Abstract
In order to study operator monoïds on ordered sets we introduce opoïds: an opoïd is a monoïd with an order verifying “ifx≤y, thenax≤ay, for alla”. We show then the existence of opoïds freely generated by generators and relations. The opoïd freely generated by a closure and an interior has exactly seven elements, whose Hasse diagram is given. The opoïd freely generated by two commuting closures and two commuting interiors is infinite. We note some links between some particular opoïds generated by closures and interiors and some others generated by closures and a decreasing involution. We give a description of the opoïd freely generated by a closure and a decreasing involution: it has fourteen elements, its Hasse diagram is given. Weakening the condition “is a closure”, we obtain infinite freely generated opoïds. Finally we examine the different quotients of the opoïd freely generated by a closure and a decreasing involution.
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Communicated by Dominique Perrin
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Garel, E., Olivier, J.P. About the finiteness of monoids generated by closures and interiors. Semigroup Forum 47, 341–352 (1993). https://doi.org/10.1007/BF02573771
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DOI: https://doi.org/10.1007/BF02573771