MATH 8900: Independent Study - Equational theories and finite axiomatizability (Spring 2022)

• term algebras, varieties, equational theories
• finite axiomatization
• undecidability in equational logic
• residually small varieties
• nilpotent and supernilpotent varieties

Schedule

1. 01/11: Introduction
2. 01/19: BS II.9 varieties, BS II.10 terms, term algebras, free algebras 10.1-10.12 without proofs except for 10.8, 10.10
   Ex II.10.1, Ex II.10.5
3. 01/26: BS II.11 Identities, Free Algebras, and Birkhoff's Theorem (in particular Thm 11.9), BS II.12 Mal'cev Conditions (Thm 12.2, 12.6), BS II.14 Equational Logic (Cor 14.10, Thm 14.19) up to Jacobson's Theorem on p 96,
   Exercises: Ex II.11.2 for X countably infinite, Ex II.11.3, Ex II.14.2, Ex II.14.10
4. 02/09: BS V.3 Principal Congruence Formulas (Lem 3.1 - Thm 3.5, Thm 3.8), BS V.4 Finite Basis Theorems
   Exercises: BS Ex V.3.1, Ex V.4.1, Ex V.4.2
5. 02/16: ALVIN review 7.1 -7.2 (new Thm 7.10), 7.3 up to Lyndon's nonfinitely based finite algebra (p 165)
   Exercises: ALVIN 7.12.12(1), 7.12.24
6. 02/23: ALVIN 7.3 inherently nonfinitely based algebras, Shift Automorphism Theorem, graph algebras
   Exercises: ALVIN 7.22.1, 7.22.8
7. 03/02: AlVIN 7.6 Lattice of Equational Theories (proofs up to Thm 7.48, afterwards without) ALVIN 7.7 Computability review
   Exercise: ALVIN Example 7.44
8. 03/09: finite basis problem for nilpotent loops
9. 03/16: ALVIN 7.8 Undecidability (proofs up to Cor 7.63, afterwards without), Ex 7.70 finite bases of finite groups are not decidable.
   Exercise: ALVIN Exercise 7.72.1, 7.72.4
10. 03/30: ALVIN 7.9 Residual bounds, 7.10 Finite algebra of residual character aleph_1 (proof of properties of these algebras can be skipped but will occur again in 7.11)
    Exercise: Give a residual bound for the variety of G-sets (unary algebras with operations corresponding to permutation actions of the group G).
11. 04/06: ALVIN 7.11 Tarski's finite basis problem
12. 04/13: ALVIN 7.4: McKenzie's Finite Basis Thm for Finite Lattice (without proof), Diamond Finite Basis Thm, Willard's Finite Basis Thm for congruence meet-semidistributive varieties
13. 04/20: Neumann: commutators 31.51-31.61, variety of c-nilpotent groups is finitely based 33.11-34.15

Reading

1. Freese, McKenzie, McNulty, Taylor. Algebras, Lattices, Varieties II.
2. Burris, Sankappanavar. A course in universal algebra. [pdf]
3. Freese, McKenzie. Commutator theory for congruence modular varieties. [pdf]
4. Neumann. Varieties of groups. Electronic copy available via CU Library