Infinite Latin Squares: Neighbor Balance and Orthogonality

Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table. Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.

Mean Proximality, Mean Sensitivity and Mean Li–Yorke Chaos for Amenable Group Actions

We consider mean proximality and mean Li–Yorke chaos for [Formula: see text]-systems, where [Formula: see text] is a countable discrete infinite amenable group. We prove that if a countable discrete infinite abelian group action is mean sensitive and there is a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li–Yorke chaotic. Moreover, we give some characterizations of mean proximal systems for general countable discrete infinite amenable groups.

Regular Homeomorphisms of Finite Order on Countable Spaces

We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following.(a) Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets.(b) If G is a countably infinite Abelian group with finitely many elements of order 2 and βG is the Stone–Čech compactification of G as a discrete semigroup, then for every idempotent p ∈ βG\﹛0﹜, the subset ﹛p,−p﹜ ⊂ βG generates algebraically the free product of one-element semigroups ﹛p﹜ and ﹛−p﹜.

A model-theoretic proof for P ≠ NP over all infinite abelian group

We give a model-theoretic proof of the fact that for all infinite Abelian groups P ≠ NP in the sense of binary nondeterminism. This result has been announced 1994 by Christine Gaßner.

The Meet Operator in the Lattice of Group Topologies

It is well known that the lattice of topologies on a set forms a complete complemented lattice. The set of topologies which make G into a topological group form a complete lattice L(G) which is not a sublattice of the lattice of all topologies on G.Let G be an infinite abelian group. No nontrivial Hausdorff topology in L(G) has a complement in L(G). If τ1 and τ2 are locally compact topologies then τ1Λτ2 is also a locally compact group topology. The situation when G is nonabelian is also considered.

On the Splitting of Modules and Abelian Groups

In a fundamental paper on torsion-free abelian groups, R. Baer [1] proved that the group P of all sequences of integers with respect to componentwise addition is not free. This means precisely that P is not a direct sum of infinite cyclic groups. However, E. Specker proved in [9] that P has the property that any countable subgroup is free. Since an infinite abelian group G is called -free if each subgroup of rank less than is free, these results are equivalent to: P is -free but not free.