The poset of retracts of a free monoid. (English) Zbl 0723.68060

Summary: The set of retracts of a free monoid F with the partial order of inclusion is investigated. This poset is a lattice if and only if F is generated by three or fewer elements. For a finite generated free monoid F it is shown non-constructively that, for every submonoid S of F, the intersection of all retracts of F containing S is regular. A regular expressions can be constructed for this intersection when S is regular. The submonoid generated by the set of all retracts of F contained in the regular submonoid S is also regular and constructable. This allows the decision to be made whether or not any given pair of retracts has a supremum or an infimum in the poset of retracts of F. The procedure yields regular expressions for such suprema and infima when they exist.


68Q45 Formal languages and automata
Full Text: DOI


[1] DOI: 10.1016/S0019-9958(77)90050-X · Zbl 0375.94022
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