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homomorfiat

Homomorfiat is a term used in theoretical discussions to describe a class of structure-preserving mappings that extend the idea of a homomorphism by requiring preservation of an additional distinguished operation called the fiat. The term is not widely standardized and appears in explorations of augmented algebraic structures and policy-preserving transformations in computer science.

Formal definition

Let A and B be algebraic structures of the same signature, each equipped with a unary fiat

Basic properties

The composition of two homomorfiats is a homomorfiat, and the identity map on A is a homomorfiat.

Examples

A simple example arises when A and B are rings with a fixed endomorphism F acting as

Applications and status

As a conceptual tool, homomorfiat helps study how algebraic structure interacts with external constraints or policies

See also

Homomorphism, Endomorphism, Automorphism, Morphism, Augmented algebra.

operator
F_A
on
A
and
F_B
on
B.
A
function
h:
A
->
B
is
called
a
homomorfiat
if
it
preserves
all
base
operations
of
the
signature
and
satisfies
h(F_A(a))
=
F_B(h(a))
for
all
a
in
A.
In
other
words,
h
is
a
homomorphism
with
respect
to
the
base
operations
and
also
commutes
with
the
fiat
operators.
The
set
of
homomorfiats
from
A
to
B
forms
a
subset
of
the
usual
homomorphisms,
consisting
of
those
maps
that
additionally
commute
with
the
fiat
operators.
If
the
fiat
operator
encodes
a
policy,
constraint,
or
projection,
then
homomorfiats
model
transformations
that
preserve
both
structure
and
the
policy.
a
fiat
operator
(for
instance,
a
projection
or
a
scalar
multiplication).
A
ring
homomorphism
h:
A
->
B
is
a
homomorfiat
if
h(F_A(a))
=
F_B(h(a))
for
all
a
in
A.
in
theoretical
computer
science
and
algebra.
It
is
primarily
used
in
toy
models
and
expository
contexts,
rather
than
as
a
widely
adopted
formal
notion.