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## Lecture Topics

 Date What we discussed/How we spent our time Aug 27 Syllabus. Policies. Sources. Algebra. Definitional equivalence. Post's lattice. Aug 29 We defined signature, algebra, term, term algebra. We sketched the explanation of why the term algebra is absolutely free. We looked at some 2-element example algebras, all equivalent to the 2-element Boolean algebra. Namely $\langle \{0,1\}; \wedge,\vee,\neg,0,1\rangle$, $\langle \{0,1\}; \vee,\neg,0,1\rangle$, $\langle \{0,1\}; \to,\neg,1\rangle$, $\langle \{0,1\}; \mid,0\rangle$. We explained why the 2-element Boolean algebra is primal, by showing that every Boolean function can be put into disjunctive normal form. Aug 31 We finished the background definitions: polynomial expansion, polynomial, polynomial operation, functionally complete, primal, identity, equational theory, variety. Using Lagrange interpolation we proved that any finite field is functionally complete. We explained why a finite field is primal iff it has prime order. We stated without proof that any finite simple group is functionally complete. Sep 5 We defined clones and gave examples. We sketched the proof of the Cayley Representation Theorem for clones, which proves that every clone has a faithful representation. Sep 7 We discussed Post's lattice (especially semilattice operations, discriminator operation, and near unanimity functions). We defined the relationship of compatibility between operations and relations, and indicated that there is an associated Galois connection. Sep 10 We talked about Galois connections in general, and the GC between operations and relations in particular. Sep 12 We showed that a set of operations on a finite set is closed in the GC iff it is a clone. We defined relational clones and stated that a set of relations on a finite set is closed in the GC iff it is a relational clone. Some details of the proof can be extracted from these notes. We started to talk about localization, following these notes. Sep 14 Today we gave most of the proof of: Theorem. Let $\mathbb A = \langle A; {\mathcal C}\rangle$ be an algebra, and let ${\mathbb A}^{\perp}=\langle A; {\mathcal C}^{\perp}\rangle$ be the associated relational structure. A subset $U\subseteq A$ has the property that the restriction map $\rho\mapsto \rho\cap U^m$ is a relational clone homomorphism if and only if there exists an element $e\in C_1$ such that $\mathbb A\models e(e(x))=e(x)$ and $U=e(A)$. The part of the proof that we did not establish was: if $U=e(A)$ for appropriate $e$, then restriction to $U$ commutes with projection onto a subset of coordinates. Sep 17 We discussed a direct construction of $e(\mathbb A)$ (avoiding reference to the Galois connection). We looked examples of modules over $R_1 = M_2(\mathbb F)$, $R_2 = UT_2(\mathbb F)$, $R_3 = \textrm{Diag}_2(\mathbb F)$. Sep 19 We showed that localization is a functor. We started talking about covers. Sep 21 We discussed how to reconstruct an algebra from its localizations to sets in a cover, using the matrix product construction. Sep 24 We showed that morphisms (in particular isomorphisms) from $\mathbb A^{\perp}|_U$ to $\mathbb A^{\perp}|_V$ are induced by terms. We explained why every finite algebra has a unique irredundant nonrefinable covewr up to isomorphism. Sep 26 We defined irreducibility of nhoods, and $\langle R,S\rangle$-irreducibility, and showed that $U$ is irreducible iff it is $\langle R,S\rangle$-irreducible for some $R\subsetneq S$. Sep 28 We defined $\langle R,S\rangle$-minimality, and showed that $U$ is $\langle R,S\rangle$-irreducible iff it is $\langle R,S\rangle$-minimal for some $S'\leq S, S'\not\subseteq R, R'\subseteq R$. We talked about the ancient history of these localization ideas. Oct 1 We discussed the Palfy-Pudlak paper, which proves that [every finite lattice is representable as an interval in the subgroup lattice of a finite group] iff [every finite lattice is representable as the congruence lattice of a finite algebra]. Oct 3 We mentioned the main result of the Finite Forbidden Lattices paper, then surveyed resulst classifying $\mathbf A|_U$ in certain situations. Oct 5 We proved half of the Twin Lemma from Kiss's notes. Oct 8 We finished the proof of the Twin Lemma. Oct 10 We proved Lemma 4.15 of Hobby-McKenzie. Oct 12 We discussed the structure of $\langle\delta,\theta\rangle$-minimal algebras of all types. We disccused Prohle's example (how to build $\langle\delta,\theta\rangle$-minimal algebras) from the end of the Easy Way notes. We examined a 6-element algebra which has all types in its congruence lattice. I asked if anyone could build one with 5 elements. Oct 15 We proved Maltsev's Congruence Generation Theorem in this form: If $G\subseteq A\times A$, then $\textrm{Cg}^{\mathbb A}(G)=\textrm{tr.cl.