Cores and retracts
An isotone map of an ordered set \(P\) into \(P\) is called a retract if it is idempotent. The set \(I(P)\) of all retracts of \(P\) is pointwise ordered. In this article necessary conditions are given (Proposition 2.1) when a retract \(r\) covers another retract \(s\) of \(P\): either (i) \(r(P) = s(P)\), (ii) \(r(P) = s(P) \cup \{a\}\), for some \(a \in P - s(P)\), or, (iii) \(s(P) = r(P) \cup \{b\}\), for some \(b \in P - r(P)\). In the conclusion (Section 3) the cores of components of \(I(P)\) (maximal connected subsets of \(I(P)\)) are investigated, where a core is defined by means of ``dismantling sequences''. E.g., it is shown (Corollary 3.6) that the cores of a component of retracts have isomorphic images.
scientific article; zbMATH DE number 683156
scientific article