Does every partially ordered set have a maximal element?

Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. Zorn’s lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element.

Can you order an infinite set?

If an infinite set is a well-ordered set, then it must have a nonempty, nontrivial subset that has no greatest element. In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself.

Can a partially ordered set be infinite?

In this paper a partially ordered set (poset) means always an arbitrary, finite or infinite one, except it is stated otherwise.

What is difference between maximal and maximum?

Originally Answered: What is the difference between maximum and maximal? There is a slight difference. Maximal is an adjective meaning the highest or greatest possible. Maximum can be an adjective meaning as great, high or intense as possible or permitted or a noun or adverb.

How do you find maximal and minimal elements?

Definition An element x of a poset is called a maximal element if there is no element z such that x < z. Definition An element x of a poset is called a minimal element if there is no element z such that x > z. ▶ For sets with inclusion, the empty set ∅ = {} is a minimal element.

Is an infinite set countable?

An infinite set is called countable if you can count it. In other words, it’s called countable if you can put its members into one-to-one correspondence with the natural numbers 1, 2, 3, .

What is the difference between maximum and maximal?

Maximum is the greatest element of a set. Maximal is an element of a subset in a partially ordered set, such that there is no other element larger in the subset.

How do you define an ordered set?

An ordered set is a relational structure (S,⪯) such that the relation ⪯ is an ordering. Such a structure may be: A partially ordered set (poset) A totally ordered set (toset)

What is the least element of a set?

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S.

What is maximal no?

An element is a maximum if it is larger than every single element in the set, whereas an element is maximal if it is not smaller than any other element in the set (where “smaller” is determined by the partial order ≤ ).