Cylindric Algebras, Part I

Volume I provides a detailed analysis of cylindric algebras, starting with a formulation of their axioms and a development of their elementary properties, and proceeding to a deeper study of their interrelationships by means of general algebraic notions such as subalgebras, homomorphisms, direct products, free algebras, reducts and relativized algebras. Contents: FOREWORD. .. . . .. . .. .. . .. .. .. . . . .. 1 PRELIMINARIES . .. . . . . . . .. .. .. .. .. .. .. 25 I. Set-theoretical notions . . . . .. .. .. .. . .. . .. .. .. 25 II. Metalogical notions . . . .. .. .. . . 39 Chapter 0. GENERAL THEORY OF ALGEBRAS. 47 0.1 Algebras and their subalgebras. . . .. .. .. .. .. .. . .. . .. 50 0.2 Homomorphisms, isomorphisms, congruence relations, and ideals. . . . .. . . .. . . . .. . .. .. .. . .. . .. . .. .. .. 67 0.3 Direct products and related notions. . .. .. . . . 83 0.4 Polynomials and free algebras.. . . ... ...... 119 0.5 Reducts.... . . .. . . . . . .. . . .. .. . .. . .. . 149 Problems. .. .. . . . . .. . . . .. .. .. . .. .. .. .. .. .. .. .. .. .. 157 Chapter 1. ELEMENTARY PROPERTIES OF CYLINDRIC ALG EB RAS .............. ..... 159 1.1 Cylindric algebras . . .. .. . . . .. .. .. .. .. .. . . .. . 161 1.2 Cylindrifications . .. . . . .. .. .. .. .. .. . .. .. .. . .. . 175 1.3 Diagonal elements .. .. . . . .. . .. . .. . .. .. .. .. .. . 179 1.4 Duality .. . . . .. . .. . .. . .. .. .. .. . .. .. .. .. . 185 1.5 Substitutions . . . . .. . .. .. .. . .. . . .. .. .. .. .. 189 1.6 Dimension sets. . .. . .. . .. . 199 1.7 Generalized cylindrifications . . .. .. .. .. .. .. .. . . .. .. 205 1.8 Generalized diagonal elements. . . .. .. .. .. .. . . . .. 209 1.9 Generalized co-diagonal elements . .. .. . .. . .. .. .. . .. .. 215 1.10 Atoms and rectangular elements . .. .. .. .. .. .. . .. .. .. . 225 1.11 Locally finite-dimensional and dimension-complemented cylin- dric algebras. . . . . . . . . . . 231 Pro blems. . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Chapter 2. GENERAL ALGEBRAIC NOTIONS APPLIED TO CYLINDRIC ALGEBRAS. .. ... 247 2.1 Suba1gebras................... 250 2.2 Relativization of cylindric algebras. . . . . . . . . . . . . 261 2.3 HomomorphislllS, isomorphisms, and ideals . . . . . . . . . 279 2.4 Direct products and related notions . . . . . . . . . . 297 2.5 Free algebras . . . . . . . . . . . . . . . . . . 335 2.6 Red ucts. . . . . . . . . . . . . . . . . . . . . . . . . 381 2.7 Canonical embedding algebras and atom structures. . . . . . 429 Problems. . . . . . . . . . . . . . . . . . . . . . . 463 BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . 467 I. Bibliography of cylindric algebras and related algebraic struc- tures. . . . . . . . . . . . . . . . .. .... 469 II. Supplementary bibliography.. ... ....... 481 INDEX OF SYMBOLS. . . . . . 489 INDEX OF NAMES AND SUBJECTS. 499