3×3 Lemma and Protomodularity

Normal Functors and Strong Protomodularity

The notion of normal subobject having an intrinsic meaning in any protomodular category, we introduce the notion of normal functor, namely left exact conservative functor which reflects normal subobjects. The point is that for the category Gp of groups the change of base functors, with respect to the fibration of pointed objects, are not only conservative (this is the definition of a protomodul...

Regular hypergraphs, Gordon's lemma, Steinitz' lemma and invariant theory

Let D(n)(D(n, k)) denote the maximum possible d such that there exists a d-regular hypergraph (d-regular k-uniform hypergraph, respectively) on n vertices containing no proper regular spanning subhypergraph. The problem of estimating D(n) arises in Game Theory and Huckemann and Jurkat were the first to prove that it is finite. Here we give two new simple proofs that D(n), D(n, k) are finite, an...

Pumping Lemma and Ogden Lemma for Displacement Context-Free Grammars

The pumping lemma and Ogden lemma offer a powerful method to prove that a particular language is not context-free. In 2008 Kanazawa proved an analogue of pumping lemma for well-nested multiplecontext free languages. However, the statement of lemma is too weak for practical usage. We prove a stronger variant of pumping lemma and an analogue of Ogden lemma for the class of tree-adjoining language...

Lecture 19 : The Vanishing Lemma and Lemma 8 . 8

A simple proof of Zariski's Lemma

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.