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We study these algebras for their own, show that they carry bidendriform structures and establish freeness properties, study their generators, dimensions, bases, and also feature their relations to the internal structure of shuffle algebras. As an application of these ideas, we give a new proof of Chapoton's rigidity theorem for shuffle bialgebras.<\/jats:p>","DOI":"10.1142\/s0218196713400183","type":"journal-article","created":{"date-parts":[[2013,4,23]],"date-time":"2013-04-23T07:27:44Z","timestamp":1366702064000},"page":"989-1009","source":"Crossref","is-referenced-by-count":15,"title":["NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS"],"prefix":"10.1142","volume":"23","author":[{"given":"LO\u00cfC","family":"FOISSY","sequence":"first","affiliation":[{"name":"Laboratoire de Math\u00e9matiques, Universit\u00e9 de Reims, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France"}]},{"given":"FR\u00c9D\u00c9RIC","family":"PATRAS","sequence":"additional","affiliation":[{"name":"UMR 6621 CNRS, Universit\u00e9 de Nice, Parc Valrose, 06108 Nice Cedex 02, France"}]}],"member":"219","published-online":{"date-parts":[[2013,6,18]]},"reference":[{"key":"rf1","unstructured":"M.\u00a0Aguiar, Hopf Algebras, Lecture Notes in Pure and Applied Mathematics\u00a0237 (CRC Press, 2004)\u00a0pp. 1\u201333."},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-07-04237-7"},{"key":"rf3","first-page":"471","volume":"165","author":"Bloh A.","year":"1965","journal-title":"Dokl. 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