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A partially ordered set P=(X,<=) is an interval order if it is isomorphic to some set of intervals on the real line ordered by left-to-right precedence. Formally, P is an ...

A theorem, also called the iteration theorem, that makes use of the lambda notation introduced by Church. Let phi_x^((k)) denote the recursive function of k variables with ...

A linear extension of a partially ordered set P is a permutation of the elements p_1, p_2, ... of P such that p_i<p_j implies i<j. For example, the linear extensions of the ...

Two totally ordered sets (A,<=) and (B,<=) are order isomorphic iff there is a bijection f from A to B such that for all a_1,a_2 in A, a_1<=a_2 iff f(a_1)<=f(a_2) (Ciesielski ...

Let (L,<=) be any complete lattice. Suppose f:L->L is monotone increasing (or isotone), i.e., for all x,y in L, x<=y implies f(x)<=f(y). Then the set of all fixed points of f ...

A totally ordered set (A,<=) is said to be well ordered (or have a well-founded order) iff every nonempty subset of A has a least element (Ciesielski 1997, p. 38; Moore 1982, ...

If S is any nonempty partially ordered set in which every chain has an upper bound, then S has a maximal element. This statement is equivalent to the axiom of choice. Renteln ...

The Latin prefix quadri- is used to indicate the number 4, for example, quadrilateral, quadrant, etc. However, it also very commonly used to denote objects involving the ...

The substitution of x=w-p/(3w) (1) into the standard form cubic equation x^3+px=q. (2) The result reduces the cubic to the equation w^3-(p^3)/(27w^3)-q=0, (3) which is easily ...

The cubic formula is the closed-form solution for a cubic equation, i.e., the roots of a cubic polynomial. A general cubic equation is of the form z^3+a_2z^2+a_1z+a_0=0 (1) ...