Upper bound of a family of sets

Question

In Wikipedia is provided the following notion of an upper bound for a set of functions with common domain and codomain:

Function $g$ defined on domain $D$ and having the same codomain $(K,\leq)$ is an upper bound of $f$, if $g(x)\geq f(x)$ for each $x$ in $D$. Function $g$ is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set.

I propose to generalize this notion for a set of functions with common only codomain by the following

Definition. Let $\left(K,\le\right)$ be a partially ordered set, $I$ be a set of indices, and a set $S=\left\{f_{i}:i \in I\right\}$ be a set of functions with a common codomain $K$. Let $D=\bigcap_{i \in I}D_{f_{i}}$ be the intersection of domains $D_{f_i}$ of functions $f_i\in S$. A function $g: D\to K$ is an upper bound of the set $S$ set on the set $D$ if $g(x)\ge f_{i}(x)$ for each $x\in D$ and $i\in I$.

I remark that if the intersection $D$ is empty, then any function from $D$ to $K$ is a subset of $D\times K=\emptyset$, so it this case the family $S$ has a trivial (and unique) upper bound $\emptyset$ on the set $D$.

In the above definition true?

Answer 1

The suggested definition is indeed a generalization of the one from Wikipedia and it agrees with it if all functions in the set have the same domain. It has one unexpected property, though -- a function can be upper bound of a set of function in this "new" sense while not being upper bound of any proper subset of it.

As a simple example, consider $f_1(x)=\sqrt{1+x}$ and $f_2(x)=\sqrt{1-x}$ defined on their respective maximal possible domains. The function $g(x)=2$ would be upper bound of the set $\{f_1, f_2\}$ but it would not be upper bound of neither $\{f_1\}$ nor $\{f_2\}$.

Is the suggested definition useful even with this counterintuitive property? Maybe...

Answer 2

Statements can be true or false. A definition is not a statement and it cannot be true or false. Thus, from a formal point of view, we can freely define any object caring only about syntactic correctness of the definition and then launch it to the world. “Hi, guys, I’m a new notion! Do you like me?”

But what is right in mathematics and what is wrong? Right statements are true, false statements are wrong. But what about definitions? Our experience teach us that a definition also can be good or bad. For a good definition usually we need some guide, some motivation. Will it be fruitful, will it direct us towards interesting, beautiful, not-trivial results? Will it help us to solve our problems? Karl Popper said

"Yet we also stress that truth is not the only aim of science. We want more than mere truth: what we look for is interesting truth -- truth which is hard to come by. And in the natural sciences (as distinct from mathematics) what we look for is truth which has a high degree of explanatory power, which implies that it is logically improbable. For it is clear, first of all, that we do not merely want truth -- we want more truth, and new truth. We are not content with 'twice two equals four', even though it is true: we do not resort to reciting the multiplication table if we are faced with a difficult problem in topology or in physics. Mere truth is not enough; what we look for are answers to our problems. The point has been well put by the German humorist and poet Busch, of Max-and-Moritz fame, in a little nursery rhyme -- I mean a rhyme for the epistemological nursery:

Twice two equals four: 'tis true,
But too empty, and too trite.
What I look for is a clue
To some matters not so light.

Only if it is an answer to a problem -- a difficult, a fertile problem, a problem of some depth -- does a truth, or a conjecture about the truth, become relevant to science. This is so in pure mathematics, and it is so in the natural sciences”.

Moreover, since mathematical thought is intuitive, we should be especially careful defining notions corresponding to intuitively clear notions, such as to be bigger. Right notions have to be natural, to correspond to the intuition. For instance, if an element $g$ is bigger than all elements of the set $S$ then $g$ should be bigger than any element of the set $S$. The function $g$ from the definition from the question satisfies this condition only for the restrictions of the functions $f_i$ of $S$ to the intersection $D$ of their domains $D_{f_i}$, but it may fail to satisfy the condition with respect to the whole domains $D_{f_i}$ of elements $f_i$ of $S$.

So, the problem is, whether we can consider the function $g$ as a (natural) upper bound of the family $S$? In fact, in the definition of an upper bound $g$ of a set $S$ from Wikipedia it is required that $g(x)\ge f(x)$ for each $f\in S$ and $x\in X$, that is an upper bound $g$ for $S$ dominates any element of $S$. This definition coincides with the definition from the question for the set $S$ of functions with a common codomain. So the second definition is a generalization of the first and the first is a partial case of the second. In fact, a function $g$ from $D$ to $K$ is an upper bound of the set $S$ with respect to the second definition if an only if $g$ is an upper bound with respect to the first definition of the set $\{f_i|D: f_i\in S \}$ of restrictions on $D$ of functions of $S$. And this is noted when we said that $g$ is an upper bound on the set $D$.

It is easy to check that the second definition of an upper bound also satisfies the following natural transitivity property. Assume that we have a partially ordered set $(K,\le)$ and for each $j\in J$ we have any set $I_j$ of functions with a codomain $K$ and any upper bound $f_j$ of the set $I_j$. Let $f$ be any upper bound of the set $\{f_j:j\in J\}$. Then $f$ is an upper bound of the set $\bigcup_{j\in J} I_j$.

Also I have to note that a notion of an upper bound of a subset $A$ of a partially ordered set $P$ is already defined.

For a subset $A$ of $P$, an element $x$ in $P$ is an upper bound of $A$ if $a\le x$, for each element $a$ in $A$. In particular, $x$ need not be in $A$ to be an upper bound of $A$.

I stress that whereas an upper bound $x$ of a subset $A$ of a partially ordered set $P$ need not be in $A$, $x$ have to belong to $P$, so it cannot be any function with given codomain.

Thus a notion of an upper bound of a set $S$ of functions is already defined, provided we specified a partially ordered set $P$ containing $S$.

So, how we can define a partial order on a set $P$ of functions with a common codomain but (possibly) different domains? One of usual definitions is the following. For any functions $f,g\in P$ we put $f\le g$ provided domain $\operatorname{dom} f$ of $f$ contains domain $\operatorname{dom} g$ of $g$ and $f(x)=g(x)$ for each $x\in\operatorname{dom} g$.

I remark that lower bounds of sets according to this partial order satisfy the above Popper’s guide, because they are used in constructions involving Martin's Axiom. This axiom was used to produce many interesting and non-trivial results, in particular, in general topology and topological algebra, where I’m working. For instance, last December I traveled from Ukraine to Austria in order to attack open problems from a “paper of my life” with two my friends, Lyubomyr Zdomskyy and Serhii Bardyla. We wrote a paper “A countably compact topological group with the non-countably pracompact square” presenting the main result of our efforts during two weeks of my stay in Vienna. It is a the first step towards the answer of my ten-year-old problem. Namely, we obtained a negative answer assuming Martin's Axiom providing us an example, in which construction was used a map $\phi$, which can be understood as a global lower bound of some set of functions with a common codomain, but different domains, see Lemma 2.1.