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 2element example algebras, all equivalent to the 2element
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 2element 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 PalfyPudlak 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 HobbyMcKenzie.

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 6element 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 (McKenzieValeriote).

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.
