SUMS OF DOUBLE SYSTEMS OF PARTIALLY ORDERED SETS
Margret Hesse Hölft · Demonstratio Mathematica · 1983
Suds of doable systems of abstract algebras and lattices were first introduced in [1] and [2] respectively.They are derived from Plonka.systems and their sums as defined in [5J, in fact, Plonka systems of abstract algebras are special double systems.In [3] the concept of a double system of lattices is reexamined and with a change in the original definition of such a system it becomes possible to represent any lattice in a very natural way as the sum of the double system of its congruence classes with respect to a given congruence relation (Theorem I and II of [3j).The approach in [3] is algebraic rather than order-theoretic.This paper focuses on the order-theoretic properties of double systems of lattices and extends the concept of a double system to arbitrary partially ordered sets.The main result here is similar to the one for lattices in [3j.It will be possible to repres-ent a partially ordered set as the sum of the double system of its equivalence classes with respect to a given acyclic equivalence relation (Theorem 2.1 and Theorem 2.2).The results of [3] for lattices will be discussed as special cases of the results for partially ordered sets (section 3). Acyclic equivalence relations on partially ordered setsIf 1 is a lattice and 6 a congruence relation on L, then the partial order on the quotient lattice L/6, whose elements are the congruence classes of L modulo 6, can be -229 -