Word problem of the Perkins semigroup via directed acyclic graphs
Kitaev, Sergey and Seif, Steven (2008) Word problem of the Perkins semigroup via directed acyclic graphs. Order, 25 (3). pp. 177194.
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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 Perkinsmonoid B½. Let ( f B½ n ) denote the free spectrum of B½, 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 B½ is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over B½ 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 B½ n ≤ 2na2 2n. A result of the second author in a companion paper states that (log an) ∈ O(n3), from which it follows that (log f B½ 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 coNPcomplete. 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.
ORCID iDs
Kitaev, Sergey ORCID: https://orcid.org/0000000333241647 and Seif, Steven;

Item type: Article ID code: 34552 Dates: DateEventAugust 2008Published30 August 2008Published OnlineKeywords: directed acyclic graph, partially order set, poset, Perkins semigroup, free semigroup, word problem, computational complexity, Probabilities. Mathematical statistics, Computational Theory and Mathematics, Algebra and Number Theory, Geometry and Topology Subjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Computer and Information Sciences Depositing user: Pure Administrator Date deposited: 13 Oct 2011 19:20 Last modified: 20 Jan 2021 19:41 URI: https://strathprints.strath.ac.uk/id/eprint/34552