The algebraic representation of posets has taken many forms, including incidence-algebras, Stanley-Reisner rings, BCK-algebras, pogroupoids, etc.. In this case the notion of down-up algebras also has its origin in considerations of this type. If \(K\) is a field, then for \(\alpha,\beta,\gamma\in K\), \(A=A(\alpha,\beta,\gamma)\) is the \(K\)-algebra with generators \(d\)(own) and \(u\)(p) and relations \(d^2u=\alpha dud+\beta ud^2+\gamma d\); \(du^2=\alpha udu+\beta u^2d+\gamma u\), whence \((ud)(du)=(du)(ud)\). From the ring-theory point of view, these algebras are interesting in and for themselves serving as examples of a variety of types. Thus \(\beta\neq 0\) is equivalent to demonstrating that: \(A\) is a right (or left) Auslander regular Noetherian domain of global dimension 3 which contains the polynomial ring \(K[ud,du]\) in two generators. In order to accomplish this the authors use some interesting techniques well worth observing for the practising ring theorist and the student ambitious to enter such a fold.