Lattices and Ordered Algebraic Structures

Cover......Page 1
Title Page......Page 5
Preface......Page 7
Contents......Page 9
1.1 The concept of an order......Page 13
1.2 Order-preserving mappings......Page 17
1.3 Residuated mappings......Page 18
1.4 Closures......Page 22
1.5 Isomorphisms of ordered sets......Page 24
1.6 Galois connections......Page 26
1.7 Semigroups of residuated mappings......Page 27
2.1 Semilattices and lattices......Page 31
2.2 Down-set lattices......Page 35
2.3 Sublattices......Page 38
2.4 Lattice morphisms......Page 40
2.5 Complete lattices......Page 41
2.6 Baer semigroups......Page 47
3.1 Ordering quotient sets......Page 51
3.2 Strongly upper regular equivalences......Page 53
3.3 Lattice congruences......Page 57
4.1 Modular pairs; Dedekind’s modularity criterion......Page 61
4.2 Chain conditions......Page 66
4.3 Join-irreducibles......Page 70
4.4 Baer semigroups and modularity......Page 73
5.1 Birkhoff’s distributivity criterion......Page 77
5.2 More on join-irreducibles......Page 81
5.3 Prime ideals and filters......Page 84
5.4 Baer semigroups and distributivity......Page 86
6.1 Complemented elements......Page 89
6.2 Uniquely complemented lattices......Page 90
6.3 Boolean algebras and boolean rings......Page 94
6.4 Boolean algebras of subsets......Page 98
6.5 The Dedekind–MacNeille completion of a boolean algebra......Page 102
6.6 Neutral and central elements......Page 104
6.7 Stone’s representation theorem......Page 107
6.8 Baer semigroups and complementation......Page 109
6.9 Generalisations of boolean algebras......Page 113
7.1 Pseudocomplements......Page 115
7.2 Stone algebras......Page 118
7.3 Heyting algebras......Page 123
7.4 Baer semigroups and residuation......Page 128
8.1 More on lattice congruences......Page 131
8.2 Congruence kernels......Page 133
8.3 Principal congruences......Page 138
8.4 Congruences on p-algebras......Page 142
8.5 Congruences on Heyting algebras......Page 146
8.6 Subdirectly irreducible algebras......Page 149
9.1 Ordering groups......Page 155
9.2 Convex subgroups......Page 159
9.3 Lattice-ordered groups......Page 162
9.4 Absolute values and orthogonality......Page 165
9.5 Convex l-subgroups......Page 170
9.6 Polars......Page 174
9.7 Coset ordering; prime subgroups......Page 176
9.8 Representable groups......Page 180
10.1 Totally ordered rings and fields......Page 183
10.2 Archimedean ordered fields......Page 189
10.3 Archimedean totally ordered groups......Page 200
11.1 Ordered semigroups......Page 205
11.2 Residuated semigroups......Page 209
11.3 Molinaro equivalences......Page 216
12.1 Anticones......Page 219
12.2 Dubreil-Jacotin semigroups......Page 224
12.3 Residuated Dubreil-Jacotin semigroups......Page 229
13.1 Regular Dubreil-Jacotin semigroups......Page 237
13.2 The Nambooripad order......Page 240
13.3 Natural orders on regular semigroups......Page 244
13.4 Biggest inverses......Page 255
13.5 Principally ordered regular semigroups......Page 263
13.6 Principally and naturally ordered semigroups......Page 267
13.7 Ordered completely simple semigroups......Page 270
14.1.1 Inverse transversals......Page 277
14.1.3 Biggest inverses......Page 281
14.2 Integral Dubreil-Jacotin inverse semigroups......Page 283
14.3 Orthodox Dubreil-Jacotin semigroups......Page 286
14.3.1 The cartesian order......Page 288
14.3.2 Unilateral lexicographic orders......Page 294
14.3.3 Bootlace orders......Page 297
14.3.4 Lexicographic orders......Page 302
14.4 Lattices for which Res L is regular......Page 303
References......Page 305
Index......Page 311