TY - JOUR
AU1 - Banaschewski, B.
AU2 - Bruns, G.
AB - VoI.XVIII, 1967 369 ~y B. BAlq*ASCHE~,VSKI ~nd G. B~u~,'s Introduction. The MacNeille completion of a partially ordered set P was first introduced by means of a particular construction which generalizes DEDEKIND'S con- struction of the totally ordered set of all real numbers from the rationals [2], [7]. Only later, characterizations were given in terms of order theoretic properties, determining the MacNeille completion of P up to isomorphism over P as an extension of p with specific properties [1], [3]. A natural problem arising in this context is that of describing the MacNeille completion in the much more confined language of order preserving mappings, i.e., in categorical terms. This we deal with in the present note, both, for the category of partially ordered sets and order preserving mappings, and for the category of Boolean lattices and Boolean homomorphisms. One of the first problems concerning categories of concrete mathematical objects of the "stl~actured set" type is to find a categorical description of the naturally given rnorphisms from subobjects to objects. In the case of partially ordered sets, GROTrIEN- a)IEcK's notion of strict monomorphism [5] provides the required description, and from this, a suitable categorical notion of essential extension furnishes the desired 
TI - Categorical characterization of the MacNeille completion
JF - Archiv der Mathematik
DO - 10.1007/BF01898828
DA - 2005-06-19
UR - https://www.deepdyve.com/lp/springer-journals/categorical-characterization-of-the-macneille-completion-5Sjq4aaaON
SP - 369
EP - 377
VL - 18
IS - 4
DP - DeepDyve
ER -