Ebook: 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