Inf- and Sup-preserving Aggregation Functions

We introduce inf-lattices, sup-lattices and convexity algebras and we define closure or interior operators and congruences in these algebraic structures. We prove that any inf-lattice can be represented as a quotient of a power set lattice with respect to a congruence. Moreover we consider closure operators defined on a poset and we provide a characterization of inf-preserving maps between inf-lattices. Inf-preserving aggregation functions are described.