Simple implementations of selected abstract algebraic structures.
Mathematical abstract algebra is built on a rich collection of algebraic structures. Learning about these structures can give non-mathematicians insights into the mathematical entities they need to work with--for example, real numbers, complex numbers, vectors, matrices, and permutations. By definition, these structures must comply with sets of axioms and properties, which are in turn a rich source of properties for generative testing.
un_algebra ("***un***derstanding algebra") is a simple
implementation of selected algebraic structures in Rust. I hope it
is useful to developers learning abstract algebra concepts for the
first time. Currently this crate provides magma, semigroup,
quasigroup, monoid, group, ring and field implementations.
un_algebra is still under pre-version 1.0 development, with a
number of outstanding design and implementation issues. Breaking
changes are likely to the crate API.
un_algebra is intended to support self-study of abstract algebraic
structures--it is not optimized for use in a production environment.
For production environments I recommend using a more sophisticated
library like alga.
un_algebra uses 2018 edition features but unfortunately requires
Rust nightly as it uses the (experimental) external documentation
I'm not a mathematician so my implementation of the various
structures and their respective axioms in
un_algebra may not be
strictly correct. Please let me know of any errors.
Please refer to the references document for more background on each structure and its associated axioms and properties.
The names of the
un_algebra structures and their respective axioms
can be long and unwieldy, for example, a "commutative
multiplicative group". To keep the exported names workable I use
these abbreviations in trait and function names:
- "Rel" for "relation"
All structures have an abstract version of the structure trait
group::Group), and, where they are commonly used,
additive and multiplicative versions of the structure trait (for
traits are defined using the terminology of addition and
multiplication rather than as abstract binary operations.
un_algebra structure traits are equipped with predicate
functions that implement the structure axioms. Some structures
also have predicate functions that implement derived properties of
These properties are not strictly necessary since they can be derived from the axioms, but they do allow richer generative testing of trait implementations, especially those using floating point numbers.
User defined traits that are implementable by Rust's built-in
numeric types seem to quickly lead to a lot of tedious, repeated
impl code, or to using tricky self-referential macros. This could
be due to missing abstractions in Rust's numeric type hierarchy or
(more likely) my lack of Rust experience.
Where a trait's
impl code is only repeated a couple of times
modules use the boilerplate code approach and in other cases they
rely on a macro to create the
un_algebra traits provide unit tests they are generally
generative tests built on the
proptest generative testing
crate. These generative tests test a selection of built-in numeric
type with every structure.
Generating test values and test functions seems to require a suprising amount of repetitive, boilerplate code items. Reducing the repeated items is possible via code generation using complex self-referential macros, but I'm not sure this is easier to maintain than the repeated items.
To avoid cluttering up module source files, the unit tests for most
modules are defined in separate files. The module source files use
path = attribute to link the source and test files.
Rust's built-in integer types (for example
i32) are finite subsets
of the natural numbers (ℕ). This means they can only satisfy
abstract structure axioms with modulo, or "wrapping" addition and
Many application data types that in theory conform to modern algebraic structures make heavy use of IEEE floating point numbers. Unfortunately, these numbers are only a finite subset of the real numbers (ℝ) and they do not reliably satisfy even the simplest real number axioms (IEEE).
For working with IEEE floating point types
"numeric" structure axioms and properties that are handy when
working with IEEE floating point types, which often require
"numeric" comparisons using an "epsilon" or error term.
un_algebra implements the relevant structure traits for all the
Rust standard library integer and floating point numeric types, for
example, an additive group for integer types
etc. It also provides rational number (ℚ) and complex number
(ℂ) examples, based on the num crate.
In addition, the crate examples directory contains abstract structure implementations of selected concepts, for example, finite fields.
Algebraic commutative group modules.
Algebraic commutative ring modules.
Algebraic field modules.
Algebraic group modules.
Helper methods for
Algebraic magma modules.
Algebraic monoid modules.
Numeric support module for floating point types.
Algebraic quasigroup modules.
Algebraic relation modules.
Algebraic ring modules.
Algebraic semigroup modules.
Generative testing support for