An order relation ~<~b on a set A is a diamond provided x<~,,hy holds exactly if x = a or y = b. A set R of diamonds on A is semirigid if the identity map on A and all constant self-maps of A are the only self-maps of A that are (jointly) isotone for all diamonds from R. The study of such sets is motivated by the classification of bases in multiple-valued logics. We give a simple semirigidity criterion. For A finite we describe all semirigid sets of diamonds of the least possible cardinality and give their number. We also give nonsemirigid sets of diamonds of the maximum possible cardinality. We find the total number of semirigid sets of diamonds and their ratio among all sets of diamonds. This ratio converges fast to 1; e.g. for a 26-element set A the probability that a randomly chosen set of diamonds is semirigid is 0.999 9992...
I. Preliminaries and a criterion
Let A be a set and p a binary relation on
Notice that the identity self-map ida on A is always an endomorphism of p; and if (a, a) E p then the constant self-map ca (with value a) is also an endomorphism of p. Denote End p the set of all endomorphisms of p. For a set R of binary relations on A set End R := NpcR End p. The set R is semirigid if EndR := {ida} t J {ca :a E A}.
Observe that a semirigid set consists of reflexive relations.