Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture.
Powers of elements in Jordan loops
A Jordan loop is a commutative loop satisfying the Jordan identity . We establish several identities involving powers in Jordan loops and show that there is no nonassociative Jordan loop of order .
A decomposition of gallai multigraphs
An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees...
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