Symbolic Logic 5, pp Lyndon, R. C.: The representation of relational algebras. Annals of Mathematics, series2, 51 (1950), pp Lyndon, R. C.: The representation of relational algebras, II. Annals of Mathematics, series2, 63 (1956), pp Lyndon, R. C.: Properties preserved under homomorphisms. Pacific Journal of Mathematics 9 (1959), pp

6 546 References 70. Lyndon, R. C.: Relation algebras and projective geometries. Michigan Mathematical Journal 8 (1961), pp Maddux, R. D.:Topics in relation algebras. Doctoral dissertation, University of California at Berkeley, Berkeley CA, 1978, iii pp. 72. Maddux, R. D.: Some sufficient conditions for the representability of relation algebras. Algebra Universalis 8 (1978), pp Maddux, R. D.: Embedding modular lattices into relation algebras. Algebra Universalis 12 (1981), pp Maddux, R. D.: Some varieties containing relation algebras. Transactions of the American Mathematical Society 272 (1982), pp Maddux, R. D., Pair-dense relation algebras. Transactions of the American Mathematical Society 328 (1991), pp Maddux, R. D.: The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica 50 (1991), pp Maddux R. D.: Relation-algebraic semantics. Theoretical Computer Science 160 (1996), pp Maddux, R. D.: Relation algebras. Studies in Logic and the Foundations of Mathematics, vol. 150, Elsevier Science, North-Holland Publishing Company, Amsterdam, 2006, xxvi pp. 79. Maddux R. D.: Arrow s theorem for incomplete relations. Journal of Logic and Algebraic Programming 83 (2014), pp Marx, M.: Relativized relation algebras. Algebra Universalis 41 (1999), pp McCoy, N. H.: Subrings of infinite direct sums. Duke Mathematical Journal 4 (1938), pp McKenzie, R. N.: The representation of relation algebras. Doctoral dissertation, University of Colorado, Boulder CO, 1966, vii pp. 83. McKenzie, R. N.: Representations of integral relation algebras. Michigan Mathematical Journal 17 (1970), pp McKenzie, R. N., McNulty, G. F., and Taylor, W. F.: Algebras, lattices, varieties. Volume 1. Wadsworth and Brooks/Cole, Belmont CA, 1987, xvi pp. 85. McKinsey, J. C. C.: The decision problem for some classes of sentences without quantifiers. Journal of Symbolic Logic 8 (1943), pp Monk, J. D.: Studies in cylindric algebras. Doctoral dissertation, University of California at Berkeley, Berkeley CA, 1961, vi + 83 pp.

7 References Monk, J. D.: On representable relation algebras. Michigan Mathematical Journal 11 (1964), pp Peirce, C. S.: Note B. The logic of relatives. In: C. S. Peirce (ed.) Studies in logic by members of the Johns Hopkins University, Little, Brown, and Company, Boston, 1883, pp [Reprinted by John Benjamins Publishing Company, Amsterdam, 1983.] 89. Pixley, A. F.: Functionally complete algebras generating distributive and permutable classes. Mathemamtische Zeitschrift 114 (1970) pp Riguet, J.: Relations binaires, fermetures, correspondances de Galois. Bulletin de la Société Mathématique de France 76 (1948), pp Riguet, J.: Quelques propriétès des relations difonctionnelles. Comptes Rendus Mathématique de l Académie des Sciences 230 (1950), pp Russell, B.: The principles of mathematics. Cambridge University Press, [Reprinted by Allen & Unwin, London, 1948.] 93. Sain, I.: Strong amalgamation and epimorphisms of cylindric algebras and Boolean algebras with operators. Preprint #1982/17, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 1982, 44 pp. 94. Sain, I.: Weak products for universal algebra and model theory. Diagrammes, vol. 8 (1982), pp. S2 S15.: 95. Schmidt, G.: Relational mathematics. Encyclopedia of Mathematics and its Application, vol. 132, Cambridge University Press, Cambridge, 2011, xiii +566 pp. 96. Schmidt, G.: Relational concepts in social choice. In: T. G. Griffin and W. Kahl (eds.) Relational and algebraic methods in computer science. 13th interational conference on relational and algebraic methods in computer science 2012, Cambridge, UK, September 17 20, Proceedings, Lecture Noties in Computer Science, vol. 7650, Springer Verlag, Berlin, 2012, pp Schmidt, G. and Ströhlein, T.: Relations and graphs Discrete mathematics for computer scientists. EATCS Monograph on Theoretical Computer Science, Springer-Verlag, Berlin, 1993, ix Schröder, E.: Vorlesungen über die Algebra der Logik (exakte Logik). Dritter Band. Algebra und Logik der Relative. Erste Abteilung. B. G. Teubner, Leipzig, [Reprinted by Chelsea Publishing Company, New York, 1966.] 99. Sikorski, R.: Cartesian products of Boolean algebras. Fundamenta Mathematicae 37 (1950), pp

