Partially ordered semigroups and groups with two mixed partial orderings

TL;DR

This paper explores the structure of partially ordered semigroups with two mixed partial orderings, their connection to mixed lattice groups, and properties like Archimedean orderings and finite order elements.

Contribution

It introduces and analyzes various types of partially ordered semigroups with dual mixed orderings and their relationship to mixed lattice groups, including properties of Archimedean groups.

Findings

Finite order elements cannot exist in a broad class of Archimedean mixed lattice groups.

An example of a non-Archimedean mixed lattice group with finite order elements is provided.

The study generalizes lattice operations to more complex ordered algebraic structures.

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.