Logic, Set Theory and Topology Ical Atom

Yonah Maissel and Matatyahu Rubin Ben Gurion University, Beer Sheva, Israel Ralph McKenzie proved that if $G$ is a group of permutations of a set $A$ with cardinality different from 6 and 1, then the action of $G$ on $A$ can be recovered from the group $G$ using first order formulas. The analogous problems for semigroups of functions from a set $A$ to itself and for clones on $A$ have not been considered (so it seems). I shall present four analogues of McKenzie’s theorem. Here is one of them. Theorem 1: Let $A$ be a set whose cardinality is different from 6 and 1, and let $S$ be a semigroup of functions from $A$ to $A$ containing all transpositions of $A$. Then the action of $S$ on $A$ can be recovered from the algebraic structure of the semigroup $S$ using first order formulas. A function $f$ from $A$ to $A$ is called a semi-transposition, if there are distinct $a,b \in A$ such that $f(a) = b$, and for every $c \in A$: if $c \neq a$, then $f(c) = c$. Theorem 2: Let $A$ be a set whose cardinality different from 1, and let $S$ be a semigroup of functions from $A$ to $A$ containing all semi-transpositions of $A$. Then the action of $S$ on $A$ can be recovered from the algebraic structure of the semigroup $S$ using first order formulas. Theorem 3: The analogues of Theorems 1 and 2 for clones are also true. I shall present several open questions both for semigroups of functions and for clones.

Yonah Maissel and Matatyahu Rubin Ben Gurion University, Beer Sheva, Israel Ralph McKenzie proved that if $G$ is a group of permutations of a set $A$ with cardinality different from 6 and 1, then the action of $G$ on $A$ can be recovered from the group $G$ using first order formulas. The analogous problems for semigroups of functions from a set $A$ to itself and for clones on $A$ have not been considered (so it seems). I shall present four analogues of McKenzie’s theorem. Here is one of them. Theorem 1: Let $A$ be a set whose cardinality is different from 6 and 1, and let $S$ be a semigroup of functions from $A$ to $A$ containing all transpositions of $A$. Then the action of $S$ on $A$ can be recovered from the algebraic structure of the semigroup $S$ using first order formulas. A function $f$ from $A$ to $A$ is called a semi-transposition, if there are distinct $a,b \in A$ such that $f(a) = b$, and for every $c \in A$: if $c \neq a$, then $f(c)=c$. Theorem 2: Let $A$ be a set whose cardinality different from 1, and let $S$ be a semigroup of functions from $A$ to $A$ containing all semi-transpositions of $A$. Then the action of $S$ on $A$ can be recovered from the algebraic structure of the semigroup $S$ using first order formulas. Theorem 3: The analogues of Theorems 1 and 2 for clones are also true. I shall present several open questions both for semigroups of functions and for clones.

Finite Ramsey Theory is nowadays quite ubiquitous in Combinatorics. This can be claimed at least to some extent for Infinite Ramsey Theory within Set Theory as well. Most considerations within Infinite Ramsey Theory, however, treat the size of a homogeneous set as given by its cardinality. Considering ordered sets can be considered as a variation of this topic but by treating cardinals as initial ordinals it can also be viewed as a generalisation.

In the first talk I will provide historical background, notations and definitions on Ramsey Theory in this generalised context. Moreover I am going to state some classical results.

In the second talk I am going to focus on linear orderings which are neither well-orderings nor anti-well-orderings, present results by Erd​ős, Milner and Rado from 1971 and present some recent results from joint work with Philipp Lücke and Philipp Schlicht.

Finite Ramsey Theory is nowadays quite ubiquitous in Combinatorics. This can be claimed at least to some extent for Infinite Ramsey Theory within Set Theory as well. Most considerations within Infinite Ramsey Theory, however, treat the size of a homogeneous set as given by its cardinality. Considering ordered sets can be considered as a variation of this topic but by treating cardinals as initial ordinals it can also be viewed as a generalisation.

