is_ideal

std.algebra.is_ideal · Level L3

Whether a subset I of a finite ring (a 0/1 mask) is a two-sided ideal, as 1 or 0: the tables form a ring; I contains 0, is closed under addition and additive inverses; and it absorbs multiplication on both sides, r·x ∈ I and x·r ∈ I for every r in the ring and x in I. Left and right absorption are checked separately, over every pair.

0 ∈ I, I + I ⊆ I, −I ⊆ I, RI ⊆ I, IR ⊆ I

Signature

is_ideal(Ad: i64[n, n], Mu: i64[n, n], I: f64[n]) → f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference zero, sums, negatives and products on both sides, every pair in exact rational arithmetic, on all 40 test cases.
All 40 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

group_axioms group_inverses is_ring

Called by

quotient_ring

sha256:d76d6e4903e50ef473daf721eb312305cbee1b5c750d50e813bf30402d8c80ad

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.is_ideal · Ω:is_ideal
Pins: sha256:96377cead22127f4168bc68c64c2ce89d9299482353394103e69f85e446ac8f1this graph and the 6 it reaches through calls
Evidence: sha256:59aece4eb2b76f452431ee89a5bbba3e5ac367747552fa7502086b24a393e850the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.