Alexander Kuklev (akuklev) wrote,
Alexander Kuklev

Conical Presemirings

Commutative Conical Presemiring is a set M equipped with two commutative monoidal operations ‹max› and + with shared neutral element 0 and
  a + (b ‹max› c) = (a + b) ‹max› (a + c)  (#distributivity).

CCPSs form a variety, in particular product of two CCPSs with component-wise + and ‹max› is a CCPS as well.

Distributivity with common neutral element makes this structure very rigid. In particular, we have:
  a ‹max› (a + x) = a + 0 ‹max› a + x = a + (0 ‹max› x) = a + x  (#naturality)
  a ‹max› a = a ‹max› (a + 0) = a + 0 = a  (#idempotency).

Assume a ‹max› c = b. By idempotency, a ‹max› a ‹max› c = b. Substituting b for a ‹max› c, we obtain a ‹max› b = b. So we also have
  a ‹max› c = b <=> a ‹max› b = b  (#saturation)

Consider a and b such that 0 = a ‹max› b. By saturation we obtain
  0 = a ‹max› 0 = a
Thus both a and b must be 0, i.e. (M, 0, +) is a conical monoid.

Consider a and b such that 0 = a + b. Then
  0 = a + (b ‹max› 0) = (a + b) ‹max› (a + 0) = 0 ‹max› a = a
Thus both a and b must be 0, i.e. (M, 0, ‹max›) also a conical monoid.

Let's define a relation a ⩽ b iff a ‹max› b = b. Commutativity of ‹max› enforces its antisymmetry, while naturality condition guarantees transitivity and reflexivity, thus it's a (partial) order. Saturation condition tells us it is precisely the canonical order generated by ‹max› operation, while distributivity condition tells us that it's at least as strong, as the canonical order generated by + operation. For CCPSs like K = ℕ, ℚ+, ℝ+, K ∪ {∞} with usual + and ‹max› and their products, orderings obviously coincide.

If + is cancelative (we rule out ∞-like objects), orderings coincide precisely when an operation ∸ (monus) can be defined, such that a ‹max› b = a + (b ∸ a) = b + (a ∸ b). Monus is a partial inverse to +, namely if a + x = b, then x = b ∸ a. (Cancellation property does non necessarily hold in any CMM (commutative monoid with monus), yet CMM forms a commutative conical presemiring with ‹max› b = a + (b ∸ a) = b + (a ∸ b) precisely when the additional condition (x + y) ∸ x = y on monus holds, which implies cancellation.) This kind of commutative conical presemirings also form a variety with simple objects isomorphic to natural numbers equipped with usual + and ‹max›.[1]

– Find a commutative conical presemiring, where canonical orders generated by + and ‹max› do not coincide. Can such a CCPS still be cancelative?
– Let C be a simple CCPS (i.e. one without nontrivial subalgebras). Is it true that ⩽ is a total order?

1. Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254

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