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AngouriMath.Functions.Algebra.MonoidAlgebra

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Classes within the AngouriMath.Functions.Algebra.MonoidAlgebra namespace

• IBasisOps`1

  Summary

  What a basis has to be able to do for its terms to be factorable: it is a monoid, and
  its elements have a greatest common part that can be divided out.
  

  Type parameter "TBasis"

  An exponent vector for a polynomial, a rational exponent for a series, a basis ket for a
  quantum state.
  

  Remarks

  Passed as an instance rather than fixed by a type parameter with new(). The
  codebase does it the other way in
  PolynomialSolver.GatherMonomialInformation<T, TPrimitive>, and that shape
  is harder to test in isolation for no gain here, since these operations are stateless
  and one instance serves every call.
  **Meet(`0,`0) is what makes factoring one operation across features.** Dividing
  out the meet of the support is simultaneously "take the common monomial out of a
  polynomial" and "detect that a quantum state is separable":
  

  x^2*y + x^2      meet of (2,1) and (2,0) is (2,0), leaving y + 1
  |001> + |011>    meet of 001 and 011 is 0-1,   leaving |0> + |1>

  
  The two look different only because a polynomial's basis is ordered -- so the meet is a
  componentwise minimum -- while a ket's is flat, and the meet keeps a position only where
  every element agrees. Both are the meet in a product of semilattices.
  

• Semiring

  Summary

  The arithmetic the coefficients of a SparseTerms`1 obey.
  

  Remarks

  Coefficients are always Entity; what changes between features is what
  adding and multiplying two of them means. A polynomial adds them in a field, a boolean
  expression joins them in a lattice, and everything else about the representation is the
  same.
  **IsIdempotent is the whole reason this is a type rather than a pair of
  delegates.** a or a = a holds in the boolean semiring and fails in the complex
  one, where |x> + |x> is 2|x> -- and that single difference is
  what separates *covering* from *superposition*. Quine-McCluskey's merge step is
  absorption, and it is sound only because a minterm may be covered twice for free; a
  procedure that assumed the same of amplitudes would double them. Anything reading this
  flag is asking whether it may cover the same basis element more than once.
  

• SparseTerms`1

  Summary

  A finitely-supported map from a basis into a coefficient semiring -- the shape shared by
  a polynomial, an asymptotic series, a boolean expression's minterms and a quantum state.
  

  Type parameter "TBasis"

  The basis element. An exponent vector, a rational exponent, a basis ket.
  **It must compare by value.** The terms are keyed on it, so a basis with reference
  equality would leave every element distinct and quietly stop collecting like terms --
  which is not an error anywhere, just a wrong answer. Use a struct or a record.
  

  Remarks

  Immutable by construction: every operation returns a new instance, and the dictionary
  handed in is copied rather than adopted. It is not
  System.Collections.Immutable because that would be a new package reference on a
  library that has none, which is a packaging decision rather than a design one.
  The invariant is that **no term carries a zero coefficient**. That is what makes the
  count of terms meaningful, and it is where an idempotent semiring quietly pays for
  itself: adding a boolean term to itself collapses to one term by
  Add(AngouriMath.Entity,AngouriMath.Entity) alone, so absorption needs no special case here.
  See *One structure under several features* in AGENTS.md for what this is for and,
  as importantly, what must not be built on it -- cover selection, factorisation into
  irreducibles and series truncation are each specific to one feature and belong in it.