A COURSE IN UNIVERSAL ALGEBRA
CONTENTS PRELIMINARIES I LATTICES §1 DE?NITIONS OF LATTICES §2 ISOMORPHIC LATTICES, AND SUBLATTICES §3 DISTRIBUTIVE AND MODULAR LATTICES §4 COMPLETE LATTICES, EQUIVALENCE RELATIONS, AND ALGEBRAIC LATTICES §5 CLOSURE OPERATORS II THE ELEMENTS OF UNIVERSAL ALGEBRA §1 DE?NITION AND EXAMPLES OF ALGEBRAS §2 ISOMORPHIC ALGEBRAS, AND SUBALGEBRAS §3 ALGEBRAIC LATTICES AND SUBUNIVERSES §4 THE IRREDUNDANT BASIS THEOREM §5 CONGRUENCES AND QUOTIENT ALGEBRAS §6 HOMOMORPHISMS AND THE HOMOMORPHISM AND ISOMORPHISM THEOREMS §7 DIRECT PRODUCTS, FACTOR CONGRUENCES, AND DIRECTLY INDECOMPOSABLE ALGEBRAS §8 SUBDIRECT PRODUCTS, SUBDIRECTLY IRREDUCIBLE ALGEBRAS, AND SIMPLE ALGEBRAS §9 CLASS OPERATORS AND VARIETIES §10 TERMS, TERM ALGEBRAS, AND FREE ALGEBRAS §11 IDENTITIES, FREE ALGEBRAS, AND BIRKHO?’S THEOREM §12 MAL’CEV CONDITIONS §13 THE CENTER OF AN ALGEBRA §14 EQUATIONAL LOGIC AND FULLY INVARIANT CONGRUENCES IIISELECTED TOPICS §1 STEINER TRIPLE SYSTEMS, SQUAGS, AND SLOOPS §2 QUASIGROUPS, LOOPS, AND LATIN SQUARES §3 ORTHOGONAL LATIN SQUARES §4 FINITE STATE ACCEPTORS IV STARTING FROM BOOLEAN ALGEBRAS §1 BOOLEAN ALGEBRAS §2 BOOLEAN RINGS §3 FILTERS AND IDEALS §4 STONE DUALITY §5 BOOLEAN POWERS §6 ULTRAPRODUCTS AND CONGRUENCE-DISTRIBUTIVE VARIETIES §7 PRIMAL ALGEBRAS §8 BOOLEAN PRODUCTS §9 DISCRIMINATOR VARIETIES §10 QUASIPRIMAL ALGEBRAS §11 FUNCTIONALLY COMPLETE ALGEBRAS AND SKEW-FREE ALGEBRAS §12 SEMISIMPLE VARIETIES §13 DIRECTLY REPRESENTABLE VARIETIES V CONNECTIONS WITH MODEL THEORY §1 FIRST-ORDER LANGUAGES, FIRST-ORDER STRUCTURES, AND SATISFACTION §2 REDUCED PRODUCTS AND ULTRAPRODUCTS §3 PRINCIPAL CONGRUENCE FORMULAS §4 THREE FINITE BASIS THEOREMS §5 SEMANTIC EMBEDDINGS AND UNDECIDABILITY RECENT DEVELOPMENTS AND OPEN PROBLEMS §1 THE COMMUTATOR AND THE CENTER §2 THE CLASSI?CATION OF VARIETIES §3 DECIDABILITY QUESTIONS §4 BOOLEAN CONSTRUCTIONS §5 STRUCTURE THEORY §6 APPLICATIONS TO COMPUTER SCIENCE §7 APPLICATIONS TO MODEL THEORY §8 FINITE BASIS THEOREMS §9 SUBDIRECTLY IRREDUCIBLE ALGEBRAS PRELIMINARIES WE HAVE ATTEMPTED TO KEEP OUR NOTATION AND CONVENTIONS IN AGREEMENT WITH THOSE OF THE CLOSELY RELATED SUBJECT OF MODEL THEORY, ESPECIALLY AS PRESENTED IN CHANG AND KEISLER’S MODEL THEORY [8] THE READER NEEDS ONLY A MODEST EXPOSURE TO CLASSICAL ALGEBRA; FOR EXAMPLE HE SHOULD KNOW WHAT GROUPS AND RINGS ARE WE WILL ASSUME A FAMILIARITY WITH THE MOST BASIC NOTIONS OF SET THEORY ACTUALLY, WE USE CLASSES AS WELL AS SETS A CLASS OF SETS IS FREQUENTLY CALLED A FAMILY OF SETS THE NOTATIONS, AI, I ?I, AND (AI)I?I REFER TO A FAMILY OF SETS INDEXED BY A SET I A NAIVE THEORY OF SETS AND CLASSES IS SU?CIENT FOR OUR PURPOSES WE ASSUME THE READER IS FAMILIAR WITH MEMBERSHIP (?), SET-BUILDER NOTATION ({—:—}), SUBSET (?), UNION (?), INTERSECTION (?), DI?ERENCE (?), ORDERED N-TUPLES (?X1, , XN?), (DIRECT) PRODUCTS OF SETS (A×B, Q I?I AI), AND (DIRECT) POWERS OF SETS (AI) ALSO, IT IS MOST USEFUL TO KNOW THAT (A) CONCERNING RELATIONS: (I) AN N-ARY RELATION ON A SET A IS A SUBSET OF AN; (II) IF N = 2 IT IS CALLED A BINARY RELATION ON A; (III) THE INVERSE R? OF A BINARY RELATION R ON A IS SPECI?ED BY ?A, B??R? I??B, A??R; (IV) THE RELATIONAL PRODUCT R?S OF TWO BINARY RELATIONS R, S ON A IS GIVEN BY: ?A, B??R?S I?FOR SOME C, ?A, C??R, ?C, B??S; (B) CONCERNING FUNCTIONS: (I) A FUNCTION F FROM A SET A TO A SET B, WRITTEN F : A ?B, IS A SUBSET OF A × B SUCH THAT FOR EACH A ?A THERE IS EXACTLY ONE B ?B WITH ?A, B??F; IN THIS CASE WE WRITE F(A) = B OR F : A 7?B; (II) THE SET OF ALL FUNCTIONS FROM A TO B IS DENOTED BY BA; (III) THE FUNCTION F ?BA IS INJECTIVE (OR ONE-TO-ONE) IF F(A1) = F(A2) ?A1 = A2; (IV) THE FUNCTION F ?BA IS SURJECTIVE (OR ONTO) IF FOR EVERY B ?B THERE IS AN A ?A WITH F(A) = B;