Adding a README

First draft of the README

README.md

@@ -1,4 +1,42 @@
 # Semirings
-[![Coverage Status](https://coveralls.io/repos/github/mcognetta/Semirings.jl/badge.svg?branch=master)](https://coveralls.io/github/mcognetta/Semirings.jl?branch=master)
+A Julia library for [semirings](https://en.wikipedia.org/wiki/Semiring)
 [![Build Status](https://travis-ci.com/mcognetta/Semirings.jl.svg?branch=master)](https://travis-ci.com/mcognetta/Semirings.jl)
-[![codecov](https://codecov.io/gh/mcognetta/Semirings.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/mcognetta/Semirings.jl)
+[![codecov](https://codecov.io/gh/mcognetta/Semirings.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/mcognetta/Semirings.jl)
+[![Coverage Status](https://coveralls.io/repos/github/mcognetta/Semirings.jl/badge.svg?branch=master)](https://coveralls.io/github/mcognetta/Semirings.jl?branch=master)
+
+This provides implementations for several important semirings as well as a small interface so that you can define your own.
+
+Provided semirings:
+- Real Semiring: `RealSemiringElement` - `(R, +, *, 0, 1)`
+- Max Plus: `MaxPlusSemiringElement` - `(R, max, +, -, -∞, 0)`
+- Min Plus: `MinPlusSemiringElement` - `(R, min, +, -, ∞, 0)`
+- Boolean: `BooleanSemiringElement` - `({False, True}, OR, AND, False, True)`
+
+Suppose we wish to define the `MySemiring` semiring. First make the type:
+```
+MySemiringElement{T} <: AbstractSemiringElement{T}
+```
+Then define the following methods for your semiring:
+1. `val(x::MySemiringElement)` - Get the value of the element
+2. `+(l::MySemiringElement, r::MySemiringElement)` - Addition for your semiring
+3. `*(l::MySemiringElement, r::MySemiringElement)` - Multiplication for your semiring
+4. `zero(::Type{MySemiringElement})` - The `0` element in your semiring
+5. `one(::Type{MySemiringElement})` - The `1` element in your semiring
+
+Implementing these gives access to all of the other operations that are present in all semirings as well as several helpful methods concering promotion, conversion, semiring type inspection, etc.
+
+Assuming the `eltype` of your semiring has the appropriate conversion and promotion rules, everything else should just work. This includes using your `MySemiringElement` in vectors and matrices (although some additional methods must be defined in order to have access to all of the `LinearAlgebra` functionality).
+
+If your semiring has additional structure (a division, complete, star, etc. semiring), you can define a few more operations to access a variety of new functionality.
+
+When applicable, one can implement:
+1. `-(x::MySemiringElement)` - additive inversion
+2. `/(x::MySemiringElement, y::MySemiringElement)` - semiring division
+3. `star(x::MySemiringElement)` - semiring star operation
+
+and the associated semiring functionality will then be defined automatically.
+
+## To do
+1. Adding `SimpleTraits` to denote semiring properties (idempotent, nilpotent, complete, division, etc)
+2. Better operator for `star`
+3. Include `conj` in the docs or work around it for `RealSemiringElements`