8 548 References 100. Sikorski, R.: Boolean algebras. Second edition. Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 25, Springer-Verlag, Berlin, 1964, x pp Skolem, T.: Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theorem über dichte Mengen. Skrifter utgitt av Videnskapsselskapet i Kristiania, I, Matematisk-naturvidenskabelig klasse 4 (1920), 36 pp Tarski, A.: Bemerkung der Redaktion. Fundamenta Mathematicae 23 (1934), p Tarski, A.: Grundzüge des Systemenkalküls. Fundamenta Mathematicae 26 (1936), pp Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6 (1941), pp Tarski, A.: Lecture notes on the theory of relation algebras. Taken by B. Jónsson, University of California at Berkeley, some time during the period 1942 to Tarski, A.: Manuscript of a book containing some of Tarski s early contributions to the theory of relation algebras, written during the period 1943 to The book was never published, but most of the results in the book were later included in [113] Tarski, A.: Some metalogical results concerning the calculus of relations. Journal of Symbolic Logic 18 (1953), pp Tarski, A.: A formalization of set theory without variables. Journal of Symbolic Logic 18 (1953), p Tarski, A.: An undecidable system of sentential calculus.journal of Symbolic Logic 18 (1953), p Tarski, A.: Contributions to the theory of models, III. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, Mathematical Sciences 58 (=Indagationes Mathematicae 17) (1955), pp Tarski, A.: Equationally complete rings and relation algebras. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, Mathematical Sciences 59 (=Indagationes Mathematicae 18) (1955), pp Tarski, A.: Lecture notes on the theory of relation algebras. Taken by S. Givant, University of California at Berkeley, Tarski, A. and Givant, S.: A formalization of set theory without variables. Colloquium Publications, vol. 41, American Mathematical Society, Providence RI, 1987, xxi pp.

9 References Tarski, A. and Vaught, R.: Arithmetical extensions of relational systems. Compositio Mathematica 13 (1957), pp Weinstein, J. M.: First-order properties preserved by direct products. Doctoral dissertation, University of Wisconsin, Madison WI, 1965, 162 pp Werner, H.: A Mal cev condition for admissible relations. Algebra Universalis 3 (1973), pp Werner, H.: Diagonal products and affine completeness. Algebra Universalis 4 (1974), pp Werner, H.: Congruences on products of algebras and functionally complete algebras. Algebra Universalis 4 (1974), pp Werner, H.: Discriminator algebras. Studien zur Algebra und ihre Andwendungen, vol. 6, Akademie Verlag, Berlin, Winter, M.: Dependencies in relational models of databases. Toappear in: Festschrift J.N. Oliveira of JLAMP, September 01, Wostner, U.: Finite relation algebras. Notices of the American Mathematical Society 23, (1976), A Zierer, H.: Relation algebraic domain constructions. Theoretical Computer Science 87 (1991), pp

10 Index 0, 38, 133 1, 38, 133 2, 16 1, 36 0, 41, 133 n, 133 ι, 79, 82, 5, 16, 36, 38, 58, 133,, 133, 36, 58, 133 +, 5, 16, 36, 58, 133, 39, 144, 16, 38,39 ;, 36, 58,+, 41, 38, 20, 22, 38 <, 38, 2 U U, 2 id U,2 di U,2, 6 1,6,79, 5, 5, 6, 6,7,79, 6, 133, 1 2,47, 2 =, 2, 58, 2, 230, 2, 230, 2, 2, 58, 59, 88, 337, 369, 380, 59, 88, 337, 369, 380, 58, 59, 59, 58, 337, 337 =, 285 n, 18 r/m, 331 r M, 330 r n, 144 rs, 82 r s mod M, 331 Springer International Publishing AG 2017 S. Givant, Introduction to Relation Algebras, DOI /

11 552 Index r s mod Θ, 313 r/θ, 319 v 0,v 1,...,58 x, y, z, 58 A/M, 331 Eq(K), 63 M R,23 M, 19 M + N, 19 M N, 22 M N, 20 M T,21 N/M, 363 R M,23 R 0, R 1,...,64 Th(K), 63 L, 58 L,64 L 3, 106 S, 63 Mo(S), 63 A, B,...,37 A/M, 331 A/Θ, 319 B C, 419 B C, 436 B I, 445 Cm(G), 80 Cm(L), 89 Cm(P ), 83 M 0, M 1, M 2, M 3,73 Ma(U), 74 Re(E), 72 Re(U), 71 U, V,...,65 (α, β), 1 α 0, α 1,...,65 ϕ(m), 332 ϕ 1 (N), 332 σ(v 0,...,v n 1 ), 59 Δ, 59 Γ,59 Γ (v 0,...,v n 1 ), 60 Θ, 313 abacus of binary relatives, 218 abelian group, 299, 487 relation algebra, 143, 158, 277, 299, 304, 311, 321, 423, 448, 489 Abian, Alexander Smbat, 541 absolute operation, see Boolean operation absorption law for lattices, addition, 16 25, 32 34, 36, 114, see also matrix addition, notation, relational addition, relative addition of functions, symbol, 58 additive operation, see distributive operation affine representation, 311 algebra, 36, see also Boolean algebra, Boolean algebra with operators, relation algebra of relations, see set relation algebra operations of, see fundamental operations of an algebra universe of, see universe of an algebra amalgamation of homomorphisms external, 485, internal, ample internal product, , 516, 518 Andréka, Hajnal Ilona, vii, 105, , 264, 372, 398, 488, 541, 543, 545 annihilator of an ideal, anthropology, xiv antisymmetric relation, 230 Aristotle, xiv arithmetic, see laws arrow diagram, 356, 366, 438 Arrow, Kenneth Joseph, vi, associative law for addition, 36, 112