In the first talk I will provide historical background, notations and definitions on Ramsey Theory in this generalised context. Moreover I am going to state some classical results.

In the second talk I am going to focus on linear orderings which are neither well-orderings nor anti-well-orderings, present results by Erd​ős, Milner and Rado from 1971 and present some recent results from joint work with Philipp Lücke and Philipp Schlicht.

Fix a model $M$. For an $M$-definable group $G$ acting definably and transitively on a definable set $X$, we can consider the induced action on the space $S_X(M)$ of types on $X$. This is an action by homeomorphisms (where $S_X(M)$ is equipped with the standard Stone space topology), making the pair $(G(M),S_X(M))$ a $G(M)$-flow in the sense of classic topological dynamics. I will discuss how various notions of topological dynamics are interpreted in the sense of model theory. I will then present the results on the universal definable flows of groups definable in an o-minimal setting (e.g. definable real Lie groups).

Ralph McKenzie proved that if $G$ is a group of permutations of a set $A$ with cardinality different from $6$ and $1$, then the action of $G$ on $A$ can be recovered from the group G using first order formulas.

The analogous problems for semigroups of functions from a set $A$ to itself and for clones on $A$ have not been considered (so it seems).

I shall present four analogues of McKenzie’s theorem.

Here is one of them.

Theorem 1: Let $A$ be a set whose cardinality is different from $6$ and $1$, and let $S$ be a semigroup of functions from $A$ to $A$ containing all transpositions of $A$. Then the action of $S$ on $A$ can be recovered from the algebraic structure of the semigroup $S$ using first order formulas.

A function $f$ from $A$ to $A$ is called a semi-transposition, if there are distinct $a,b$ in $A$ such that $f(a)$ equals $b$, and for every $c$ in $A$: if $c\neq a$, then $f(c) = c$.

Theorem 2: Let A be a set whose cardinality different from $1$, and let $S$ be a semigroup of functions from $A$ to $A$ containing all semi-transpositions of $A$. Then the action of $S$ on $A$ can be recovered from the algebraic structure of the semigroup $S$ using first order formulas.

Theorem 3: The analogues of Theorems 1 and 2 for clones are also true.

I shall present several open questions both for semigroups of functions and for clones.

In the note “Quasi-Ordered Fields” by S. M. Fakhruddin [JPAA 45 (1987) 207-210] the author introduces the notion of a quasi-ordered (q.o.) field and shows the following dichotomy: a q.o. field is either an ordered field or a Krull valued field. We take this approach further to exhibit a theory of q.o. convex valuations. Classical results on (order) convex valuations can be reformulated for q.o. convex valuations in a natural way. In particular, this provides an elegant and uniform treatment of lifting of orderings, coarsening and composition of valuations. In this talk, I will explain the above, focusing on a new concept of “q.o.-minimality” generalizing several existing minimality notions.

The Lusternik-Schnirelmann category (LS-category) is a topological invariant that has historically been studied for absolute neighbourhood retracts. I will discuss how the theory of the LS-category can be extended to general metric spaces. One can obtain upper bounds for the LS-category of general spaces by using dimension-theoretic techniques to prove analogues to the Grossman-Whitehead theorem and Dranishnikov’s theorem. One can also obtain lower bounds in terms of cup-length, category weight and Bockstein maps. These results can be used to calculate the LS-category for some compacta like the Menger spaces and Pontryagin surfaces. I will also compare this definition with Borsuk’s shape theoretic LS-category. Finally, I will talk about potential applications of this work to geometric group theory, specifically the possibility of obtaining an analogue to the Bestvina-Mess formula in terms of LS-category.

Coarse geometry has been a very effective tool in proving very powerful conjectures like the Baum-Connes conjecture (which includes the Novikov and Borel conjectures) and Gromov’s positive scalar curvature conjecture. The aim of my talk is to give motivation for the study of coarse geometry/uniform space theory and to discuss big ideas in the application of these two dual geometries.