properties of certain monotone functions

In the definitions of some partially ordered algebraic systems such as po-groups and po-rings, the multiplicationPlanetmathPlanetmath is set to be compatible with the partial ordering on the universePlanetmathPlanetmath in the following sense:

abaciffbc   and   abcbiffac

This is no coincidence. In fact, these “definitions” are actually consequences of properties concerning monotone functions satisfying certain algebraic rules, which is the focus of this entry.

Recall that an n-ary function f on a set A is said to be monotone if it is monotone in each of its variables. In other words, for every i=1,2,,n, the function f(a1,,ai-1,x,ai+1,,an) is monotone in x, where each of the aj is a fixed but arbitrary element of A. We use the notation ,, to denote the monotonicity of each variable in f. For example, (,,) denotes a ternary isotone function, whereas (,) denotes a binary function which is antitone with respect to its first variable, and both isotone/antitone with respect to the second.

Proposition 1.

Let f be an n-ary commutativePlanetmathPlanetmathPlanetmath monotone operationMathworldPlanetmath on a set A. Then f is either isotone or antitone.


Suppose f is isotone (or antitone) in its first variable. Since f(x,a1,,an-1)=f(a1,x,,an-1)==f(a1,,an-1,x), f is isotone (or antitone) in each of its remaining variables. ∎

Proposition 2.

Let f be an n-ary monotone operation on a set A with an identity elementMathworldPlanetmath eA. In other words, f(x,e,,e)=f(e,x,,e)==f(e,e,,x)=x. Then f is either strictly isotone or strictly antitone.


The proof is the same as the one before. Furthermore, if f is isotone and a<b, then f(a,e,,e)=a<b=f(b,e,,e), so the strict ordering is preserved. The same holds true if f is antitone. ∎

Proposition 3.

Let f be a binary monotone operation on a set A such that it is isotone (antitone) with respect to its first variable. Suppose g is a unary operation on A such that f(x,g(x)) is a fixed element of A. Then g is antitone (isotone).

Proposition 4.

Let f be an n-ary associative monotone operation on a set A. Then

  • f is isotone if n is even

  • f is either isotone, or is (,,m,,,,m), if n is odd, say n=2m+1.


Suppose first that n=2m, im, and g(x)=f(a1,,ai-1,x,,am+1,,a2m) is antitone. Then g(g(x)) is isotone. By the associativity of f, g(g(x)) is

= f(a1,,ai-1,a1,,ai-1,x,ai+1,,f(a2m-i+1,,a2m,ai+1,,am+1,,a2m)).

In the second expression, the position of x is 2i-22m-1<2m, therefore implying that g(g(x)) is antitone, which is a contradictionMathworldPlanetmathPlanetmath! Therefore, g(x) is isotone. Now, if i>m, and h(x)=f(b1,,bm,,bi-1,x,bi+1,,b2m) is antitone, then h(h(x)) is isotone. But

= f(f(b1,,bm,,bi-1,b1,,b2m-i+1),,bi-1,x,bi+1,,b2m,bi+1,,b2m)),

and the position of x is the second expression is (i-1)-(2m-i+1)+2=2i-2m>1, therefore implying that h(h(x)) is antitone, again a contradiction. As a result, f is isotone for all i=1,,n.

The argument above also works when n is odd, say n=2m+1 and im+1. Finally, since f is monotone, it is monotone with respect to the i-th variable when i=m+1, so f is one of the following three forms:


the first two of which imply that f is isotone. ∎

An example of an associative function that is, say (,,), is given by

f:3  where  f(x,y,z)=x-y+z.

f is associative since f(f(r,s,t),u,v)=f(r,f(s,t,u),v)=f(r,s,f(t,u,v))=r-s+t-u+v.

Title properties of certain monotone functions
Canonical name PropertiesOfCertainMonotoneFunctions
Date of creation 2013-03-22 19:03:31
Last modified on 2013-03-22 19:03:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Result
Classification msc 06F99
Classification msc 08C99
Classification msc 08A99