Finite congruence-distributive P.O. systems
A P.O. system is a set P on which is defined a transitive, antisymmetric (but not necessarily reflexive) relation \(\triangleleft\). In this case, \(\leq =\triangleleft \cup \Delta_ P\) is a partial ordering of P. The structure of P.O. systems is richer than that of the corresponding partially ordered set, in that \(\triangleleft\) singles out a distinguished subset of P, namely, \(\{\) \(s\in P:\) \(s\triangleleft s\}\). P.O. systems arise in the study of certain Boolean algebras, and in this context, there is a natural definition of morphisms between P.O. systems: a mapping f: \(P\to P'\) is a morphism if \(p\triangleleft q\) implies \(f(p)\triangleleft f(q)\), and \(f(p)\triangleleft q'\) implies \(q'=f(q)\) for some q such that \(p\triangleleft q\). In particular, P is a subsystem of \(P'\) if \(P\subseteq P'\) and the inclusion is a morphism, and a congruence on P is the kernel of a morphism. The congruences on P are characterized as equivalences \(\theta\) with convex classes such that if \(p\theta q\triangleleft r\), then \(p\triangleleft s\theta r\) for some \(s\in P\). The partially ordered set Con(P) of all congruences on P is a complete lattice, and P is congruence distributive if Con(P) is a distributive lattice. The main accomplishment of this paper is the construction of an \(\aleph_ 0\)-universal congruence distributive system D(\(\infty)\). Explicitly, D(\(\infty)\) is a P.O. system such that for each \(p\in D(\infty)\), \([p)=\{q\in D(\infty):\) \(p\leq q\}\) is a finite, congruence distributive P.O. system; and every finite, congruence distributive P.O. system is isomorphic to a set of the form [p) for a unique \(p\in D(\infty)\).
scientific article; zbMATH DE number 4175001
scientific article