12 Index 553 for composition, 10 for direct products, 490, 493 for groups, 79 for lattices, for relational addition, 10 for relative addition, 119, 134 for relative multiplication, 36, 134 infinite, see infinite associative law atom, 40, 68 69, 80 81, , 195, , 227, 277, 286, , , , , 376, , , 414, 418, 427, 443, 449, 464, 507, Atomic Decomposition Theorem, , 488 Isomorphism Theorem, , 303, , 533 Monomorphism Theorem, , 303, Subalgebra Theorem, atomic Boolean algebra, 46, 48 57, 68, , 376, formula in language of relation algebras, 59 in language of theory of relations, 65 relation algebra, 40, 71 74, , 286, , 307, , , 376, , 418, 427, 449, 464, , 507, see also finite relation algebra, small relation algebra subalgebra, , term, 58 atomless Boolean algebra, 68, , 488 of ideal elements, relation algebra, 40, 107, 270, 277, 286, 305, 307, 376 Augenquaderrelativ, 220 automorphism, , 303, 311, see also Boolean automorphism, inner automorphism, trivial automorphism axioms, see also associative law, Boolean axioms, commutative law, distributive law, distributivity axiom, Huntington s law, identity law, independence of axioms, involution law, logical axiom, Pasch Axiom, Tarski s law of projective geometry, 81 of relation algebra, 35 41, 51 57, 69, , 112, , 230, 321, 373, , 489, 493 of theories of minimal relation algebras, 538 base case of definition by induction, 58 of proof by induction, 59 clause, see base case isomorphic relation algebras, 287 isomorphism, , 303, 308, set of an algebra, 72 van Benthem, Johannes Franciscus Abraham Karel, 541 bifunctional element, see bijection bijection, , 304, bijectional element, see bijection binary operation on relations, see operation relation, see relation tree, Bird, Richard Simpson, 541 Birkhoff, Garrett, 487, 516, 542 Boole, George, xi, xiv xvi, 27, 542 Boolean addition, see addition algebra, xi xiv, 16 27, 32, 42 57, 77 79, 113, 376, 496, 516, 518,

13 554 Index see also atomic Boolean algebra, atomless Boolean algebra, Boolean part of a relation algebra of ideal elements, , , 370, , , 488, 492 of right-ideal elements, , with complete operators, 50, , 262, , 303 with normal operators, 401 with operators, xvii, 41 50, with quasi-complete operators, 50, 267 automorphism, 40, , axioms, 51 element, see ideal element epimorphism, 40, filter, 367, 370 flag, 87 88, 104 group, 38, 143, 318 homomorphism, 40, 273 ideal, 325, , , 376, 386 inequality, see partial order isomorphism, 40 law, 113, 134 matrix, see matrix monomorphism, 40 multiplication, see multiplication operation, 5 8, 36, 280, see also addition, multiplication, complement, equivalence, implication, intersection, one, subtraction, symmetric difference, union, zero part of a relation algebra, 37 relation algebra, 77 79, 103, 107, 182, 265, 299, 389, 417, , 522, 530, 536 subalgebra, 376, 382 bounded relation, 30 Boyd, John Paul, 542 Brams, Steven J., 542 Brink, Chris, 542 Burris, Stanley Neal, 542 calculus of relations, x xvii, 1 32, cancellation law for direct products, 496 canonical homomorphism, see projection isomorphism, , , 491, 495, , 517 cardinality of a relation algebra, 37, Cartesian factor, see factor product, see direct product Cayley Arthur, xii representation, 296, 303 chain of congruences, 317, 372 of elementary subalgebras, 260 of ideals, 350 of subalgebras, 237, 240, Chang, Chen-Chung, 487, 542 characterization of equivalence elements, , , 218 of functions, , , 221 of ideal elements, , , 219 of rectangles, , 220 of right-ideal elements, , , 219 of squares, , 220 of symmetric elements, of transitive elements, Chin, Louise Hoy, xvii xix, 66, , 542 closure, see also one-step closure equivalence, see equivalence closure of an element

