. Don Monk

# Don Monk

My present research interests are the theory of infinite Boolean algebras and related set-theoretic topics, such as continuum cardinals and pcf theory. My previous research was in algebraic logic (cylindric algebras and relation algebras), and I also did some work in pure logic, set theory, and universal algebra. The files below are in postscript or pdf form, gzipped, tarred if several files are combined.
Boolean algebra bibliography (updated monthly, roughly)
Publications:
[01] On the representation theory for cylindric algebras. Pacific J. Math. 11, 1961, 1447--1457.
[02] On pseudo-simple universal algebras. Proc. Amer. Math. Soc. 13, 1962, 543--546.
[03] (with A. Daigneault) Representation theory for polyadic algebras. Fund. Math. 52, 1963, 151--176.
[04] (with D. Scott) Additions to some results of Erd\"os and Tarski. Fund. Math. 53, 1964, 335--343.
[05] Singulary cylindric and polyadic algebras. Trans. Amer. Math. Soc. 112, 1964, 185--205.
[06] On representable relation algebras. Mich. Math. J. 11. 1964, 207--210.
[07] The substitutionless predicate logic. Archiv f. Math. Logik und Grundlagenforschung 7, 1964, 102--121.
[08] Model-theoretic methods and results in the theory of cylindric algebras. in The Theory of Models, Proc. of a Symposium. 1965, 238--250.
[09] (with F. M. Sioson) $m$-semigroups, semigroups, and function representations. Fund. Math. 59, 1966, 233--241.
[10] Non-trivial $m$-injective Boolean algebras do not exist. Bull. Amer. Math. Soc. 73, 1967, 526--527.
[11] Introduction to Set Theory, McGraw-Hill, 1969 , ix + 193pp; Italian translation: Introduzione alla teorioa degli insiemi, Boringhieri 1972, 241 pp.
[12] Nonfinitzability of classes of representable cylindric algebras. J. Symb. Logic 34, 1969, 331--343.
[13] On an algebra of sets of finite sequences. J. Symb. Logic 35, 1970, 19--28.
[14] On the foundations of set theory. Amer. Math. Monthly 77, 1970, 703--711.
[15] Completions of Boolean algebras with operators. Math Nachr. 46, 1970, 47--55.
[16] On equational classes of algebraic versions of logic, I. Math. Scand. 27, 1970, 53--71.
[17] (with L. Henkin, A. Tarski) {\bf Cylindric Algebras, Part I.} North--Holland, 1971 vi $+$ 508pp.
[18] Provability with finitely many variables. Proc. Amer. Math. Soc. 27, 1971, 353--358.
[19] (with F. M. Sioson) On the general theory of m-groups. Fund. Math. 72, 1971, 233--244.
[20] (with R. Solovay) On the number of complete Boolean algebras. Algebra Universalis 2, 1972, 365--368.
[21] (with G. Meisters) Construction of the reals via ultrapowers. Rocky Mountain J. Math. 3, 1973, 141--158.
[22] (with R. McKenzie) On automorphism groups of Boolean algebras. Proc. Erd\"os Symposium 1973, 951--988.
[23] Connections between combinatorial theory and algebraic logic. Algebraic Logic, Math. Assoc. Amer. 1974, 58--91.
[24] (with L. Henkin) Cylindric algebras and related structures. Proc. Symp. in honor of A. Tarski 1974, 105--122.
[25] Some cardinal functions on algebras. Algebra Universalis 5, 1975, 76--81.
[26] On the automorphism groups of denumerable Boolean algebras. Math. Ann. 216, 1975, 5--10.
[27] Some cardinal functions on algebras, II. Algebra Universalis 5, 1975, 361--366.
[28] Some problems in algebraic logic. Proc. Summer Logic Conf., Clermont--Ferrand, CNRS 49, 1975, 83--88.
[29] Mathematical Logic. Springer--Verlag, 1976 , x $+$ 531pp.
[30] On free subalgebras of complete Boolean algebras.} Arch. d. Math. 29, 1977, 113--115.
[31] Omitting types algebraically. Ann. Sci. Univ. Clermont, Ser. Math. Fasc. 16, 1978, 101--105.
[32] (with W. Rassbach) The number of rigid Boolean algebras. Algebra Universalis 9, 1979, 207--210.
[33] A very rigid Boolean algebra.} Israel J. Math. 35, 1980, 135--150.
[34] (with E. K. van Douwen, M. Rubin) Some questions about Boolean algebras. Algebra Universalis 11, 1980, 220--243.
