I'm wondering if anyone has compiled a summary of commonly used algebraic laws or axioms (e.g. closure, associativity, commutativity, distributivity)?

The reason I ask is that these laws or axioms are very useful in expressing properties to automatically test in generative software testing. Having a summary of these laws handy would help in selecting effective tests.

I've found some useful summaries of algebraic structures and have started to extract the laws or axioms those articles list but it seems like the kind of summary that may well already exist somewhere.

The functions and data structures being tested don't always seem to neatly belong to an algebraic structure I recognise but they are often subject to individual laws. This situation probably reflects my lack of algebraic intuition more than anything.

These references have been useful so far:


Algebraic structure cheat sheet anyone?

[not enough reputation points to post the other links sorry]

Any advice much appreciated,


  • $\begingroup$ You can post other links you need as a plain text: I think the post will be edited then to make them alive. And if it is about software testing, I'm not sure this is a post for math.SE and not StackOverflow. I don't understand what kind of links you are asking about, if you are not satisfied with the given two. $\endgroup$
    – Wolfram
    Commented Feb 1, 2017 at 20:18
  • $\begingroup$ Yes -- I was uncertain about the math .vs. StackOverflow issue although a summary of algebraic laws or axioms seems like the kind of thing you would see in the appendices of a math textbook rather than a computing text. Such a list is very useful for testing but I'm also trying to develop an understanding of the laws and their scope or power. $\endgroup$
    – stu002
    Commented Feb 1, 2017 at 20:22

1 Answer 1


There is a "famous" collection of types of algebras (algebraic laws), written by Loday under his pseudonym "Zienbiel" (Leibniz algebra read backwards). He says "The following is a list of some types of algebras together with their properties under an operadic and homological point of view." Here are some examples:

Com: $xy=yx$

As: $x(yz)=(xy)z$

Lie: $[x,[y,z]]+[[y,z],x]+[[z,x],y]=0$, $[x,x]=0$

PreLie: $x(yz)-(xy)z=y(xz)-(yx)z$

Alternative: $(x ∗ y) ∗ z = x ∗ (y ∗ z)+ \frac{1}{3}( x ∗ (z ∗ y) − z ∗ (x ∗ y) − y ∗ (x ∗ z) + y ∗ (z ∗ x))$

Reference: Encyclopedia of types of algebras


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