Tolerances and orderings on semilattices
To everz partiallz ordered set a certain groupoid is assigned. A tolerance on it is defined similarlz as a congruence, onlz the requirement of transitivitz is omitted. Some theorems concerning these tolerances are proved.
In this paper we find a one-to-one correspondence between transitive relations and partial orders. On the basis of this correspondence we deduce the recurrence formula for enumeration of their numbers. We also determine the number of all transitive relations on an arbitrary -element set up to .