Page 7 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press

Table of Contents

I: PRELIMINARIES ................................................................................................................ 11
1.1 Lattice ........................................................................................................................................ 13
1.2 Bands ......................................................................................................................................... 20
1.3 Noncommutative lattices – initial observations ........................................................... 28
References ........................................................................................................................................ 39

II: SKEW LATTICES .............................................................................................................. 41
2.1 Fundamental results .............................................................................................................. 43
2.2 Instances of commutative behavior .................................................................................. 47
2.3 Normal skew lattices .............................................................................................................. 53
2.4 Primitive skew lattices and skew lattice structure ..................................................... 59
2.5 Partial skew lattices and coset projections .................................................................... 71
2.6 Decompositions of normal, symmetric skew lattices ................................................. 76
References ........................................................................................................................................ 86

III: QUASILATTICES, PARALATTICES & THEIR CONGRUENCES ........................... 87
3.1 Congruences on quasilattices. ............................................................................................. 89
3.2 Antilattices that are simple as algebras .......................................................................... 94
3.3 Regular quasilattices ............................................................................................................. 96
3.4 Paralattices and refined quasilattices ............................................................................. 98
3.5 The effects of the distributive identities ....................................................................... 105
3.6 Deriving simple antilattices from magic squares ....................................................... 108
References ...................................................................................................................................... 117

IV: SKEW BOOLEAN ALGEBRAS .................................................................................... 119
4.1 Skew Boolean algebras ........................................................................................................ 121
4.2 Finiteness, orthosums and free algebras ...................................................................... 126
4.3 Connections with strongly distributive skew lattices .............................................. 134
4.4 Skew Boolean algebras with intersections ................................................................... 138
4.5 Omega algebras and skew Boolean covers ................................................................... 152
References ...................................................................................................................................... 161

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