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Arithmetic Semirings (algebrax.semiring.arithmetic)

The arithmetic semirings provide classical numerical and discrete modulo fields:

  1. StandardSemiring[T]: Classical arithmetic \((+, \cdot, 0, 1)\) over numeric types (float, int, complex, or custom rings).
  2. ModularSemiring: Integer arithmetic modulo \(n\) \((\mathbb{Z}_n, +_n, \times_n, 0, 1)\).

Standard Semiring (Linear Algebra)

The default semiring uses standard arithmetic (\(+, \times\)).

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import algebrax as ax

# Sparse Matrices
A = {0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}
B = {0: {0: 5, 1: 6}, 1: {0: 7, 1: 8}}

# Standard Matrix Multiplication
C = ax.matrix.dot(A, B, semiring=ax.semiring.StandardSemiring())
print(C)
# output: {0: {0: 19.0, 1: 22.0}, 1: {0: 43.0, 1: 50.0}}

Modular Integer Ring \(\mathbb{Z}_p\)

The ModularSemiring in algebrax.semiring represents the modular integer ring \(\mathbb{Z}_p = (\{0, 1, \dots, p-1\}, +\bmod p, \times\bmod p, 0, 1)\).

Ring Operations

  • Additive Identity: 0
  • Multiplicative Identity: 1 % p
  • Addition: \((a + b) \bmod p\)
  • Multiplication: \((a \times b) \bmod p\)
  • Exponentiation: \(a^n \bmod p\)

It serves as a foundational coefficient ring for finite field representations (\(\text{GF}(p^m)\)) and modular matrix arithmetic.

Python Example

import algebrax as ax

# Create Z_5 modular ring
z5 = ax.semiring.ModularSemiring(p=5)

# Ring operations in Z_5
print("3 + 4 mod 5:", z5.add(3, 4))    # 2
print("3 * 4 mod 5:", z5.mul(3, 4))    # 2
print("2^4 mod 5:  ", z5.power(2, 4))  # 1

# Matrix multiplication over Z_5
A = {0: {0: 3, 1: 4}, 1: {0: 2, 1: 1}}
B = {0: {0: 2, 1: 1}, 1: {0: 4, 1: 3}}

C = ax.matrix.dot(A, B, semiring=z5)
print("A @ B over Z_5:", C)