Schreier theorem on groups which split over free abelian groups

Implicit function theorem over free groups

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the Implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lift...

Intuitionistic Free Abelian Groups

Hindman’s Theorem in Abelian Groups

Hindman’s theorem tells us that for every finite colouring/partition of the natural numbers, there is an infinite subset X for which every finite subset X0 Ď X has the same colour for ř X0. Certain generalisations of this theorem to abelian groups are known to fail in the uncountable case but an example exists for boolean groups when the desired homogeneous set is finite. In this paper, I will ...

Quantum Error-Correction Codes on Abelian Groups

We prove a general form of bit flip formula for the quantum Fourier transform on finite abelian groups and use it to encode some general CSS codes on these groups.

Free abelian lattice-ordered groups

Let n be a positive integer and FA`(n) be the free abelian latticeordered group on n generators. We prove that FA`(m) and FA`(n) do not satisfy the same first-order sentences in the language L={+,−, 0,∧,∨} if m 6= n. We also show that Th(FA`(n)) is decidable iff n ∈ {1, 2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices. A. M. S. C...