n-closure systems and n-closure operators

It is very well known and permeating the whole of mathematics that a closure operator on a given set gives rise to a closure system, whose constituent sets form a complete lattice under inclusion, and vice-versa. Recent work of Wille on triadic concept analysis and subsequent work by the author on polyadic concept analysis led to the introduction of complete trilattices and complete n-lattices, respectively, that generalize complete lattices and capture the order-theoretic structure of the collection of concepts associated with polyadic formal contexts. In the present paper, polyadic closure operators and polyadic closure systems are introduced and they are shown to be in a relationship similar to the one that exists between ordinary (dyadic) closure operators and ordinary (dyadic) closure systems. Finally, the algebraic case is given some special consideration. 1. Background: Closure operators and n-ordered Sets This section contains a brief account of the well-known correspondence between closure operators and closure systems and of the notion of an n-ordered set. Our main source for the former is [3] (see also [2]) and for the latter [10] (see also [9] for the triadic case). Given a set X, a family L of subsets of X, such that • X ∈ L and • {Ai : i ∈ I} ⊆ L implies ⋂ i∈I Ai ∈ L, is said to be a closure system or a topped intersection structure and it forms a complete lattice under inclusion. The meet and join, respectively, are given by ∧