Word Problem of the Perkins Semigroup via Directed Acyclic Graphs

For a word w in an alphabet Γ, the alternation word digraph Alt(w), a certain directed acyclic graph associated with w, is presented as a means to analyze the free spectrum of the Perkins monoid B2. Let (f B2 n ) denote the free spectrum of B2, let an be the number of distinct alternation word digraphs on words whose alphabet is contained in {x1, . . . , xn}, and let pn denote the number of distinct labeled posets on {1, . . . , n}. The word problem for the Perkins semigroup B2 is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over B2 if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: pn ≤ an ≤ f 1 2 n ≤ 2a2n. A result of the second author in a companion paper states that (log an) ∈ O(n3), from which it follows that (log f 1 2 n ) ∈ O(n3) as well. Alternation word digraphs are of independent interest combinatorially. It is shown here that the computational complexity problem that has as instance {u, v} where u, v are words of finite length, and question “Is Alt(u) = Alt(v)?”, is co-NP-complete. Additionally, alternation word digraphs are acyclic, and certain of them are natural ∗Institute of Mathematics, Reykjav́ık University, Ofanleiti 2, IS-103 Reykjav́ık, Iceland; e-mail: [email protected] †Mathematics Department, University of Louisville, Louisville KY 40292, USA; e-mail: [email protected] ‡The authors thank the Mathematics Department at the University of Kentucky for its support of both authors during the preliminary stages of this work.