14 Index 555 properties of the set of equivalence elements, , , 218 of functions, , 221 of ideal elements, , , 225 of rectangles, , 220 of right-ideal elements, , 219 of symmetric elements, , 217, 222 of transitive elements, , 217, symmetric-transitive, see equivalence closure of an element transitive, see transitive closure of an element under operations, 229 cofinite set, 264 cokernel coset, 368, 378 of a congruence, 368, 378 of a homomorphism, 368, 380 collinear points, collineation, 309 column of a matrix, 18 Comer, Stephen Daniel, 104, 398, common refinement, 496 commutative law, x for addition, 36, 112 for direct products, 491, 493 for lattices, for relative multiplication, 83 87, 276 relation algebra, see abelian relation algebra compact element, , 317, 342, 346, 370, 373, 432, 491 compactly generated lattice, 239, 317, 319, 343, 346, 373, 432, 491 Compactness Theorem, 63, 258 complement, ix, 6, 16 20, 24, 32 34, 38, 306 of a function, relative, see relative complement symbol, 58 complementation, see complement complete distributivity law for converse over addition, for converse over multiplication, for functions, for relative multiplication over addition, embedding, see complete monomorphism epimorphism, 407, 450 equational theory, 536, 538 filter, 380 generators, 240, 285 homomorphic image, 283 homomorphism, , 307, 378, 426, 429, 449, 480, 485 ideal, 378, 380 lattice, 236, 239, 319, 337, 343, 346 monomorphism, , , 302 operation, see completely distributive operation relation algebra, 39, 286, 307, 411, 428, 450, 458, 464, subalgebra, , 262, , 517 subuniverse, 240, 285 Complete Decomposition Theorem, , 488 completely distributive operation, for atoms, representable relation algebra, 103 Completeness Theorem for equational logic, 64 for first-order logic, 63 complex algebra

15 556 Index of a geometry, 81 88, 92 93, , 143, 224, , 267, 307, , 395, 400, of a group, 79 81, 103, 158, 218, , , , 286, 307, 309, 395, 400, 405 of a lattice, 88 94, 104, 108, 224, 309, 400, 417 inverse, 80 multiplication, 80 product of filters, 369 of ideals, sum, 369 of ideals, component, 436, 459 composition, see functional composition, relational composition of homomorphisms, 272 computer science, xiii conditional equation, 60, 422, , equational theory, 63 congruence, , , 378, 381, class, , generated by a set, see generators of a congruence induced by a homomorphism, 322 relation, see congruence conjunction symbol, 59 constant almost everywhere, term, conventions, see order of operations converse, ix, 36, see also definability of converse, notation, relational converse complement, 30, 133 symbol, 133 symbol, 58 conversion, see converse coordinate sequence, 420, 446 coplanar lines, 82 Correspondence Theorem, 365, 371, 381, 391 coset, 330, 408 countably complete relation algebra, 39, 147, 223 generated ideal, 375 subalgebra, 235 Couturat, Louis, xvi cycle law, 41, 68, cylinder, 168, 172 Decomposition Theorem for Re(E), 473 Dedekind, Julius Wilhelm Richard, 371, 543 definability of complement, 280 of converse, , 135, 280, 302 of identity element, 123, 138 of inequality, 276 definition by induction on formulas, 60 on terms, 59 degenerate relation algebra, 40, 74, 105, 139, 306, 386, 433, 445, 448, 491, 520, 528 De Morgan Augustus, xiv, 27, 28, 217, 543 Tarski laws, 11 12, 41, 68, , , , 138, 164, , 221 dense linear order, 106 without endpoints, 105 relation, 222 Desargue s theorem, xviii Déscartes, René, 486 diagonal embedding, 510 difference, see subtraction, symmetric difference

16 Index 557 direct decomposition external, , internal, , , 483, , 537 of homomorphisms, power, product, 314, 385, , binary external, 435, binary internal, , general external, general internal, , of set relation algebras, , 459, , 491, 494 sum, 516, see also weak direct product directed edge, 5 graph, 4 set of elements, , 204 system of congruences, 317, of elementary subalgebras, of elementary substructures, 263 of filters, 370 of homomorphisms, 305 of ideals, of regular subalgebras, 242 of subalgebras, , 265 triangle, directly indecomposable algebra, 399, 401 relation algebra, , 433, 492 Dirichlet, Johann Peter Gustav Lejeune, 486, 543 discriminator, see also unary discriminator function, 399 term, 401 variety, 399, 401, 497 disjoint domains, elements, 40, rectangles, set, 40 system of elements, 40 disjunction symbol, 59 dissimilar homomorphisms, 400 distinguished constant, 5 distributive lattice, , 346 law, x, 376, see also complete distributivity law, general complete distributivity law, general finite distributivity law, general quasi-complete distributivity law for bijections, 216 for composition over union, 11 for converse, 112 for converse over addition, 36, 114, 134 for converse over complement, 114 for converse over intersection, 11 for converse over multiplication, 134 for converse over subtraction, 114 for converse over symmetric difference, 114 for converse over union, 11 for direct product over intersection, 491 for direct product over join, 491 for functions, , 221 for ideal elements, , 220, 225, 282 for relational addition over intersection, 11 for relative addition over multiplication, 119, 134 for relative multiplication over addition, 36, 68, 112, 116, 134

