I'll use and example to explain my question:

Example: Let there be two distinct types $a$ and $b$ and each type has equal number of sets.

  • There are two sets of type $a$: $\{1,2\}$ and $\{3,4\}$.
  • There are two sets of type $b$: $\{2,4\}$ and $\{1,3,4\}$.

I want to work with the following objects: $\{1,2,2,4\}$, $\{1,1,2,3,4\}$, $\{2,3,4,4\}$ and $\{1,3,3,4,4\}$.

One important property: $\{1,2\}=\{1,1,2,2\}$, $\{1,2,2,4\}=\{1,2,2,2,4,4\}$, etc.

So basically, each of these objects is constructed by conjoining exactly one set of type $a$ with exactly one set of type $b$ and the elements can have repetitions. Also, multiplying the elements of one set before conjoining it with another doesn't affect the object; e.g. $\{1,2\}=\{1,1,2,2\}$ is because RHS is just two copies of the LHS conjoined together, and $\{1,2,2,4\}=\{1,2,2,2,4,4\}$ because RHS is two copies of $\{2,4\}$ - a set of type $b$ conjoined with one copy of $\{1,2\}$ - a set of type $b$.

So my questions are:

  1. Is there a name for these objects? They're not sets, not exactly tuples (but the cardinality of distinct objects is the same as that of tuples).
  2. More importantly, how do you define a general rule to explain what they are?
  3. Ultimately, I want to compare these objects. Specifically, I want to check whether one object is a subset (I know it's not a set but bear with me) of another. So I also want set relations to be meaningful, i.e. $\{1,2,2,4\}\subset\{1,1,2,2,3,4\}$ etc.

I hope it makes enough sense that somebody will help me. Thank you!

  • $\begingroup$ The definition of equality is not very clear here. If repetition of elements doesn't matter at all, they're just sets and there's no reason to consider repetitions. But maybe that's not what you have in mind. Is there any example of two of these objects with the same underlying set that are not equal? $\endgroup$ – Jair Taylor May 21 at 22:31
  • $\begingroup$ A copy of a set of a certain type doesn't matter but if an element comes from two different sets, then repetition matters. So $\{1,2\}=\{1,1,2,2\}$ because RHS is two copies of the LHS (a set of some type) but $\{1,2,2,4\}\neq\{1,2,4\}$ because RHS is not constructed from sets of any type. $\endgroup$ – Curious George May 21 at 22:39
  • $\begingroup$ You're thinking of multisets, I think. $\endgroup$ – Asaf Karagila May 22 at 7:48
  • $\begingroup$ Having repetitions etc. points to using multisets but there's also the point of taking exactly one set of each type to construct the object, so it's kind of like a tuple without order. $\endgroup$ – Curious George May 22 at 12:40

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