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To prove the existence of a mathematical object that can be viewed as a maximal element in some partially ordered set in some way, one can try proving the existence of such an object by assuming there is no maximal element and using transfinite induction and the assumptions of the situation to get a contradiction. Zorn's lemma tidies up the ...
When the mind is cognizing an object, the object presents itself to the senses or to thought in one of a number of different forms (e.g. a picture is a work of art, a saleable commodity, a representation of a house, etc.). Presentation is thus essentially a cognitive process.
A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.
The philosopher Reinhardt Grossmann has observed that On the Content and Object of Presentations greatly influenced the course of philosophy. The philosopher Alexius Meinong adopted Twardowski's distinction between the individual mental act, its content and its object, and his contention that there are many objects of acts that do not exist.
An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest ...
A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes ...
A nontrivial poset satisfying the descending chain condition is said to have deviation 0. Then, inductively, a poset is said to have deviation at most α (for an ordinal α) if for every descending chain of elements a 0 > a 1 >... all but a finite number of the posets of elements between a n and a n+1 have deviation less than α. The deviation ...
The poset P is a continuous poset if it has some base. Especially, P itself is a base in this situation. In many applications, one restricts to continuous (d)cpos as a main object of study. Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of finite elements. Such a poset is called ...