Simple and subdirectly irreducible finitely supported Cb-sets

The aim of this paper is to study and characterize simple, subdirectly irreducible, and indecomposable finitely supported Cb-sets. To study and find their relations, we introduce the notion of fix-simple Cb-sets, and completely describe them. Among other things, we show that under some conditions on supports, fix-simple finitely supported Cb-sets are exactly the ones whose congruence lattices are the two or three element chains. Also, it is shown that an infinite finitely supported Cb-set is simple if and only if it is a special fix-simple one. Furthermore, we observe that all the elements of a simple or subdirectly irreducible infinite Cb-set are finitely supported if we have enough elements with finite support.