Abstract. We associate a certain lattice with every relational set. We characterize finite irreducible relational sets by the property that their associated lattice leaves a lattice if its top element is removed. This characterization is somewhat dual to that of subdirectly irreducible algebras by their congruence lattices. As a corollary we prove that if the idempotent clone related to a finite relational set $P$ is trivial then $P$ is irreducible. A stronger version of irreducibility is also explored.

AMS Subject Classification (1991): 08C05, 08A05, 08B25

Keyword(s): relational sets, irreducible relational sets, retract, product

Received March 10, 1998 and in final form June 16, 1998. (Registered under 3299/2009.)