Partially ordered semigroups and groups with two mixed partial orderings
Jani Jokela
TL;DR
The paper extends the theory of mixed lattices to semigroups and groups with two mixed partial orderings, formalizing A- and B-structures and their weak and strong lattice variants. It establishes translation-invariance properties of mixed envelopes, provides several equivalent characterizations via minimum/maximum envelopes, and connects semigroup structures to mixed lattice groups through substructure and subgroup results. A detailed taxonomy of regularity classes is developed, clarifying when substructures yield mixed lattice subgroups and how Archimedean properties constrain finite-order elements. The results offer a cohesive framework for analyzing ordered algebraic systems with dual, potentially non-commutative orderings and highlight the necessity of Archimedean assumptions in controlling order and torsion phenomena.
Abstract
A mixed lattice is a partially ordered set with two mixed partial orderings that are linked by asymmetric upper and lower envelopes. These notions generalize the join and meet operations of a lattice. In the present paper, we study different types of partially ordered semigroups with two mixed orderings, and investigate their relationship to subsemigroups of mixed lattice groups, which are partially ordered groups with a similar order structure. We also consider Archimedean orderings, and we show that elements of finite order cannot exist in a rather general class of Archimedean mixed lattice groups. Moreover, we give an example of a non-Archimedean mixed lattice group that contains an element of finite order.