On a New Construction of Pseudo Effect Algebras

TL;DR

This paper introduces kite pseudo effect algebras, a new class linked to generalized pseudo effect algebras without a guaranteed greatest element, expanding the algebraic framework and exploring their properties.

Contribution

It defines kite pseudo effect algebras and investigates their connection with Riesz Decomposition Properties and normal ideals, broadening the understanding of pseudo effect algebra structures.

Findings

Kite pseudo effect algebras can be non-commutative even from commutative starting points.

Conditions for the existence of the least non-trivial normal ideal are established.

Connections with various Riesz Decomposition Properties are demonstrated.

Abstract

We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which is connected not necessarily with partially ordered groups, but rather with generalized pseudo effect algebras where the greatest element is not guaranteed. Starting even with a commutative generalized pseudo effect algebra, we can obtain a non-commutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz Decomposition Properties. We find conditions when kite pseudo effect algebras have the least non-trivial normal ideal.