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

Abstract

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 ∈ L _n , 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′ ∈ L _n , then compute the fundamental sequence ${f = \alpha \wedge \alpha\prime \in L_{m}}$ and conclude for ${m \nmid n}$ that α is adjacent to α′ in L _n . Whereas for m | n, the sequence α is adjacent to a sequence by f and the least element of L _d , where ${{d = \frac{n}{m}}}$. Thus, while right sequences control the lexicality property of an α-sequence, it turns out that left sequences control the adjacency property of lexical and nonlexical sequences.