1 QUASIGROUPS AND LOOPS
1.1 Latin squares
1.2 Equational quasigroups
1.3 Conjugates
1.4 Semisymmetry and homotopy
1.5 Loops and piques
1.6 Steiner triple systems I
1.7 Moufang loops and octonions
1.8 Triality
1.9 Normal forms
1.10 Exercises
1.11 Notes
2 MULTIPLICATION GROUPS
2.1 Combinatorial multiplication groups
2.2 Surjections
2.3 The diagonal action
2.4 Inner multiplication groups of piques
2.5 Loop transversals and right quasigroups
2.6 Loop transversal codes
2.7 Universal multiplication groups
2.8 Universal stabilizers
2.9 Exercises
2.10 Notes
3 CENTRAL QUASIGROUPS
3.1 Quasigroup congruences
3.2 Centrality
3.3 Nilpotence
3.4 Central isotopy
3.5 Central piques
3.6 Central quasigroups
3.7 Quasigroups of prime order
3.8 Stability congruences
3.9 No-go theorems
3.10 Exercises
3.11 Notes
4 HOMOGENEOUS SPACES
4.1 Quasigroup homogeneous spaces
4.2 Approximate symmetry
4.3 Macroscopic symmetry
4.4 Regularity
4.5 Lagrangean prcperties
4.6 Exercises
4.7 Notes
5 PERMUTATION REPRESENTATIONS
5.1 The category ]FSQ
5.2 Actions as coalgebras
5.3 Irreducibility
5.4 The covariety of Q-sets
5.5 The Burnside algebra
5.6 An example
5.7 Idempotents
5.8 Burnside's Lemma
5.9 Exercises
5.10 Problems
5.11 Notes
6 CHARACTER TABLES
6.1 Conjugacy classes
6.2 Class functions
6.3 The centralizer ring
6.4 Convolution of class functions
6.5 Bose-Mcsner and Hecke algebras
6.6 Quasigroup character tables
6.7 Orthogonality relations
6.8 Rank two quasigroups
6.9 Entropy
6.10 Exercises
6.11 Problems
6.12 Netcs
7 COMBINATORIAL CHARACTER THEORY
7.1 Congruence lattices
7.2 Quotients
7.3 Fusion
7.4 Induction
7.5 Linear characters
7.6 Exercises
7.7 Problems
7.8 Notes
8 SCHEMES AND SUPERSCHEMES
8.1 Sharp transitivity
8.2 More no-go theorems
8.3 Superschemes
8.4 Superalgebras
8.5 Tenser squales
8.6 Relation algebras
8.7 The Reconstruction Theorem
8.8 Exercises
8.9 Problems
8.10 Notes
9 PERMUTATION CHARACTERS
9.1 Enveloping algebras
9.2 Structure of enveloping algebras
9.3 The canonical representaticn
9.4 Commutative actions
9.5 Faithful homogeneous spaces
9.6 Characters of homogeneous spaces
9.7 General permutation characters
9.8 The Ising model
9.9 ExeI cises
9.10 Problems
9.11 Nctes
10 MODULES
10.1 Abelian groups and slice categories
10.2 Quasigroup modules
10.3 The Fundamental Theorem
10.4 Differential calculus
10.5 Representations in varieties
10.6 Group representations
10.7 Exercises
10.8 Problems
10.9 Notes
11 APPLICATIONS OF MODULE THEORY
11.1 Nonassociative lowers
11.2 Exponents
11.3 Steincr triple systems Ⅱ
11.4 The Burrside Problem
11.5 A free commutative Mcufang loop
11.6 Extensions aid cohomology
11.7 Exercises
11.8 Problems
11.9 Notes
12 ANALYTICAL CHARACTER THEORY
12.1 Functions on finite quasigroups
12.2 Periodic functions on groups
12.3 Analytical character theory
12.4 Ahnost periodic functions
12.5 Twisted translation operators
12.6 Proof of the Existence Theorem
12.7 Exercises
12.8 Problems
12.9 Notes
A CATEGORICAL CONCEPTS
A.1 Graphs and categories
A.2 Natural transformations and functors
A.3 Limits and colimits
B UNIVERSAL ALGEBRA
B.1 Combinatorial universal algebra
B.2 Categorical universal algebra
C COALGEBRAS
C.1 Coalgebras and covarieties
C.2 Set functors
References
Index
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