Cores and retracts
type: Item
instance of: scholarly article
title: Cores and retracts
zbMATH Open document ID: 0814.06002
DOI: 10.1007/BF01462224
published in: Order
publication date: 1994-11-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 683156
zbMATH Keywords: maximal connected subsets
zbMATH Keywords: retract
zbMATH Keywords: cores of components
author: Micheale Susan Roddy
reviewed by: Ladislav Skula
MaRDI profile type: Publication
cites work: Q3356339
cites work: Q3853684
cites work: Q3853685
cites work: A structure theory for ordered sets
cites work: Retracts and the Fixed Point Problem for Finite Partially Ordered Sets
cites work: Fixed points of products and the strong fixed point property
cites work: The uniqueness of the core
cites work: On the strong fixed point property
cites work: The strong fixed point property for small sets
cites work: A fixed point theorem for finite partially orderes sets
cites work: Q3952142
cites work: Isotone relations and the fixed point property for posets
description: scientific article; zbMATH DE number 683156