17 558 Index for relative multiplication over multiplication, operation, distributivity axiom, 41, 51 diversity element, 41, , 153, see also notation matrix, 19 relation, 2, 10, domain algebra, 271 of a function, , 214, 216 of a relation, 164 of an element, , 220, 226, , 412 downward Löwenheim-Skolem-Tarski Theorem, see Löwenheim- Skolem-Tarski Theorem dual filter, , ideal, , 379 of symmetric difference, duality principle, 134, see also first duality principle, second duality principle, third duality principle Düntsch,Ivo,543 economics, xiv element, see compact element, diversity element, domain, equivalence element, function, identity element, left-ideal element, range, rectangle, reflexive element, right-ideal element, square, subdiversity element, subidentity element, symmetric element, transitive element field of, see field of an element elementary diagram, 258 embedding, , 304, 307 extension, 254, 258 language, see first-order language monomorphism, see elementary embedding subalgebra, , 270, , see also directed system of elementary subalgebras substructure, 263, see also directed system of elementary substructures theory, 63 of a class of models, 63 elimination of quantifiers, 263 embedding, 272, see also complete monomorphism, diagonal embedding, elementary embedding, monomorphism emotional relation, ix empty relation, 2, 10 set, 39 subspace, 92 entry of an array, 18 epimorphism, 271, , 305, , 353, 374, see also Boolean epimorphism equality between relations, 2 symbol, 58 equation, 59, , , 518 equational theory, 63 64, see also complete equational theory, inconsistent equational theory of a class of algebras, 63 of relation algebras, 538 equivalence, 29 class, 32, 178, 319 closure of an element, 152 element, , , , , 276, 304, , 424, 449, 469, 489, 495, 536, , see also characterization of equivalence elements, closure properties of the set of equivalence elements,

18 Index 559 modular law for equivalence elements, reflexive equivalence element, type of an equivalence element logical, see logical equivalence modulo a set of formulas, 75 relation, 32, 72, 149, 177, 223, , 285, 313, 316, 319, 327, 405, , 526, 528 symbol, 59 equivalent relation algebras, see base isomorphic relation algebras Exchange Principle, 258, , 303, 311, 362, 439 Existence Theorem for internal products, 463 binary version, 439 for weak internal products, existential formula, 60 quantification symbol, 59 sentence, 60 exponentiation, 144, see also first law of exponentiation for direct powers,secondlawofexponentiation for direct powers, third law of exponentiation for direct powers, expression, 58 extension, see also regular extension of a homomorphism, 272 of a relation algebra, 230, external product, see direct product factor external, 420, homomorphism, 449, 481 internal, , 459 field of a relation, 148, 185 of an element, 185 filter, , , see also Boolean filter, proper filter element, 369 generated by a set, see generators of a filter finite Boolean algebra, , cofinite subalgebra, 264 join property, 374, 377 meet property, relation algebra, , , 458, 474, 537, see also small relation algebra finitely generated Boolean algebra, 236 congruence, filter, 369, 379 ideal, relation algebra, 234 subalgebra, 234, First Homomorphism Decomposition Theorem, , 488, 497 Isomorphism Theorem, 362, 371, 381, , for congruences, , 430, 451 for ideals, , 512 first dual of a law, , 164 of a notion, 179 duality principle, involution law, 112 law of exponentiation for direct powers, 494 first-order language, see language of relation algebras, language of relations logic, theory, see elementary theory Fishburn, Peter C., 542 formula, see also atomic formula, conditional equation, equation, existential formula, open formula, positive formula, quantifier-free formula, uni-

