Local completeness
by
I.G. Rosenberg
Mathematiques statistiques, Universite de Montreal
Coauthors: D. Schweigert (U. Kaiserslautern)

A clone C on an infinite universe A is local if it contains every operation f on A such that each restriction of f to a finite subset of A extends to an operation of C. An algbera on A is locally complete if the clone O of all operationson A is the only local clone containing the operations of the algebra. For a local completeness criterion we search for the locally maximal clones and the increasing chains of proper local clones whose union is locally complete, called towers. The knowledge of both locally maximal clones and towers could be used to classify local clones.

To complete the search it remains to shift through the following 5 families of locally bounded relations on A: 1) graphs, 2) reflexive digraphs of diameter 2, 3) paraorders, 4) ternary relations containing (a,a,b) for all a,b in A and some triples with distinct coordinates and 5) the totally reflexive and totally symmetric h-ary incomplete relations ( h = 1,2,... ). We present partial results for all the 5 families.

Date received: June 8, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caee-93.