Basic algebraic categories

The basic algebraic categories provided by $\Sigma{}^{{\rm it}}$ are shown in Figure 1. While it follows quite closely the usual algebraic structures, a few points needs to be mentioned.

• The multiplication $\ast$ is in general not assumed to be commutative, except in the subtree starting at CommutativeRing, which appears in blue in Figure 1. This means that code written for the other categories should be careful and not assume commutativity of $\ast$. Addition is on the other hand always assumed commutative.
• The $\Sigma{}^{{\rm it}}$ category tree has a unique root SumitType to which almost all the types provided by $\Sigma{}^{{\rm it}}$ belong. The category SumitType inherits PrimitiveType from SALLI, which means that types in $\Sigma{}^{{\rm it}}$ must export an equality. That equality however does not need to be complete, as described in the $=$ page of the SALLI reference manual. SumitType plays also an important role in input/output as explained below.
• The SALLI categories ArithmeticType and AdditiveType already export the basic arithmetic operations provided respectively by rings and abelian groups, and in fact Ring and AbelianGroup do inherit from them. There is however a fundamental difference: while ArithmeticType and AdditiveType export the usual operations $=$, $+$, $-$, $\ast$,..., they make no assumption about their algebraic properties whatsoever. On the other hand, the corresponding algebraic categories in $\Sigma{}^{{\rm it}}$ do assume that $=$ is a complete equality test, and that the arithmetic operations satisfy the algebraic axioms of their mathematical structure. So for example, while Integer is extended by $\Sigma{}^{{\rm it}}$ to be a Ring (among other things), SingleFloat remains an ArithmeticType but is not a Ring. There are similar correspondances between LinearCombinationType and Module, and between LinearArithmeticType and Algebra. Mathematical types such as DenseMatrix$(R)$ or DenseUnivariatePolynomial$(R)$ allow their argument $R$ to be an ArithmeticType rather than a Ring, but those among their functions that require a complete equality on $R$ are not exported when $R$ is not a Ring. Similarly, while DenseUnivariatePolynomial$(R)$ is always an ArithmeticType, it is a Ring only when $R$ itself is a Ring. For example,
  
  
  DenseMatrix DenseUnivariatePolynomial SingleFloat
  is a valid type in $\Sigma{}^{{\rm it}}$, but Gaussian elimination is not available for such matrices, and the Euclidean algorithm is not available for their polynomial entries. Basic arithmetic, as provided by ArithmeticType is however available.