Every synaptic algebra has the monotone square root property

TL;DR

This paper proves that every synaptic algebra possesses the monotone square root property, meaning the order relation between positive elements is preserved under taking square roots.

Contribution

It establishes that all synaptic algebras have the monotone square root property, generalizing known results from specific algebraic structures.

Findings

Synaptic algebras have the monotone square root property.

Order relations between positive elements are preserved under square roots.

Generalizes the property to a broad class of algebraic structures.

Abstract

A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW*-algebra. In this paper we prove that a synaptic algebra A has the monotone square property, i.e., if a and b are positive elements, then if a is less or equal than b, then the square root of a is less or equal than the square root of b.