Partially ordered sets 论文

摘要

if Postulates 1.1 and 1.2 hold for any elements of K.1.1 Postulate, a c b and bee imply acc.1.2 Postulate, aca.Such a set need not possess simple or linear order and for this reason is commonly called a partially ordered set.J A large part of the theory of simply ordered sets applies, with little or no change, to partially ordered sets.Although the principal objectives of the paper are certain properties of partial order which have either a trivial or no counterpart in the theory of simple order, considerable use is made of the parallelism between the two theories without a systematic development of it.This parallelism makes it natural to call any set which satisfies Postulates 1.1 and 1.2 an ordered set, dropping the word "partially."This convention leads to no confusion in this paper as but scant reference is made to simple order, a c b is read a is contained in b.The relations of equality and equivalence order any set in which they occur.Being a subset of is a relation ordering the subsets of a given set.The relation less than or equal to orders the set of real numbers.A set of propositions is ordered by the relation of implication.An ordering relation can be assigned to any system, such as a Boolean algebra, which has an associative operation with respect to which every element is idempotent.The relations of being homomorphic to and being a subsystem of order any aggregate of classes with a common set of operations.These examples do not begin to exhaust even the