MATH 8090: Independent Study - General Algebra and Applications (Spring 2020)

• term algebras, varieties, equational theories
• finite axiomatization
• Mal'cev conditions
• commutator theory for general algebras
• circuit complexity for equation solving

Online meetings via Zoom

Schedule

1. 02/04: BS II.5 Congruences and Quotient Algebras (in particular Thm 5.9, 5.10 on congruence modularity), BS II.7 Direct Products (in particular Thm 7.5 on direct decompositions)
   Exercises: Show groups are not necessarily congruence distributive, lattices not necessarily congruence permutable.
   Ex II.7.2 (with additionally finding A_1 x A_2 whose congruences are not just product congruences), Ex II.7.5
2. 02/11: BS II.8 Subdirect Products (in particular Thm 8.4, 8.6), BS II.9 Varieties, BS II.10 Terms, Term Algebras and Free Algebras (no proofs except for Thm 10.16), BS II.11 Identities, Free Algebras, and Birkhoff's Theorem (in particular Thm 11.9)
   Exercises: Ex II.8.6, Ex II.10.2, Ex II.11.2 for X countably infinite
3. 02/18: BS II.12 Mal'cev Conditions (Thm 12.2, 12.6), BS II.14 Equational Logic (Cor 14.10, Thm 14.19), BS IV.1 Boolean Algebras (Cor 1.9, Cor 1.12), BS IV.10 Quasiprimal Algebras (Lem 10.1, 10.2, 10.4)
   Exercises: Ex II.14.2, Ex IV.10.3,
   (cf. Baker-Pixley Thm) Let A be an algebra with majority term and B,C be subalgebras of A^n such that for any i < j the projections of B and of C onto the pair of coordinates (i,j) is equal. Show that B=C.
4. 02/25: BS IV.11 Functionally Complete and Skew Free Algebras (Lemma 11.3, Thm 11.12 without proof, Cor 11.13), BS V.3 Principal Congruence Formulas (Lem 3.1 - Def 3.4), BS V.4 Finite Basis Theorems (Thm 4.3, Thm 4.18 without proof)
   Exercises: BS Ex IV.11.3, Ex V.3.1,
5. 03/03: FM 1 Commutator in Groups and Rings, 2 Universal Algebra (Thm 2.2 Day terms for CM varieties, Lem 2.4 Shifting Lemma), 3 Several Commutators
   Exercises: FM Ex 2.2, 3.1, 3.2
6. 03/10: FM 4 One Commutator in Modular Varieties (up to and including Remark 4.6), 5 Fundamental Theorem on Abelian Algebras 6 Permutability (up to Theorem 6.2)
   Exercises: FM Ex 5.1, 5.7, 5.10
7. 03/31: FM 7 Nilpotent Algebras (up to and including Cor 7.7), 8 Congruence identities only (C1), Theorem 8.1, 10.3 Residually Small Varieties (Definition and Theorem 10.15 without proof)
   Exercises: FM Ex 7.3, 7.4, 8.1, 8.5
8. 04/09 (Thursday 11-12): FM 14 Finite Basis for Nilpotent Algebras of Prime Power Order (Thm 14.9 is the main result, the rest of the chapter comprises its proof.)
   Exercises:
   1. Before reading FM 14: Give a finite basis for a finite abelian group (A,+).
   2. How do commutators as defined on p 124 specialize to groups? How do they relate to the classical group commutators?
9. 04/14: Ai: Bounding the free spectrum of nilpotent algebras of prime power order. Sections 1 Intro, 2 Prelims about supernilpotency, 3 Commutators and nilpotency (without proofs since mainly contained in FM).
   Exercises:
   1. Determine the free spectrum of a finite abelian group (A,+) explicitly.
   2. As above for a finite 2-nilpotent group (G,*). Hint: Rewrite an arbitrary term t(x_1,...,x_k) over (G,*) in a normal form using powers of x_i and commutators [x_i,x_j].
10. 04/21: Ai: Bounding the free spectrum of nilpotent algebras of prime power order. Sections 4-7.
    Exercises:
    1. Expand the cyclic group of order p^2 (p prime) with an elementary abelian group operation as in Ai: Theorem 4.2. What is the nilpotence degree of this expansion?
    2. What are the homovariate polynomials H for F = { x_1^3+x_2 } over Z_5 in Ai: Theorem 5.5?
11. 04/28: Ke: Congruence modular varieties with small free spectra.
    Exercises:
    1. Determine the unary polynomial functions on the dihedral group D_8 with 8 elements. What is the twin monoid of D_8?

Reading

1. Aichinger. Bounding the free spectrum of nilpotent algebras of prime power order. Israel J. of Math. 230 (2019), 919-947. Available via CU Library and on arxiv.
2. Burris, Sankappanavar. A course in universal algebra. [pdf]
3. Freese, McKenzie. Commutator theory for congruence modular varieties. [pdf]
4. Kearnes, Congruence modular varieties with small free spectra. Algebra Universalis 42 (1999), 165--181. Available via CU Library and from Keith's website.