Vorderings

Vorderings, often referred to simply as orderings in mathematical literature, are binary relations that compare elements of a set in a way that imposes a notion of "before" or "larger than." In many contexts, especially algebra, orderings are required to be compatible with the algebraic structure, producing what are called left-orderings, right-orderings, or bi-orderings on groups.

Formally, a left-ordering on a group G is a total order ≤ on G with the property that a ≤ b implies ca ≤ cb for all a,b,c in G. In other words, left multiplication preserves the order. A right-ordering requires preservation under multiplication on the right, and a bi-ordering requires preservation on both sides. Equivalently, these orders can be described via positive cones: the set P = {g in G : e < g} is a semigroup (closed under multiplication) and G is the disjoint union of P, P^{-1}, and {e}. The existence of such a positive cone characterizes left-orderability.

Examples include the additive group of real numbers with the usual order, which is both left- and right-invariant. Free groups are left-orderable, and braid groups admit left-invariant orders (notably the Dehornoy order). Lexicographic and other concrete orders can yield invariant orders on direct products of groups.

Properties and consequences vary by context, but common themes include the ability to compare elements consistently, to study dynamics of group actions on ordered spaces, and to investigate structural features such as torsion-freeness and archimedean properties. Well-orderings can be considered in related contexts, though compatibility with group operations is not automatic and may require the axiom of choice.

See also: ordered group, left-orderable group, right-ordering, bi-ordering, positive cone, Dehornoy order.