Quasi-inverse monoids
Résumé
The notion of quasi-recognizability has been defined in a former work as a remedy to the collapse of standard algebraic recognizability on languages of one-dimensional overlapping tiles. There quasi-recognizability makes an essential use of a peculiar class of ordered monoids. Our purpose here is to provide a presentation and a study of these monoids. Called well-behaved ordered monoids in that former work, they can also be seen as monoids S equipped with two unary mappings x → xL and x → xR such that, if S is an inverse monoid then xL just equals x−1x and xR just equals xx−1, and if S is not an inverse monoid then, as much as possible, both xL and xR still behave the same way. These quasi-inverse monoids are thus presented and studied as such in this paper. It is shown in particular that quasi-inverse monoids are Lawson's U-semiadequate monoids and that stable quasi-inverse monoids are well-behaved ordered monoids. The link between quasi-inverse monoids and prehomomorphisms is also studied a little further. We show in particular that the extension of arbitrary monoid S into a quasi-inverse monoid Q(S), used in former work, is actually an expansion in the sense of Birget and Rhodes in the category of (ordered monoid) prehomomorphisms instead of (monoid) homomorphisms.
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