Least upper bound in the context of Well-ordered

In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.

Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order.

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the ordering is then called a well-ordered set (or woset). In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering.

Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements, besides the least element, that have no predecessor (see § Natural numbers below for an example). A well-ordered set S contains for every subset T with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of T in S.

In mathematics, the extended real number system is obtained from the real number system ${\displaystyle \mathbb {R} }$ by adding two elements denoted ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence ${\displaystyle (1,2,\ldots )}$ of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has ${\displaystyle +\infty }$ as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis, the use of ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ as actual limits extends significantly the possible computations. It is the Dedekind–MacNeille completion of the real numbers.

The extended real number system is denoted ${\displaystyle {\overline {\mathbb {R} }}}$, ${\displaystyle [-\infty ,+\infty ]}$, or ${\displaystyle \mathbb {R} \cup \left\{-\infty ,+\infty \right\}}$. When the meaning is clear from context, the symbol ${\displaystyle +\infty }$ is often written simply as ${\displaystyle \infty }$.