Flatness of projection and separated projection algebras (Q1881607)

Let \({\mathcal {PRO}}\) be a category whose objects are projection algebras (a projection algebra is a pair \((X,\lambda )\) such that \( X\) is a set and \(\lambda :X\times \mathbb{N}^{\infty}\to X\) is a mapping where \( \mathbb{N}^{\infty}\) is the set of all natural numbers with the greatest element \(\infty\) satisfying \(\lambda (x,\infty )=x\) and \(\lambda (x,\min\{ m,n\})=\lambda (\lambda (x,m),n)\) for all \(x\in X\) and all \(m,n\in \mathbb{N}^{\infty}\)) and morphisms are projection morphisms (a mapping \(f:X\to Y\) is a projection morphism from \((X,\lambda )\) into \((Y,\mu )\) if \(f(\lambda (x,n))=\mu (f(x),n)\) for all \( x\in X\) and \(n\in \mathbb{N}^{\infty}\)). A projection algebra \( \)\(\mathbb X=(X,\lambda )\) is called flat if the tensor \(\mathbb X\otimes -:{\mathcal {PRO}}\to {\mathcal {PRO}}\) preserves finite limits. The paper characterizes flat projection algebras in the category \({\mathcal {PRO}}\). It is shown that any subalgebra of a flat projection algebra is flat. We say that a projection algebra \((X,\lambda )\) is separated if \(x=y\) for \(x,y\in X\) whenever \(\lambda (x,n)=\lambda (y,n)\) for all natural numbers \(n\). It is proved that any flat projection algebra is separated.