Le théorème de Jordan-Hölder dans certains groupoïdes ordonnés
Lexicographic product decompositions of half linearly ordered loops
In this paper we prove for an hl-loop an assertion analogous to the result of Jakubík concerning lexicographic products of half linearly ordered groups. We found conditions under which any two lexicographic product decompositions of an hl-loop with a finite number of lexicographic factors have isomorphic refinements.
Linear extensions of orders invariant under abelian group actions
Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a linear preorder ≤ on the powerset 𝓟X invariant under G and such that if A is a proper subset of B, then A < B...