[35] (with L. Henkin, A. Tarski, H. Andr\'eka, I. N\'emeti) Cylindric Set Algebras.} Springer-Verlag, Lecture Notes in Mathematics 883, 1981, 323pp.
[36] (with L. Henkin, A. Tarski) Cylindric set algebras and related structures . in {\bf Cylindric Set Algebras.} Springer--Verlag, Lecture Notes in Mathematics 883, 1981, 1--129.
[37] (with R. McKenzie) Chains in Boolean Algebras. Annals of Math. Logic 22, 1982, 137--175
[38] Independence in Boolean Algebras. Periodica Mathematica Hungarica 14, 1983, 269--308.
[39] (with S. Koppelberg) Homogeneous Boolean algebras with very nonsymmetric subalgebras. Notre Dame J. Formal Logic 24, 1983, 353--356.
[40] (with G. Brenner) Tree algebras and chains. in Springer Lecture Notes in Mathematics 1004, 1983, 54-66.
[41] (with B. Koppelberg, R. McKenzie) Cardinality and cofinality of homomorphs of products of Boolean algebras. Algebra Universalis 19, 1984, 38--44.
[42] Cardinal functions on Boolean algebras. Proc. Ordered Sets Conf., Orders: ~Descriptions and Roles, North--Holand 1984, 9--37.
[43] (with L. Henkin, A. Tarski) {Cylindric Algebras, Part II. North--Holland, 1985, 302pp.
[44] On endomorphism bases. Algebra Universalis 20, 1985, 264--266.
[45] (with L. Henkin, A. Tarski) Representable cylindric algebras. Ann. Pure Appl. Logic 31, 1986, 23--60
[46] Tarski's contributions to algebraic logic. J. Symb. Logic 51, 1986, 899--906.
[47] (editor, with the cooperation of R. Bonnet) Handbook of Boolean Algebras, vol. 1. North--Holland, 1989, xix + 312pp.
[48] (editor, with the cooperation of R. Bonnet) Handbook of Boolean Algebras, vol. 2. North--Holland, 1989, 313--716pp.
[49] (editor, with the cooperation of R. Bonnet) Handbook of Boolean Algebras, vol. 3. North--Holland, 1989, 717--1366pp.
[50] The number of Boolean algebras. in {\bf Handbook of Boolean algebras.} North--Holland, 1989, 469--489.
[51] Endomorphisms of Boolean algebras. in {\bf Handbook of Boolean algebras.} North--Holland, 1989, 491--516.
[52] Automorphism groups. in {\bf Handbook of Boolean Algebras.} North-Holland, 1989, 517--545.
[53] Appendix on set theory. in {\bf Handbook of Boolean algebras.} North--Holland 1989 1213--1233.
[54] Bibliography. in {\bf Handbook of Boolean algebras}. North--Holland 1989, 11269--1342.
[55] Cardinal functions on Boolean algebras.} Birkh\"auser--Verlag, 1990, 152pp.
[56] Structure problems for cylindric algebras.} Colloq. Math. Soc. J. Bolyai 54, 1991, Budapest, Hungary, 413--429.
[57] (editor, with H. Andr\'eka, I. N\'emeti) {\bf Algebraic Logic.} North--Holland, 1991, 746pp.
[58] {\it Remarks on the problems in the books Cylindric Algebras, Part I and Part II and Cylindric Set Algebras.} Colloq. Math. Soc. J. Bolyai 54, 1991, Budapest, Hungary, 723--726.
[59] {\it Corrections for the books Cylindric Algebras, Part I and Part II and Cylindric Set Algebras.} Colloq. Math. Soc. J. Bolyai 54, 1991, Budapest, Hungary, 719--722.
[60] (with S. Koppelberg) Pseudo-trees and Boolean algebras. Order 8, 1992 359--374.
[61] Lectures on cylindric set algebras. Banach Center Publications 28, 1993, 253--290.
[62] Problems in the set theory of Boolean algebras. Mathematica Japonica 42, 1995, 179--185.
[63] Cardinal Invariants on Boolean algebras. Birkh\"auser--Verlag, 1996, 298pp.
[64] Minimum-sized infinite partitions of Boolean algebras. Mathematical Logic Quarterly 42, 1996, 537--550.
[65] (with A. Dow) Depth, $\pi$-character, and tightness in superatomic Boolean algebras. Topology and its Applications 75, 1997, 183--199. There is a Correction to this.
[66] (with P. Nyikos) On cellularity in homomorphic images of Boolean algebras. Topology Proceedings 22, 1998, 341--362.
[67] The spectrum of partitions of a Boolean algebra. Archive for Mathematical Logic 40 (2001), 243--254.