19 560 Index versal Horn formula, universal formula in language of relation algebras, in language of relations, 65 relation algebra, 75 77, 103, , 400 Frias, Marcelo Fabián, 543 Frobenius algebra, see complex algebra of a group Ferdinand Georg, 486, 543 full set relation algebra on a set, 71 72, 74 75, 103, 287, 389, 395, 405, 522, , 539 on an equivalence relation, 72, 389, 405, , function, 32, , , , 276, 304, , 424, 449, 489, 495, see also characterization of functions, closure properties of the set of functions, distributive law for functions, modular law for functions inverse, 304 functional composition, 7, 205, 304 element, see function part of an element, 201 fundamental operations of an algebra, 36 Galvin, Frederick William, 488, 543 general complete distributivity law, finite distributivity law, monotony law, 43 quasi-complete distributivity law, generalized relativization, generated, see also finitely generated ideal element, see ideal element generated by an element right-ideal element, see right-ideal element generated by an element generating set, see generators generators complete, see complete generators of a congruence, of a filter, 369, 379 of a relation algebra, of a subalgebra, , , 286, 305, , , of an ideal, , 357, geometric complex algebra, see complex algebra of a geometry relation algebra, see complex algebra of a geometry geometry, 81 88, 224 order of, see order of a geometry Givant, Steven Roger, 66 67, 105, 113, 136, , , 303, , 398, 416, , 516, 536, , 548 Goldblatt, Robert Ian, vii graph of a relation, 3 10, Grätzer, George, 543 greatest lower bound, see infimum Griffen, Timothy G., 547 group, 40 41, 79 81, , 299, , 487, 500, 516, see also abelian group, Boolean group complex algebra, see complex algebra of a group isomorphism, 292 Halmos, Paul Richard, 516, Henkin, Leon Albert, 303, 544 Hirsch, Robin David, xii xiii, 262, 302, 544 Hodkinson, Ian Martin, vii, xii xiii, 262, 302, 544 homomorphic image, 272, 280, , , , 385, , , 418, see

20 Index 561 also complete homomorphic image Homomorphism Decomposition Theorem, see First Homomorphism Decomposition Theorem, Second Homomorphism Decomposition Theorem Extension Theorem, , 383, 399 Theorem, 371, 381 for congruences, 322 for ideals, 355 homomorphism, , , , , 386, , , , 449, , 490, 497, see also Boolean homomorphism, complete homomorphism, complete epimorphism, complete monomorphism, epimorphism, dissimilar homomorphisms, extension of a homomorphism, isomorphism, monomorphism, Peircean homomorphism, restriction of a homomorphism on an algebra, 272 Huntington s law, 36, 112 Huntington, Edward Vermilye, xvi ideal, 164, 220, 313, , , , , 414, , 427, 490, , see also annihilator of an ideal, Boolean ideal, complete ideal, dual ideal, finitely generated ideal, improper ideal, interval of ideals, maximal ideal, non-principal ideal, non-trivial ideal, orthogonal pair of ideals, orthogonal system of ideals, principal ideal, proper ideal, trivial ideal element, , , 222, 225, 228, 276, 282, 304, , , , 374, 382, , , , , , , 449, , , , , see also Boolean algebra of ideal elements, characterization of ideal elements, closure properties of the set of ideal elements, distributive law for ideal elements, modular law for right-ideal elements, orthogonal system of ideal elements atom, , , , 507 generated by an element, 179 generated by a set, see generators of an ideal idempotent element, 175 law for lattices, 88 identity, 61 element, 36, 115, 228, , see also definability of identity element, notation law for composition, 10 for direct products, 491 for groups, 79 for relational addition, 10 for relative addition, 119, 135 for relative multiplication, 36, 112, 116, 135 matrix, 19 relation, 2, 315, 317, 372 singleton, , 539 symbol, 58 image, see also homomorphic image, inverse image set algebra, 272 set, , , 332, , 368, 374 implication, 29 symbol, 58

21 562 Index improper filter, 369 ideal, 333, 335, 348, 374, 386 subalgebra, 230, 234 incidence relation, 81 inclusion between algebras, see subalgebra between relations, see inequality between relations proper, see inequality between relations inconsistent equational theory, 538 independence of axioms, 112 independent axiom, 100 set of axioms, 100 individual constant symbol, 64 induction base case, see base case clause, see induction step definition, see definition by induction on formulas, 60, 255 on positive formulas, on quantifier-free formulas, on terms, 59, 231, , , , step, 59 inequality, see also partial order between relations, 2, 12 infimum, 39, 67, 427, see also meet infinitary associative law, 67 distributive law, see complete distributivity law injection, see bijection injectional element, see bijection inner automorphism, 300, 311 cylindrification, 172 ideal closure, 181 integral relation algebra, 104, , 270, 386, , , , 423, 489 Integrality Theorem, , 412, 533 internal product, see direct product interpretation in language of relation algebras, in language of relations, 65 intersection, ix, 5, 10 of a system of congruences, of filters, 379 of ideals, 333, 374 of subalgebras, of lines, 82 interval of ideals, inverse, see also converse complex, see complex inverse image ideal, 374 image set, , 305, 332, , 368, 374 law for groups, 79 of an isomorphism, 272 Inversion Theorem, 135 involution, 40 law first, 10, 36, 40, 134 for relative addition, 119 second, 10 11, 36, 40, 135, 272 isomorphic algebras, 272 image, 272 isomorphism, 115, , , , 302, 304, , , , see also Atomic Isomorphism Theorem, automorphism, base automorphism, base isomorphism, Boolean isomorphism, Correspondence Theorem, First Isomorphism Theorem, group isomorphism, lattice isomorphism, Peircean isomorphism, Second Isomorphism Theorem, Third Isomorphism Theorem type, 285

