properties of certain monotone functions
In the definitions of some partially ordered algebraic systems such as po-groups and po-rings, the multiplication is set to be compatible with the partial ordering on the universe in the following sense:
Recall that an -ary function on a set is said to be monotone if it is monotone in each of its variables. In other words, for every , the function is monotone in , where each of the is a fixed but arbitrary element of . We use the notation to denote the monotonicity of each variable in . 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.
Suppose is isotone (or antitone) in its first variable. Since , is isotone (or antitone) in each of its remaining variables. ∎
Let be an -ary monotone operation on a set with an identity element . In other words, . Then is either strictly isotone or strictly antitone.
The proof is the same as the one before. Furthermore, if is isotone and , then , so the strict ordering is preserved. The same holds true if is antitone. ∎
Let be a binary monotone operation on a set such that it is isotone (antitone) with respect to its first variable. Suppose is a unary operation on such that is a fixed element of . Then is antitone (isotone).
Let be an -ary associative monotone operation on a set . Then
is isotone if is even
is either isotone, or is , if is odd, say .
Suppose first that , , and is antitone. Then is isotone. By the associativity of , is
In the second expression, the position of is , therefore implying that is antitone, which is a contradiction! Therefore, is isotone. Now, if , and is antitone, then is isotone. But
and the position of is the second expression is , therefore implying that is antitone, again a contradiction. As a result, is isotone for all .
The argument above also works when is odd, say and . Finally, since is monotone, it is monotone with respect to the -th variable when , so is one of the following three forms:
the first two of which imply that is isotone. ∎
An example of an associative function that is, say , is given by
is associative since .
|Title||properties of certain monotone functions|
|Date of creation||2013-03-22 19:03:31|
|Last modified on||2013-03-22 19:03:31|
|Last modified by||CWoo (3771)|