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Let $S$ be a set and let $S=\displaystyle\bigcup_{a\in A}S_a$ be a partition of $S$. Let $P$ be a property regarding ordered pairs $(x,y)\in S\times S$. I am wondering about the name of such a property:

$(x,y)$ satisfies $P$ if there exists $a\in A$ such that $x,y\in S_a$.

What is the name of this property? Is it called "piecewise $P$"?

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    Isn't your $P$ just the equivalence relation induced by the partition, or am I missing something? – 2017-01-05
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    I mean, a property $P$ that is pre-defined that agrees with this partition, not defined by this partition. Thus I wrote "$(x,y)$ satisfies $P$ if there exists $a\in A$ such that $x,y\in S_a$" instead of "$(x,y)$ satisfies $P$ if and only if there exists $a\in A$ such that $x,y\in S_a$". – 2017-01-05
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    In that case it looks like your $P$ is really a relation among the $S_a$s, simply reinterpreted to work on their elements instead. – 2017-01-05
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    @HenningMakholm, for example, if $G$ is a group and $G=S_1\cup\cdots\cup S_n$, where $x$ commutes with $y$ whenever $x$, $y$ are in the same $S_i$ but $G$ may not be abelian. In this case should I call $G$ "piecewise Abelian"? – 2017-01-05
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    Associated to a partition is an equivalence relation $R$. It sounds like you have $(x,y)\in R\Rightarrow (x,y)\in P$ (i.e. $R\subseteq P $). Do you know anything else about $P$? If $P $ is transitive and symmetric then you have a "coarser equivalence relation than the one associated to the partition", but if $P$ is pretty arbitrary then I don't know of any tidy phrasing. – 2017-04-26
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    Thank you@MarkS. for your comment! It is very clear and helpful. – 2017-04-26

2 Answers 2

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In general, a property of ordered pairs of elements from $S$ is called a relation (Wikipedia) on $S$, and a property $P$ for which there is a partition $S=\bigcup_{a\in A}S_a$ with

$(x,y)$ has property $P$ $\iff$ $x,y\in S_a$ for some $a$

is called an equivalence relation (Wikipedia), and they are of fundamental importance in mathematics.

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    Thanks for your answer! I should have asked my question more clearly: I mean, a property $P$ that is pre-defined that agrees with this partition, not defined by this partition. Thus I wrote "$(x,y)$ satisfies $P$ if there exists $a\in A$ such that $x,y\in S_a$" instead of "$(x,y)$ satisfies $P$ if and only if there exists $a\in A$ such that $x,y\in S_a$". – 2017-01-05
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Your property is (or induces) a relation on $S$, and the relation is the 'is in the same partition block'-relation. That is, for $x,y\in S$, $x$ is related to $y$ if and only if $(x,y)$ satisfy property $P$. This occurs if and only if $x$ and $y$ are in the same partition block.

This relation will be an equivalence relation.

Does that answer your question?

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    I was thinking about some examples such as $G$ being a group and $G=S_1\cup\cdots\cup S_k$, where $x$ commutes with $y$ whenever $x,y$ are in the same $S_i$ but $G$ may not be abelian. Maybe I should have phrased my question more clearly. – 2017-01-05
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    That wouldn't partition a nonabelian group. For instance, the identity commutes with everything. – 2017-01-05
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    I mean we make a partition first (a trivial example would be $\mathbb{Z_k}$ with the discrete partition; that is, every subset consists of one element). – 2017-01-05