monounary algebra; 2-homogeneous; 2-set-homogeneous; partially-2-homogeneous; partially-2-set-homogeneous
This paper is a continuation of , where $k$-homogeneous and $k$-set-homogeneous algebras were defined. The definitions are analogous to those introduced by Fraïssé  and Droste, Giraudet, Macpherson, Sauer  for relational structures. In  we found all 2-homogeneous and all 2-set-homogeneous monounary algebras when the homogenity is considered with respect to subalgebras, to connected subalgebras and with respect to connected partial subalgebras, respectively. The results of , where all homogeneous monounary algebras were characterized, were applied in  for 1-homogeneity. The aim of the present paper is to describe all monounary algebras which are 2-homogeneous and 2-set-homogeneous with respect to partial subalgebras, respectively; we will say that they are partially-2-homogeneous and partially-2-set-homogeneous.
 R. Fraïssé: Theory of Relations
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 D. Jakubíková-Studenovská: On homogeneous and 1-homogeneous monounary algebras
. Contributions to General Algebra 12. Proceedings of the Wien Conference, June 1999, Verlag J. Heyn, 2000, pp. 221–224. MR 1777661