Abstract
The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aîzenštat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.
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Communicated by John M. Howie
Supported by an Australian Postgraduate Research Award and the University of Sydney.
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Solomon, A. Catalan monoids, monoids of local endomorphisms, and their presentations. Semigroup Forum 53, 351–368 (1996). https://doi.org/10.1007/BF02574150
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DOI: https://doi.org/10.1007/BF02574150