least upper bound

theory (lub or "join", "supremum") The least upper bound of

two elements a and b is an upper bound c such that a #@= c and

b #@= c and if there is any other upper bound c' then c #@= c'.

The least upper bound of a set S is the smallest b such that

for all s in S, s #@= b. The lub of mutually comparable

elements is their maximum but in the presence of incomparable

elements, if the lub exists, it will be some other element

greater than all of them.

Lub is the dual to greatest lower bound.

(In LaTeX, "#@=" is written as sqsubseteq, the lub of two

elements a and b is written a sqcup b, and the lub of set S

is written as bigsqcup S).

(1995-02-03)