Extensions of rich words

In [X. Droubay et al, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has at most |w|+1 many distinct palindromic factors, including the empty word. The unified study of words which achieve this limit was initiated in [A. Glen et al, Palindromic richness, Eur. Jour. of Comb. 30 (2009)]. They called these words rich (in palindromes). This article contains several results about rich words and especially extending them. We say that a rich word w can be extended richly with a word u if wu is rich. Some notions are also made about the infinite defect of a word, the number of rich words of length n and two-dimensional rich words.