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A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. These things seem to come up in quantum logic, for example:

D. J. Foulis, M. K. Bennet, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1 331–1 352.

and

Peter Hines, A categorical analogue of the monoid semiring construction

(Probably the same definition has come up in other places too - if so, please tell me about it.)

  • Question Does the category PCM have a closed symmetric monoidal product?

This might sound a bit random, so I'll try to explain a bit of background and my motivation.

PCMs sit somewhere in between abelian monoids (aka $\mathbb{N}$-modules) and pointed sets (which one might call modules over the field with one element $\mathbb{F}_1$). Commutative monoids and pointed sets both form closed symmetric monoidal categories, and the commutative monoids in these categories are commutative semirings ($\mathbb{N}$-algebras) and commutative monoids with an absorbing element (the corresponding notion of $\mathbb{F}_1$-algebras) respectively. A commutative monoid in PCM should, I expect, be something like a semiring, but with a multiplication operation that is always defined and an addition operation that is only sometimes defined.

Various people, including Toen-Vaquie, have studied the variants of scheme theory built from these things as their affine local models. For various reasons, I find myself wanting a category of schemes that fits in between $\mathbb{F}_1$ and $\mathbb{N}$, and so commutative monoids in PCM seem to fit the bill. The input for the general Toen-Vaquie machine is a closed symmetric monoidal category which one thinks of as the category of modules over the algebraic object we are considering.

I've tagged this as reference-request in the hope that I can simply cite something, but feel free to also post an argument/construction/counter-example.

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Yes, it does; see Theorem 11 of this paper.

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