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We study the algebraic properties of binary relations whose underlyingdigraph is smooth, that is has no source or sink. Such objects have beenstudied as surjective hyper-operations (shops) on the corresponding vertex set,and as binary relations that are defined everywhere and whose inverse is alsodefined everywhere. In the latter formulation, they have been calledmultipermutations. We study the lattice structure of sets (monoids) ofmultipermutations over an n-element domain. Through a Galois connection, thesemonoids form the algebraic counterparts to sets of relations closed underdefinability in positive first-order logic without equality. The first side ofthis Galois connection has been elaborated previously, we show the other sidehere. We study the property of inverse on multipermutations and how it connectsour monoids to groups. We use our results to give a simple dichotomy theoremfor the evaluation problem of positive first-order logic without equality onthe class of structures whose preserving multipermutations form a monoid closedunder inverse. These problems turn out either to be in Logspace or to bePspace-complete. We go on to study the monoid of all multipermutations on ann-element domain, under usual composition of relations. We characterise itsGreen relations, regular elements and show that it does not admit a generatingset that is polynomial on n.

The lattice and semigroup structure of multipermutations