}(P_1(G)\cup P_1(G^{\cup}).$ We began discussing Theorem 2.8 of HM. Oct 17 We continued discussing Thm 2.8 (including a proof that elements of $M_{\mathbb A}(\delta,\theta)$ are $\langle\delta,\theta\rangle$-minimal/irreducible when $\delta\prec\theta$). Oct 19 We finished Thm 2.8. Then we talked about $E$-minimal algebras. Oct 22 We talked about the structure of the $E$-minimal algebras ${\mathbb E}(q,k)$ and ${\mathbb E}(\delta_0,\ldots,\delta_k)$. Oct 24 We introduced $C(\alpha,\beta;\delta)$ and $[\alpha,\beta]$. Oct 26 We began discussing the relationship between diagonal congruences, $3$-nets, and compatible Maltsev operations. We proved that if $\mathbb A$ has a Maltsev term $f(x,y,z)$ and also a compatible Maltsev operation $g(x,y,z)$, then $f=g$ and $\mathbb A$ is polynomially equivalent to a module. Oct 29 We proved that if $\mathbb A$ has a Maltsev term operation and $\mathbb A^2$ has a congruence $\Delta$ with the diagonal as a class, then $\Delta$ along with the tweo coordinate projection kernels form a $3$-net. Oct 31 We finished the proof that an algebra is Maltsev + abelian iff it is polynomially equivalent to a module. We discussed a categorical definition of the commutator and a definition based on the modular law. Nov 2 We revisited the Twin Lemma, to connect commutator theory to types of minimal sets. We began a proof that if $U$ is a $\langle 0,\theta\rangle$-minimal set where body twins have the same character, and $N$ is a trace in $U$, then $\mathbb A|_N$ is polynomially equivalent to a $G$-set or vector space. Nov 5 We sketched the end of the proof that if $U$ is a $\langle 0,\theta\rangle$-minimal set where body twins have the same character, and $N$ is a trace in $U$, then $\mathbb A|_N$ is polynomially equivalent to a $G$-set or vector space. Nov 7 We discussed definability and interpretability, and sketched the reason that every axiomatizable class of structures in a finite language is interpretable into the class of graphs in a way that preserves finiteness of the structures. We also showed that the class of graphs is interpretable in the class of semilattices. Nov 9 We talked about interpretations, and ($\omega$-)unstructured model classes. We mentioned the hereditary undecidability of the class of finite graphs (Lavrov), the decidability of the theory of two functions (Ehrenfeucht), the decidability of any class of abelian groups (Szmielew), and the decidability of any class of Boolean algebras (Tarski). We stated the SAD Theorem (McKenzie-Valeriote). Nov 12 We showed how to interpret the class of subalgebras of powers of $\mathbb A|_U$ into the class of subalgebras of powers of $\mathbb A$. We proved that if a variety has a finite algebra with a type $5$ prime quotient, then it is possible to interpret graphs into the variety. Nov 14 We began a proof that if a variety has a finite algebra with a type $4$ prime quotient, then it is possible to interpret graphs into the variety. Nov 16 We finished a sketch of a proof that if a variety has a finite algebra with a type $4$ prime quotient, then it is possible to interpret graphs into the variety. Nov 26 We began to discuss other properties of varieties which do not interpret finite graphs, namely empty tails in types 2 and 3 and the transfer principles. Nov 28 We started working through the proof that the class of finite graphs can be interpreted in the class of subalgebras of finite powers of any finite algebra with a type 3 quotient whose minimal sets have nonempty tail. Nov 30 Modulo one small point, we finished the proof that the class of finite graphs can be interpreted in the class of subalgebras of finite powers of any finite algebra with a type 3 quotient whose minimal sets have nonempty tail. Dec 3 We finished the proof that the class of finite graphs can be interpreted in the class of subalgebras of finite powers of any finite algebra with a type 3 quotient whose minimal sets have nonempty tail. Dec 5 We discussed clone homomorphisms and products of varieties. Dec 7 We discussed the lattice of interpretability types of varieties and (strong) Maltsev filters/conditions. In particular, we discussed the conditions: having a Maltsev term, having an NUF, having a Taylor term. We started on the TCT characterization of congruence distributivity. Dec 10 We proved most of the theorem that asserts that a locally finite variety is congruence distributive iff it has type set in $\{{\bf 3}, {\bf 4}\}$ and all minimal sets have empty tail.