[68] Generalized free products. Colloquium Mathematicum 88 (2001), 175--192.
[69] Continuum cardinals generalized to Boolean algebras. J. Symb. Logic 66 (2001), 1928--1958.
[70] An introduction to cylindric set algebras (with an appendix by H. Andreka. Logic Journal of the IGPL 8 (2000), 451--506. (Reprint, with corrections and an added appendix, of [61].)
[71] Boolean algebras. Entry in Stanford Encyclopedia of Philosophy (electronic) (2002).
[72] An atomless interval Boolean algebra $A$ such that ${\frak a}(A)<{\frak t}(A)$.} Alg. Univ. 47 (2002), 495--500.
[73] The spectrum of maximal independent subsets of a Boolean algebra. Annals of Pure and Appl. Logic 126 (2004), 335--348.
[74] (with R. McKenzie) On some small cardinals for Boolean algebras. J. Symb. Logic 69 (2004), no. 3, 674--682.
[75] Generalized ${\frak b}$ and ${\frak d}$. Notre Dame J. Formal Logic 45 (2004), no. 3, 129--146.
[76] The size of maximal almost disjoint families. Dissert. Math. 437, 47pp. Inst. of Math., Polish Acad. Sci. (2006)
[77] Towers and maximal chains in Boolean algebras. Alg. Univ. 56 (2007), 337--347.
[78] Maximal irredundance and maximal ideal independence in Boolean algebras. J. Symb. Logic 73, no. 1 (2008), 261-275.
[79] On the existence of towers in pseudo-tree algebras. Order 26 (2009), 163-175.
[80] Leon Albert Henkin (1921--2006).} Bull. Symb. Logic 15, no. 3 (2009), 326-331.
[81] Special subalgebras of Boolean algebras. Math. Logic Quarterly 56, no. 2 (2010), 148-158.
[82] Maximal free sequences in a Boolean algebra. Comment. Math. Univ. Carol. 52, 4 (2011), 593--611.
[83] Remarks on continuum cardinals on Boolean algebras. Math. Log. Quarterly 58, no. 3 (2012), 159-167.
[84] Cardinal invariants on Boolean algebras. Second revised edition. (2014) vii + 573pp.
[85] Leon Henkin and cylindric algebras. In "The Life and Work of Leon Henkin.", (2014) Birkhauser, 59-66.
[86] A large list of small cardinal characteristics of Boolean algebras. Math. Log. Quarterly 64, no. 4-5, 336-348 (2018)
Abstracts:
[1] Nonrepresentable polyadic algebras of finite degree. Notices of the AMS, 7 (1960), 735.
[2] Polyadic Heyting algebras. Notices of the AMS, 7 (1960), 735.
[3] Representation theorems for cylindricc algebras. Notices of the AMS, 7 (1960), 1000.
[4] Relation algebras and cylindric algebras. Notices of the AMS, 8 (1961), 358.
[5] On finite dimensional cylindric algebras. Notices of the AMS, 9 (1962), 470.
[6] (with F. M. Sioson) m-semigroups, semigroups, and function representations. Notices of the AMS, 12 (1965), 549-550.
[7] Nonfinitizability of classes of cylindric algebras. Notices of the AMS, 15 (1968), 353.
[8] On the lattice of equational classes of one- and two-dimensional polyadic algebras. Notices of the AMS, 16 (1969), 183.
[9] (with P. Sparks) Counting Boolean algebras. Notices of the AMS, 18 (1971), 551
[10] Independent subsets of subsets of free Boolean algebras. Notices of the AMS 25 (1978), A-3.
[11] (With R. McKenzie) Chains in Boolean algebras. Abstracts of papers presented to the AMS 2(985), 235.
[12] Remarks on Boolean algebras: complete embeddings, independence, tightness Abstracts of papers presented to the AMS 2 (1981), 236.
[13] (with H. Andreka, I. Nemeti) Homomorphic images of weak cylindric algebras with infinite bases. Abstracts of papers presented to the AMS 5 (1984), 470.
[14] (with A. Dow) Compact scattered spaces, tightness, depth, and pi- character. Abstracts of papers presented to the AMS 15 (1994), 558.

Lecture notes, etc.:
(1) Unpublished notes on subalgebras of interval algebras.
(2) Continuum cardinals
(3) Basic pcf theory. (files are put in a subdirectory "tmp")
(4) Notes on Cardinal invariants on Boolean algebras, second revised edition.
(5) Lectures on set theory; also, solutions for exercises
(6) Lectures on model theory
(7) Lectures on logic (completeness and incompleteness)