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There was this question in my Discrete Maths exam:

Let S={1,2,3,4,5,6}. 2 partitions P1 and P2 are given as:

P1={(1,2,3),(4,6),(5)}

P2={(1,2),(3,4,5),(6)}

Define and find the sum and product of these 2 partitions P1 and P2.

I simply thought it was just the unions and intersections of these 2 sets. But after going through many books and online resources, I cannot find anything to validate my method. Please tell how are these 2 operations defined on partitions of sets.

EDIT: Please don't just randomly downvote the question if you dont know the answer. It increases the probability that someone who might be knowing the answer may end up ignoring it.

Aman Kumar
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  • Welcome to MSE. I've never heard of such operations. Was there more to the exam question? Alternatively, are you sure these operations weren't defined in your course? – saulspatz Mar 10 '20 at 17:35
  • Yes I am quite sure. That's the reason I resorted to asking it here, after I had given up all hope. Our teacher has asked us to search the answer on our own, and I am really stuck there – Aman Kumar Mar 10 '20 at 17:37

1 Answers1

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I've found a definition of the product here.

The product of two set partitions B and C is defined as the set partition whose parts are the nonempty intersections between each part of B and each part of C. This product is also the infimum of B and C in the classical set partition lattice (that is, the coarsest set partition which is finer than each of B and C). Consequently, inf acts as an alias for this method.

Look at that page for an example. Presumably, the sum is the dual of the product, namely the finest partition coarser than both.

I know these operations by the names "meet" and "join". If you have difficulty figuring out what's going on, look up lattice orders. I think you'll be able to figure things out from the link I posted, but feel free to ping me if you have questions.

saulspatz
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  • Thanks a lot @saulspatz . At least now I know the product. And I think your hypothesis is correct, regarding the sum. It is quite intuitive... – Aman Kumar Mar 10 '20 at 18:11