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.
|