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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{B_2^1}$\end{document}. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(f_n^{\mathbf{B_2^1}})$\end{document} denote the free spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{B_2^1}$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{B_2^1}$\end{document} is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{B_2^1}$\end{document} if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_n \leq a_n \leq f_n^{\mathbf{B_2^1}} \leq 2^{n}a_{2n}^2$\end{document}. A result of the second author in a companion paper states that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\operatorname{log} \; a_n)\in O(n^3)$\end{document}, from which it follows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\operatorname{log} f_n^{\mathbf{B_2^1}})\in O(n^3)$\end{document} 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 extensions of posets; each realizer of a finite poset determines an extension by an alternation word digraph.