, Volume 25, Issue 3, pp 177–194 | Cite as

Word Problem of the Perkins Semigroup via Directed Acyclic Graphs

  • Sergey Kitaev
  • Steve Seif


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 \(\mathbf{B_2^1}\). Let \((f_n^{\mathbf{B_2^1}})\) denote the free spectrum of \(\mathbf{B_2^1}\), let a n be the number of distinct alternation word digraphs on words whose alphabet is contained in {x 1,..., x n }, and let p n denote the number of distinct labeled posets on {1,..., n}. The word problem for the Perkins semigroup \(\mathbf{B_2^1}\) is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over \(\mathbf{B_2^1}\) if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: \(p_n \leq a_n \leq f_n^{\mathbf{B_2^1}} \leq 2^{n}a_{2n}^2\). A result of the second author in a companion paper states that \((\operatorname{log} \; a_n)\in O(n^3)\), from which it follows that \((\operatorname{log} f_n^{\mathbf{B_2^1}})\in O(n^3)\) 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.


Directed acyclic graph Partially order set Poset Perkins semigroup Free semigroup Word problem Computational complexity \(\mathbf{B_2^1}\) 


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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institute of MathematicsReykjavík UniversityReykjavíkIceland
  2. 2.Mathematics DepartmentUniversity of LouisvilleLouisvilleUSA

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