Let $\Gamma$ be a group and $\mathscr{C}$ class of groups endowed with bi-invariant metrics. We say that is $\mathscr{C}$-stable if every $\varepsilon$-homomorphism $\Gamma \rightarrow G$, $(G,d) \in \mathscr{C}$, $\delta_\varepsilon$-close to homomorphism, $\delta_\varepsilon\to 0$ when $\varepsilon\to 0$. If $\delta_\varepsilon < C \varepsilon$ for some $C$ we $ \mathscr{C} $-stable linear ra...

For digraphs D and H, a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected graph input digraph D admits a homomorphism to H. The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V (...

We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a first-order formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existential-positive formula. This result, which contrasts with the known failure of other classical preservat...

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which r...

1. [10 points] Determine whether the following statements are true or false (you have to include proofs/counterexamples): (a) Let R be an integral domain, F – a free R-module of finite rank, and M – a torsion R-module. Then there is no injective homomorphism from F to M . Solution: True. Suppose there was an injective homomorpism φ : F → M . Then let N = φ(F ); N is a submodule of M , and there...