Not that this answers your question, but keep an eye out for the related forms X x (A + Y) --> (X x A) + Y and (X + A) x Y --> X + (A x Y) in case you run into either one. These are half of the weak distributivity laws for linear logic studied by Cockett and Seely a decade or so ago. Although full-blown category theory didn't exist in the 19th century it did have its posetal fragment, and the above first appears in C.S. Peirce "Note B: The Logic of Relatives", 1883, see p. 456 of Vol. 4 of Kloesel's "Writings of C.S. Peirce" where x and + are respectively relative product (i.e. composition in Rel) and its De Morgan dual (with respect to Boolean complement) relative sum. Peirce describes them as "two formulae that are so constantly used that hardly anything can be done without them." (They also hold for x,+ as logical or Boolean conjunction,disjunction whence "hardly anything" could well extend to brushing one's teeth etc, though only your subconscious would know that.) The posetal case of the internal hom, what Ward and Dilworth called "residuation" in 1939, goes back even further, namely to De Morgan's Theorem K in his "On the Syllogism: IV", 1860, first pointed out by Roger Maddux, see http://boole.stanford.edu/pub/ocbr.pdf (evening LICS talk I gave 6 weeks after my quintuple bypass). Vaughan Pratt On 5/9/2012 4:35 PM, Ondrej Rypacek wrote:

Could anyone kindly help me with the following: How much is known about categories with

- two monoidal structures - and a natural transformation (but not isomorphism) X x Y -> X + Y

I believe this isn't called a bimonoidal category, as we don't have an iso above (?)

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