22 Index 563 Jevons, William Stanley, xiv xvi, 27 Jipsen, Peter, vii, 105, 398, 416, 541, 544 join, 88 of a system of congruences, of filters, 370 of ideals, , 341 of subalgebras, Jónsson, Bjarni, vii, xii xiii, xvii xix, 66, 104, 218, 263, 302, , 416, , 516, 536, 542, , 548 Kahl, Wolfgang, 542, 545, 547 Kamel, Hyman, 105, 136, 545 kernel of a congruence, 324, , 381 of a homomorphism, 331, , 490 kinship relation, ix x Korselt, Alwin Reinhold, 28 Köthe, Gottfried Maria Hugo, 516, 545 Kramer, Richard Lynn, vii, 67, 417, 545 Kronecker, Leopold, 486 Lambek, Joachim, 545 Langford, Cooper Harold, 545 language, see also expression, formula, sentence, term of relation algebras, 57 65, 113, , , 386, , 500 of relations, 64 65, 75, 400 Lattice of Ideals Theorem, 346, , , 380, 386, 518 lattice, 88, see also compactly generated lattice, complex algebra of a lattice, complete lattice, distributive lattice, modular lattice isomorphism, 343, of Boolean filters, 370, 380 of Boolean ideals, , 386 of congruences, , of filters, , 380 of ideals, , , 391 of subalgebras, law, x, 10 12, 28, 31, 37, , 139, , see also absorption law for lattices, associative law, Boolean law, commutative law, complete distributivity law, cycle law, De Morgan- Tarski laws, distributive law, first law of exponentiation for direct powers, general complete distributivity law, general monotony law, general quasi-complete distributivity law, Huntington s law, identity law, idempotent law for lattices, infinitary associative law, involution law, modular law, second law of exponentiation for direct powers, Tarski s law, third law of exponentiation for direct powers least upper bound, see supremum left ideal, 164 element, 164, , 191, 220 Lewis, Clarence Irving, 28, 545 line, 81 linguistics, xiv Liu, Kexin, vii logic, see first-order logic logical axiom, 63 equivalence, 75 symbol, 58, 64 Lovász, László, 487, 545 Löwenheim Leopold, xv, 28, 263, 545 Skolem-Tarski Theorem, , 263 relational version, 270 Lukács, Erzsbébet, 398, 544

23 564 Index Lyndon algebra, see complex algebra of a geometry Roger Conant, xii, xvii xix, , , 398, 488, Maddux algebra, see complex algebra of a lattice Roger Duncan, vii, xii xiii, 28, 104, 220, 263, 398, , 488, 536, 541, 544, 546 maps, Marx, Maarten Johannes, 417, 546 material relation, ix matrix, 16, 18 27, addition, 21 22, 33 algebra, 74 75, 105 on a set, 74 associated with a relation, 23 27, 33 multiplication, 20 21, 33 notation, transposition, 21, 24, 33 maximal congruence, 518 equational theory, see complete equational theory filter, 381 ideal, , , 385, 391, 490, 515 in an ideal, Maximal Ideal Theorem, , 372, , 381 McCoy, Neal Henry, 516, 546 McKenzie, Ralph Nelson Whitfield, vii, xviii xix, 104, 398, 546 McKinsey, John Charles Chenoweth, xii, , 220, 398, 487, 546 McNulty, George Frank, 546 meet, 88 of a system of congruences, 317 of filters, 370 of ideals, of subalgebras, 236 Mikulás, Szabolcs, 372, 541 minimal relation algebra, 72 74, 105, 131, 389, 395, 494, 519, , , subalgebra, 234, , 278, 500, 517 model, 62 modular lattice, 88 94, , 224, 319, 417, 432, 491 law, 129, see also standard modular law, strong modular law for equivalence elements, , 182, 218, 225 for functions, , 221, 227 for lattices, 88 91, 129 for multiplication over relative multiplication, 131 for right-ideal elements, , 182, 219, 225 Monk, James Donald, vii, xviii xix, 221, 303, 516, monoid, 40 monomorphism, 272, 301, , 386, see also Atomic Monomorphism Theorem, Boolean monomorphism monotone operation, 43 monotony law, see also general monotony law for converse, 114 for relative addition, 119 for relative multiplication, 117 de Moor, Oege, 541 multiplication, 16, 20, 24, 32 34, 38, 114, 315, see also matrix multiplication, notation, relative multiplication of functions, 17

