monotoniset

Monotoniset is a term that appears only rarely in mathematical literature and does not have a universally accepted definition. In practice, it is sometimes used as a neologism to refer to objects that exhibit a monotone, or order-preserving, behavior with respect to some underlying order. Because there is no standard usage, the meaning of monotoniset depends on the author and context.

One common interpretation encountered in informal discussions is that a monotoniset refers to an upward-closed subset (an up-set) of a partially ordered set. If (P, ≤) is a poset, a subset S ⊆ P is upward closed if x ∈ S and x ≤ y imply y ∈ S. Such sets are sometimes described in standard terminology as up-sets or filters, and in some texts they may be referred to, informally, as monotone sets. This usage emphasizes the preservation of order under extension.

A second interpretation frames a monotoniset as a monotone family of subsets with respect to inclusion. Let X be a fixed set and F ⊆ 2^X a family of subsets. F is monotone if whenever A ∈ F and A ⊆ B ⊆ X, we have B ∈ F. In this sense, a monotoniset corresponds to an upward-closed family under the inclusion order on subsets. Examples include the family of all supersets of a fixed subset of X or any family defined by a monotone property.

Because the term is not standardized, it is essential to consult the specific source for the intended definition. See also up-sets (or filters), monotone functions, and the general concept of monotonicity in order and lattice theory.