hom_kernel_image

std.algebra.hom_kernel_image · Level L3

The kernel of a homomorphism f : G → K, the full preimage of K's identity, as a 0/1 mask; its image in K as a 0/1 mask; and whether f is injective, which holds exactly when the kernel is the identity alone. The kernel is a normal subgroup of G, and the nonempty fibres of f are its cosets.

ker f = f⁻¹(e_K); injective ⇔ ker f = {e}

Signature

hom_kernel_image(TG: i64[n, n], TK: i64[m, m], f: i64[n]) → f64[n], f64[m], 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 the preimage of e_K, the set of values, and |ker| = 1 in exact rational arithmetic, on all 40 test cases.
All 367 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

Called by

first_isomorphism

sha256:4873ec7e1f8f379da059a7df77d436da9114299c2f75d60d9cbd6c0d1a3a545b

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.hom_kernel_image · Ω:hom_kernel_image
Pins: sha256:2a3902157ac6587a3a158aa370c8df028e3e8ae32e848e48e711d499e3dcbc8dthis graph and the 1 it reaches through calls
Evidence: sha256:204f20676f380e15a390937279c9811f12193096b6ec871b7db0c0061cd8f9a9the 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.