Search results
Results from the WOW.Com Content Network
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.
In mathematics, specifically category theory, a posetal category, or thin category, [1] is a category whose homsets each contain at most one morphism. [2] As such, a posetal category amounts to a preordered class (or a preordered set, if its objects form a set).
A presentation on information flow in living systems. Living systems are life forms (or, more colloquially known as living things) treated as a system. They are said to be open self-organizing and said to interact with their environment. These systems are maintained by flows of information, energy and matter. Multiple theories of living systems ...
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 ...
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.
For example, the ideal completion of a given partial order P is the set of all ideals of P ordered by subset inclusion. This construction yields the free dcpo generated by P . An ideal is principal if and only if it is compact in the ideal completion, so the original poset can be recovered as the sub-poset consisting of compact elements.
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.
[5] [6] [7] This is complicated by a lack of knowledge of the characteristics of living entities, if any, that may have developed outside Earth. [8] [9] Philosophical definitions of life have also been put forward, with similar difficulties on how to distinguish living things from the non-living. [10]