A Non-Boolean Version of Feferman-Vaught's Theorem

✍ Scribed by R. Lavendhomme; Th. Lucas

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 623 KB
• Volume: 31
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19850311904

✦ Synopsis

COMER 11.1. MAKSFIELD [BJ) has lead t o consider t'liat theorem as giving a relation between global sections of a sheaf and its local propert'ies. The aim of t>liis paper is to shon-that similar relations hold for cert'ain sheaves. to be called liere "niellon-". over other types of spaces. Our niain theorem: theorem 1 , describing that rrtatiori in Fefermari-~au~lit's style comes from our analysis 1 4 1 of MACINTYRE'S [5] : it is essentially a c o r n h a t i o n of some aspects of MACINTYRE'S proof with -'mellowness ", a natural logical cxtemioii of softness. It is not excluded that theorem 1 can be interpreted as giving sufficient conditions to reduce certain noii-Boolean situations to Boolean ones, I t is interesting however to give a direct description uf a connection hetn-ren -'local * ' and ~~p l o h a l ' ~ i n noii-Boolean cases and t.lie examples of 5 3 show that our theorein properl>-rsteiids the Boolean case without isolated points. In coniparison v.itli [S]. it should I)? eiiipliasized that although 1c.e restrict our attention t o regular formulas (thus giving rise to a Boolean-valued structure). no finite completeness i h assunied here. and the partition argument is replaced by a n ext>ension argument l x ~~l on mcllowiPss. \Ye also show in 3 3.2 lion; to turn a sheaf of struct,urea over a metriza1)le space into a mcllon. one over the same space. Possible refinements arc suggested in the la,st paragraph.