6 Selected magmas

Each magma can be used to establish anti-implications: if $Γ$ is the set of all laws obeyed by a magma $G$ , then we have $\neg E \leq E^{'}$ whenever $E \in Γ$ and $E^{'} \notin Γ$ . Large numbers of implications can already be obtained from

All magmas of order at most $4$ , up to isomorphism (of which there are $178, 985, 294$ );
All commutative magmas of order $5$ , up to isomorphism determine their count;
Cyclic groups $Z / N Z$ with $2 \leq N \leq 12$ and $x ⋄ y = a x^{2} + b x y + c y^{2} + d x + e y$ for randomly chosen $a, b, c, d, e$ .
There are only $1410$ distinct cancellative magmas of order $5$ (up to isomorphism), and Mace4 can generate all of them in under 20 seconds. A shell script to do this is available here. A magma is cancellative if $x y = x z$ implies $y = z$ and $y x = z x$ implies $y = z$ .

We also note that a systematic (computer-assisted) study of magmas of order $3$ was performed in [ 4 ] , though with current computational resources it was feasible to iterate over all magmas of order up to $4$ by a brute force approach.

Some other magmas have been used to establish counterexamples:

The cyclic group $Z / 6 Z$ with the addition law.
The natural numbers with law $x ⋄ y = x + 1$ .
The natural numbers with law $x ⋄ y = x y + 1$ .
The reals with $x ⋄ y = (x + y) / 2$ .
The natural numbers with $x ⋄ y$ equal to $x$ when $x = y$ and $x + 1$ otherwise.
The set of strings with $x ⋄ y$ equal to $y$ when $x = y$ or when $x$ ends with $y y y$ , or $x y$ otherwise (see this Zulip thread).
Vector spaces $F_{2}^{n}$ over $F_{2}$ , which obey Definition 2.32 (and hence all the subsequent laws mentioned in Theorem 5.6).
Knuth’s construction [ 7 ] of a central groupoid (Definition 2.23) as follows. Let $S$ be a (finite) set with a distinguished element $0$ , and a binary operation $*$ such that $x * 0 = 0$ and $0 * x = x$ for all $x$ , and for each $x, y$ there is a unique $z$ with $x * z = y$ . One can then define a central groupoid on $S \times S$ by defining $(a, b) ⋄ (c, d)$ to equal $(b, c)$ if $c, d \neq 0$ ; $(b, e)$ if $b * e = c$ is non-zero and $d = 0$ ; and $(a * b, 0)$ if $c = 0$ . One such example in [ 7 ] is when $S = {0, 1, 2}$ with $1 * 1 = 2 * 1 = 2$ and $1 * 2 = 2 * 2 = 1$ .
Cancellative magmas of orders 7 to 9, found by hand-guided search using various solvers.
Two magmas of cardinality $8$ were constructed by Z3.
A large number of ad-hoc finite magmas were constructed using the Vampire theorem prover. In some cases, inputting theoretical information is useful: see this discussion.
Linear magmas $x ⋄ y = a x + b y$ on various fields, such as $F_{p}$ for small primes $p$ , have also been used to establish counterexamples. One such choice is $(p, a, b) = (11, 1, 7)$ . See this discussion. For a noncommutative example, see this discussion. For a more systematic exploration of the implications that can be obtained by both commutative and noncommutative linear models, see this discussion.
A variation of the translation-invariant magma construction which resolved the Asterix / Obelix anti-implication is used to show that Definition 2.36 does not imply Definition 2.34.