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A continuous lattice is a complete lattice that is continuous as a poset. An algebraic lattice is a complete lattice that is algebraic as a poset. Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities.
In mathematics, a differential poset is a partially ordered set (or poset for short) satisfying certain local properties. (The formal definition is given below.) This family of posets was introduced by Stanley (1988) as a generalization of Young's lattice (the poset of integer partitions ordered by inclusion), many of whose combinatorial properties are shared by all differential posets.
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms ...
A poset D is said to be a directed complete poset, or dcpo, if every directed subset of D has a supremum. Distributive. A lattice L is called distributive if, for all x, y, and z in L, we find that x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). This condition is known to be equivalent to its order dual.
An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler . Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics , such as various restrictions on f -vectors of convex simplicial polytopes ...
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n.This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank.