Algorithms for Producing and Ordering Lexical and Nonlexical Sequences out of one Element

This paper deals with algorithms for producing and ordering lexical and nonlexical sequences of a given degree. The notion of ”elementary operations” on positive α-sequences is introduced. Our main theorem answers the question of when two lexical sequences are adjacent. Given any lexical sequence, α ∈ Ln, we can produce its adjacent successor as follows; apply one elementary operation on the tail of the longest left sequence, of even length, which gives a lexical successor α′ ∈ Ln, then compute the fundamental sequence f = α ∧ α′ ∈ Lm and conclude for m ∤ n that α is adjacent to α′ in Ln. Whereas for m | n , the sequence α is adjacent to a sequence generated by f and the least element of Ld, where d = n m . Thus, while the longest right sequence controls the lexicality property of an α-sequence, it turns out that the longest left sequence controls the adjacency property of lexical and nonlexical sequences.