Definition 1. A monoid is a set M with an element e and an associative multiplication M×M −→ M for which e is a two-sided identity element: em = m = me for all m ∈ M . A group is a monoid in which each element m has an inverse element m, so that mm = e = mm. A homomorphism f : M −→ N of monoids is a function f such that f(mn) = f(m)f(n) and f(eM ) = eN . A “homomorphism” of any kind of algebrai...

Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | TorRm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This r...

In 2006, P. J. Cameron and J. Nešetřil introduced the following variant of homogeneity: we say that a structure is homomorphismhomogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we classify finite homomorphism-homogeneous graphs where some vertices may have loops, but only up to a certain point. We focus on ...

We show that certain canonical realizations of the complexes Hom(G,H) and Hom+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of ...

in this paper we introduce a generalization of m-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . we will use the structure of the radical of a module in  and get some suitable results about this class of modules. also the relation between injective hull in  and this kind of modules will be investigated in this article.   for a module  we show...

Let $M$ be a manifold, $N$ 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $\rho: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has standard form: necessarily is $1$-dimensional, and there are countably many embeddings $\phi_i: M\to N$ with disjoint images such the action of $\rho$ conjugate (via product $\phi_i$) to diagonal $\text{Diff}^r_c(M)$ on $M \ti...

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Let Gn,k denote the Kneser graph whose vertices are the n-element subsets of a (2n + k)-element set and whose edges are the disjoint pairs. In this paper we prove that for any non-negative integer s there is no graph homomorphism from G4,2 to G4s+1,2s+1. This confirms a conjecture of Stahl in a special case.

We give a systematic definition of the fundamental groups of gropes, which we call grope groups. We show that there exists a nontrivial homomorphism from the minimal grope group M to another grope group G only if G is the free product of M with another grope group.

We give a systematic definition of the fundamental groups of gropes, which we call grope groups. We show that there exists a nontrivial homomorphism from the minimal grope group M to another grope group G only if G is the free product of M with another grope group.