document: The contiguity in R/M An r.e. degree c is contiguous if deg/sub wtt / -LRB- A -RRB- = deg/sub wtt / -LRB- B -RRB- for any r.e. sets A , B in c . In this paper , we generalize the notation of contiguity to the structure R/M , the upper semilattice of the r.e. degree set R modulo the cappable r.e. degree set M . An element -LSB- c -RSB- in R/M is contiguous if -LSB- deg/sub wtt / -LRB- A -RRB- -RSB- = -LSB- deg/sub wtt / -LRB- B -RRB- -RSB- for any r.e. sets A , B such that deg/sub T / -LRB- A -RRB- , deg/sub T / -LRB- B -RRB- in -LSB- c -RSB- . It is proved in this paper that every nonzero element in R/M is not contiguous , i.e. , for every element -LSB- c -RSB- in R/M , if -LSB- c -RSB- not = -LSB- o -RSB- then there exist at least two r.e. sets A , B such that deg/sub T / -LRB- A -RRB- , deg/sub T / -LRB- B -RRB- in -LSB- c -RSB- and -LSB- deg/sub wtt / -LRB- A -RRB- -RSB- not = -LSB- deg/sub wtt / -LRB- B -RRB- -RSB-
doc_bio_tags: O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: contiguity upper semilattice nonzero element
abstractive_keyphrases: turing degree recursively enumerable set recursion theory