is_abelian

std.algebra.is_abelian · Level L3

Whether a finite Cayley table is an abelian group, as 1 or 0: a group in which a·b = b·a for every pair, checked over the whole table. A few commuting pairs, or commuting generators not shown to generate, are not enough.

group and ab = ba ∀a, b

Signature

is_abelian(T: i64[n, 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 a group, and T equal to its transpose 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

is_group

Called by

ring_axioms

sha256:c465cf93115716aa1565a42c1801d54dc409423520b16239e5c0ae082ca56990

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_abelian · Ω:is_abelian
Pins: sha256:e80cde67297d9b89bcc11d039b6e22663502d36125141ef0f06e1681dbef5ea3this graph and the 2 it reaches through calls
Evidence: sha256:eae2df2b113dedadd9977e632ca45c95b476b6514cb49f769e880c93364f65cfthe 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.