Minimal element in the context of Transitive closure

⭐ Core Definition: Minimal element

In mathematics, especially in order theory, a maximal element of a subset $S$ of some preordered set is an element of $S$ that is not smaller than any other element in $S$ . A minimal element of a subset $S$ of some preordered set is defined dually as an element of $S$ that is not greater than any other element in $S$ .

The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset $S$ of a preordered set is an element of $S$ which is greater than or equal to any other element of $S,$ and the minimum of $S$ is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.

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👉 Minimal element in the context of Transitive closure

In mathematics, the transitive closure $R$ of a homogeneous binary relation $R$ on a set $X$ is the smallest relation on $X$ that contains $R$ and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets $R$ is the unique minimal transitive superset of $R$ .

For example, if $X$ is a set of airports and $x R y$ means "there is a direct flight from airport $x$ to airport $y$ " (for $x$ and $y$ in $X$ ), then the transitive closure of $R$ on $X$ is the relation $R$ such that $x R y$ means "it is possible to fly from $x$ to $y$ in one or more flights".

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⭐ Core Definition: Minimal element
👉 Minimal element in the context of Transitive closure
Minimal element in the context of Well-founded
Minimal element in the context of Tree (set theory)

Minimal element in the context of Well-founded

In mathematics, a binary relation $R$ is called well-founded (or wellfounded or foundational) on a set or, more generally, a class $X$ if every non-empty subset (or subclass) $S \subseteq X$ has a minimal element with respect to $R$ ; that is, there exists an $m \in S$ such that, for every $s \in S$ , one does not have $s R m$ . More formally, a relation is well-founded if: $(\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].$ Some authors include an extra condition that $R$ is set-like, i.e., that the elements less than any given element form a set.

Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence $x 0, x 1, x 2, ...$ of elements of $X$ such that $x n +1 R x n$ for every natural number $n$ .

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Minimal element in the context of Tree (set theory)

In set theory, a tree is a partially ordered set $(T,<)$ such that for each $t\in T$ , the set $\{s\in T:s<t\}$ is well-ordered by the relation $<$ . Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.