Inverse Monoids: Decidability and Complexity of Algebraic Questions

• Published in: Lecture notes in computer science pp. 664 - 675
• Main Authors: Lohrey, Markus, Ondrusch, Nicole
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2005; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Theoretical computing; Decidability; Computer theory; Free monoid; Regular language; Cayley graph; Second order theory; Monadic logic; Inverse problem; Polynomial time
• ISBN: 9783540287025; 3540287027
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11549345_57

The word problem for inverse monoids generated by a set Γ subject to relations of the form e = f, where e and f are both idempotents in the free inverse monoid generated by Γ, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, it is shown that the Cayley-graph of the free inverse monoid has an undecidable monadic second-order theory.