0 Products of all elements in a loop and a framework for non - associative analogues of the Hall - Paige conjecture
نویسنده
چکیده
For a finite loop Q, let P (Q) be the set of elements that can be represented as a product containing each element of Q precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal implications between the following conditions: (A) Q has a complete mapping, i.e. the multiplication table of Q has a transversal, (B) there is no N EQ such that |N | is odd and Q/N ∼= Z2m for m ≥ 1, and (C) P (Q) intersects the associator subloop of Q. We prove (A) =⇒ (C) and (B) ⇐⇒ (C) and show that when Q is a group, these conditions reduce to familiar statements related to the Hall-Paige conjecture (which essentially says that in groups (B) =⇒ (A)). We also establish properties of P (Q), prove a generalization of the Dénes-Hermann theorem, and present an elementary proof of a weak form of the Hall-Paige conjecture.
منابع مشابه
Products of All Elements in a Loop and a Framework for Non-Associative Analogues of the Hall-Paige Conjecture
For a finite loop Q, let P (Q) be the set of elements that can be represented as a product containing each element of Q precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal implications between the following conditions: (A) Q has a complete mapping, i.e. the multiplication table of Q has a transversal, (B) there is no N E Q such that |N | is odd...
متن کاملPreferred Lexical Access Route in Persian Learners of English: Associative, Semantic or Both
Background: Words in the Mental Lexicon (ML) construct semantic field through associative and/ or semantic connections, with a pervasive native speaker preference for the former. Non-native preferences, however, demand further inquiry. Previous studies have revealed inconsistent Lexical Access (LA) patterns due to the limitations in the methodology and response categorization. Objectives: To f...
متن کاملFrankl's Conjecture for a subclass of semimodular lattices
In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...
متن کامل$L^p$-Conjecture on Hypergroups
In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$. Among the other things, we also show that if $K$ is a locally compact hyper...
متن کاملOn two problems concerning the Zariski topology of modules
Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...
متن کامل