Semirigid sets of diamond orders

An order relation ⩽ab on a set A is a diamond provided x ⩽aby 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 …