General mixed lattices

TL;DR

This paper explores the fundamental properties of mixed lattices, a generalization of lattices with two partial orders, establishing their algebraic and order-theoretic equivalence and introducing new definitions.

Contribution

It introduces an alternative definition of mixed lattices and mixed lattice groups, connecting algebraic and order-theoretic perspectives.

Findings

Established properties like associativity, distributivity, and modular laws.

Proved equivalence of algebraic and order-theoretic definitions.

Provided new frameworks for understanding mixed lattice structures.

Abstract

A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and semigroups, while the more general notion of a mixed lattice remains unexplored. In this paper, we study the fundamental properties of mixed lattices and the relationships between the various properties, such as the one-sided associative, distributive and modular laws. We also give an alternative definition of mixed lattices and mixed lattice groups as non-commutative and non-associative algebras satisfying a certain set of postulates. The algebraic and the order-theoretic definitions are then shown to be equivalent.