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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.
The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is vacuously true. However, its opposite poset has deviation 1. Let k be an algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in one variable. Since the deviation of this poset is the ...
2. An inductive definition is a definition that specifies how to construct members of a set based on members already known to be in the set, often used for defining recursively defined sequences, functions, and structures. 3. A poset is called inductive if every non-empty ordered subset has an upper bound infinity axiom See Axiom of infinity.
Base.See continuous poset.; Binary relation.A binary relation over two sets is a subset of their Cartesian product.; Boolean algebra.A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.
Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value. [1] [2] Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram.
In mathematics, a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is a graded poset, or; a poset with the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or; a poset in which all maximal chains have the same ...
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.
In the context of proofs, this phrase is often seen in induction arguments when passing from the base case to the induction step, and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence. necessary and sufficient