24 Index 565 negation symbol, 58 Németi, István, vii, 105, 264, 372, 541, 545 Noether, Amalie Emmy, 371, 516 non-degenerate relation algebra, 40, 386, 391, , 433, 476, , 520, non-desarguesian plane, 103 non-logical symbol, 58, 64 non-principal ideal, non-trivial filter, 369 ideal, 332, 375 normal operation, 45 46, 50, see also Boolean algebra with normal operators subgroup, 158 notation, 5 7, 36 41, nullary operation on relations, see operation one, 38, 114, see also notation one-step closure, 235, extension, open formula, see quantifier-free formula Horn formula, 61, , operation, ix x, 5 10, 19 28, 36 58, 61, see also Boolean operation, completely distributive operation, distributive operation, monotone operation, normal operation, operator, order of operations, Peircean operation, preservation of operations, quasi-completely distributive operation operator, 50, see also Boolean algebra with complete operators, Boolean algebra with operators, Boolean algebra with quasi-complete operators, operation order, see also partial order of a geometry, 82, of logical connectives, of operation symbols, 58 of operations, 9, 37 partial, see partial order ordered pair, 1 Orlowska,Ewa,545 orthogonal pair of congruences, , 453 of ideals, system of congruences, of ideal elements, , , , 526 of ideals, outer cylindrification, 168 ideal closure, 179 pair, see ordered pair parentheses, 58, see also order of operations partial order, 38, 230 partition of an element, 40 Pasch Axiom, 82 Peirce, Charles Sanders, xi, xiv xvi, 27 28, , 219, 398, 547 Peircean homomorphism, 273 isomorphism, 273 operation, 5 28, 36, 280, see also composition, converse, converse-complement, diversity element, diversity relation, identity element, identity relation, relational addition, relative addition, relative multiplication part of a relation algebra, 40 permutation, 106, , 221, 300, 304

25 566 Index permutational element, see permutation Pixley, Alden French, 399, 547 plane, 82 point, 81, see also identity singleton polynomial, 61 62, 142, 146, 152, 167, 179, 192, 201, see also projection positive formula, 60, sentence, 60 power of a relation algebra, 445, of an element, see exponentiation preservation of operations, 271, , , 323, , 343, 373, on atoms, of properties under congruences, under direct products, , , under homomorphisms, under isomorphisms, 286 under subalgebras, under subdirect products, , 518, 535 of the supremum property, principal filter, , 379 ideal, 335, , 346, , , , principle of induction, see definition by induction, induction product, see also direct product, infimum, multiplication, subdirect product, weak direct product homomorphism, 449, sequence, 420, 446 Product Decomposition Theorem, 475, , 492, 495, 503, 519 binary external version, 433 binary internal version, general external version, 458 general internal version, , , 482, 526 projection, , , , 491, 494, of a homomorphism, 486 projective geometry, see geometry proper filter, 369, 379 ideal, 333, , 357, 374, relation algebra, see set relation algebra subalgebra, 230 quantifier elimination, see elimination of quantifiers quantifier-free formula, 60, , quasi-atom, 46 48, quasi-complete operation, see quasi-completely distributive operation operator, see quasi-completely distributive operation quasi-completely additive operation, see quasicompletely distributive operation distributive operation, 43 50, 267, for quasi-atoms, quasi-dense set, 375 quotient algebra, , , , 368, 385, 391, , 414, 416, , , 490, homomorphism, , 331, , 358, , 378, , , range algebra, see image algebra

26 Index 567 of a function, of a relation, 164 of an element, , 214, 220, 392, 412 Rauszer, Cecilia, 544 rectangle, , 220, 226, 276, 304, 388, 489, , see also characterization of rectangles, closure properties of the set of rectangles, symmetric rectangle refinement property, 487, 496 reflexive element, 144, 149 equivalence element, 149, , 162, , , 539 relation, 144, 230, 316 transitive closure, 147 regular extension, 241 subalgebra, , 262, 266, 284, , 489, 493, see also directed system of regular subalgebras relation, ix xi, 1 32, see also antisymmetric relation, bounded relation, calculus of relations, dense relation, diversity relation, emotional relation, empty relation, equivalence relation, equality between relations, graph of a relation, identity relation, incidence relation, inequality between relations, kinship relation, material relation, partial order, reflexive relation, symmetric relation, ternary relation, transitive relation, universal relation associated with a matrix, 23 27, 34 field of, see field of a relation symbol, 64 relation algebra, x xxv, 36, passim, see also abelian relation algebra, atomic relation algebra, atomless relation algebra, axioms of relation algebra, Boolean part of a relation algebra, Boolean relation algebra, cardinality of a relation algebra, complete relation algebra, completely representable relation algebra, complex algebra, countably complete relation algebra, degenerate relation algebra, directly indecomposable relation algebra, extension of a relation algebra, finite relation algebra, finitely generated relation algebra, integral relation algebra, minimal relation algebra, non-degenerate relation algebra, Peircean part of a relation algebra, representable relation algebra, restricted formula relation algebra, set relation algebra, simple relation algebra, small relation algebra, subdirectly irreducible relation algebra, symmetric relation algebra relational addition, ix, 6, 10 composition, ix x, 6 9, 287, converse, 6 structure, 65 relative addition, 41, , 315, , see also notation symbol, 41 complement of an element, 404 multiplication, 36, passim, see also notation symbol, 58 operation, see Peircean operation