Regularity of residuated mappings (Q2277498)

A principal ideal of a poset P is a subset of the form \(J_ x=\{t\in P\); \(t\leq x\}\) for some fixed element \(x\in P\). If P and Q are posets, a mapping f: \(P\to Q\) is said to be residuated if it has the property that for each \(q\in Q\), \(f^{-1}(q)\) is necessarily a principal ideal of P. Let Res(P) denote the monoid of all residual mappings f: \(P\to P\) with composition as the operation. The author studies complete lattices L for which Res(L) is regular. He shows that Res(L) is regular provided L is a complete chain or the horizontal sum of a certain class of complete chains. A complete characterization of lattices L of finite length with regular Res(L) is given.