Logic Seminar Abstracts


Title: Satisfiability of linear equations on [0,1].
Speaker:  Walter Taylor
Affiliation:  CU
Time:  3pm, Monday, January 26
Location:  Math 220

Abstract:  
We consider the satisfiability of a set $ \Sigma$ of equations on a space $ A$, i.e. the existence of continuous $ A$-operations that satisfy the equations $ \Sigma$. In my Kempner Colloquium talk on 12/1/03, I outlined a proof that there is no algorithm to decide if a finite $ \Sigma$ is satisfiable on $ \mathbb{R}$ .

In the seminar, we shall restrict our attention to linear $ \Sigma$. (Here an equation is called linear if each of its terms contains either no function symbols or a single function symbol.) In this context we prove that if $ \Sigma$ is consistent (i.e. it has a model of more than one element), then $ \Sigma$ is satisfiable on [0,1] and on $ \mathbb{R}$ . (We shall prove the result for [0,1] in detail, while offering only a sketch for $\mathbb{R}$.)

How much logic will one need to follow the proof? Very little: at a certain point in the proof I will use a few linear equations that have been seen to lie in $ \Sigma$ and deduce the inconsistent equation x = y, thereby finishing off a proof by contradiction. And how much topology? Not a whole lot: we will study a metric on a quotient space that is defined by a closed equivalence relation, and to the resulting space we will apply Tietze's Extension Theorem.


Title: Satisfiability of linear equations on [0,1].
Speaker:  Taylor
Affiliation:  CU
Time:  3pm, Monday, February 2
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title: Satisfiability of linear equations on [0,1].
Speaker:  Taylor
Affiliation:  CU
Time:  3pm, Monday, February 9
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Countable choice and excluded middle.
Speaker:  Fred Richman
Affiliation:  Florida Atlantic University
Time:  3pm, Monday, February 16
Location:  Math 220

Abstract:  
A mostly expository talk touching on The Weyl-Polya bet, The Myhill-Goodman proof that full choice implies excluded middle, Arguments for and against countable choice, Examples of life without countable choice (and excluded middle): Cauchy reals versus reals, The Hilbert syzygy theorem, Trace-class operators, The fundamental theorem of algebra.


Title:   Recently discovered properties of the group of order preserving permutations of the real line.
Speaker:  Charles Holland
Affiliation:  Bowling Green State University
Time:  3pm, Monday, February 23
Location:  Math 220

Abstract:  
A lattice-ordered group is both a group and a lattice in which the lattice operations are respected by all group translations. The group Aut(R) of all order preserving permutations of the real line R is a lattice-ordered group when given the pointwise order, and it is universal in the sense that every countable lattice-ordered group embeds into it. The group Aut(R), therefore, has been much studied by those interested in lattice-ordered groups. Only recently has it been discovered that Aut(R) has some remarkable "cofinality" properties. I will discuss a theorem of which the following two are corollaries:

1. Aut(R) is not the union of a countable tower of proper subgroups.

2. If S is any set of generators of Aut(R) then there is a positive integer n such that every member of Aut(R) is a product of no more than n elements of S and their inverses.


Title:   Recently discovered properties of the group of order preserving permutations of the real line.
Speaker:  Charles Holland
Affiliation:  Bowling Green State University
Time:  3pm, Monday, March 1
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Which T0  topological algebras are Hausdorff?
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Monday, March 8
Location:  Math 220

Abstract:  
The variety of groups has the property that every T0  topological algebra in the variety is Hausdorff ( = T2). In this talk we discuss what is known about the characterization of the class of varieties for which the implication "T0 implies T2" is valid.


Title:   Which T0  topological algebras are Hausdorff?
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Monday, March 15
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Which groups arise as Aut(A2)? 
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Monday, April 12
Location:  Math 220

Abstract:  
It is not known which groups arise as Aut(A2) when A is a nontrivial finite algebra. Clearly, any such group has an element of order 2, namely the automorphism (x,y) |-> (y,x). Matt Gould showed that if G is a finite group with a retraction onto a 2-element subgroup, then G = Aut(A2) for some finite algebra A, but it is known that there are finite groups without such a retraction that arise as Aut(A2). We will discuss Steve Tschantz's proof that the cyclic group Z4 is NOT the automorphism group of the square of any finite algebra. (Z4 is the only finite group with an element of order 2 that is known not to be the automorphism group of the square of a finite algebra.)


Title:   Which groups arise as Aut(A2)? 
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Monday, April 19
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Which groups arise as Aut(A2)? 
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Monday, April 26
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Which groups arise as Aut(A2)? 
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Wednesday, April 28
Location:  Math 220

Abstract:  
A continuation of the previous talk.



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