latticeordered

Lattice-ordered, often written lattice-ordered group or l-group, is a mathematical structure that combines group and lattice order properties. It consists of a group G equipped with a partial order ≤ that makes G into a lattice, i.e., any two elements have a meet (infimum) and a join (supremum). The order is compatible with the group operation: for all a, b, c in G, if a ≤ b then a + c ≤ b + c (and, in the non-abelian case, left and right translations preserve the order as well). When the underlying group is abelian, the structure is typically called a lattice-ordered abelian group.

Key features include the positive cone G+ = {g ∈ G : g ≥ e}, which is closed under the group operation, and every element can be expressed as a difference of two elements of G+: a = a+ − a− with a+ = a ∨ e and a− = (−a) ∨ e. The lattice operations provide pointwise definitions of meet and join, and the lattice order interacts with the algebraic operations in a way that generalizes familiar ordered number systems.

Common examples are the real numbers under addition with the usual order, the integers, and spaces of real-valued functions on a set with pointwise addition and order. More generally, any totally ordered abelian group is a lattice-ordered group, and function spaces or other ordered algebraic structures often yield l-groups.

Related concepts extend to lattice-ordered rings and fields, and to vector-lattice or Riesz spaces when compatible scalar multiplication is present. Lattice-ordered groups play a role in areas such as functional analysis, measure theory, and the study of ordered algebraic systems.