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Group Structures

Let \(X\) be a set. This section outlines the traits Algebraeon provides for defining group-like structures on \(X\).

Composition

  • Composition : Set for a set with a binary operation called composition denoted by \(\circ\). \[\circ : X \times X \to X : (a, b) \mapsto a \circ b\] .compose(a: X, b: X) -> X is used to compute \(a \circ b\).
  • AssociativeComposition : Composition when \(\circ\) is associative \[a \circ (b \circ c) = (a \circ b) \circ c \quad \forall a, b, c \in X\]
  • CommutativeComposition : Composition when \(\circ\) is commutative \[a \circ b = b \circ a \quad \forall a, b \in X\]
  • LeftCancellativeComposition : Composition when \[a \circ x = a \circ y \implies x = y \quad \forall a, x, y \in X\] In that case the solution (or lack thereof) to \(a = b \circ x\) for \(x\) given \(a\) and \(b\) is unique whenever it exists and is computed using .try_left_difference(a: X, b: X) -> Option<X>.
  • RightCancellativeComposition : Composition when \[x \circ a = y \circ a \implies x = y \quad \forall a, x, y \in X\] In that case the solution (or lack thereof) to \(a = x \circ b\) for \(x\) given \(a\) and \(b\) is unique whenever it exists and is computed using .try_right_difference(a: X, b: X) -> Option<X>.
  • CancellativeComposition : Composition. .try_difference(a: X, b: X) -> Option<X>.

Identity

  • Identity : Set for a set with a special element called the identity element which we’ll denote by e.
  • TryLeftInverse : Identity + Composition when the solution to \[x \circ a = e\] for \(x\) given \(a\) is unique whenever it exists. The solution (or lack thereof) is computed using .try_left_inverse(a: X) -> Option<X>.
  • TryRightInverse : Identity + Composition when the solution to \[a \circ x = e\] for \(x\) given \(a\) is unique whenever it exists. The solution (or lack thereof) is computed using .try_right_inverse(a: X) -> Option<X>.
  • TryInverse: Identity + Composition when the solution to \[x \circ a = a \circ x = e\] for \(x\) given \(a\) is unique whenever it exists. The solution (or lack thereof) is computed using .try_inverse(a: X) -> Option<X>.
  • Monoid : Identity + AssociativeComposition when \[a \circ e = e \circ a = a \quad \forall a \in X\]
  • Group : TryInverse + TryLeftInverse + TryRightInverse + LeftCancellativeComposition + RightCancellativeComposition when every element has an inverse. Left-, right-, and two-sided-inverses all coencide in this case and are computed using .inverse(a: X) -> X.
  • AbelianGroup := Group + CommutativeComposition + CancellativeComposition.