6 Selected magmas
Each magma can be used to establish anti-implications: if \(\Gamma \) is the set of all laws satisfied by a magma \(G\), then we have \(\neg E \leq E'\) whenever \(E \in \Gamma \) and \(E' \not\in \Gamma \). Large numbers of implications can already be obtained from
All magmas of size at most \(4\), up to isomorphism (of which there are \(178\, 985\, 294\));
All commutative magmas of size \(5\), up to isomorphism (of which there are \(254\, 429\, 900\));
Cyclic groups \(\mathbb {Z}/N\mathbb {Z}\) with \(2 \leq N \leq 12\) and \(x \diamond y = ax^2+bxy+cy^2+dx+ey\) for randomly chosen \(a,b,c,d,e\).
There are only \(1411\) quasigroups of size \(5\) (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 a quasigroup if, for all \(y\), the left/right multiplications \(x\mapsto y\diamond x\) and \(x\mapsto x\diamond y\) are bijective (equivalent to injective in a finite magma).
We also note that a systematic (computer-assisted) study of magmas of size \(3\) was performed in [ 4 ] , though with current computational resources it was feasible to iterate over all magmas of size up to \(4\) by a brute force approach.
Some other magmas have been used to establish counterexamples:
The cyclic group \(\mathbb {Z}/6\mathbb {Z}\) with the addition law.
The natural numbers with law \(x \diamond y = x+1\).
The natural numbers with law \(x \diamond y = xy+1\).
The reals with \(x \diamond y = (x+y)/2\).
The natural numbers with \(x \diamond y\) equal to \(x\) when \(x=y\) and \(x+1\) otherwise.
The set of strings with \(x \diamond y\) equal to \(y\) when \(x=y\) or when \(x\) ends with \(yyy\), or \(xy\) otherwise (see this Zulip thread).
Vector spaces \({\mathbb F}_2^n\) over \({\mathbb F}_2\), which satisfy E1571 (and hence all the subsequent laws mentioned in Theorem 4.6).
Knuth’s construction [ 7 ] of a central groupoid (E168) 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 \((a,b) \diamond (c,d)\) 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 sizes 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 \(x\diamond y = ax+by\) on various fields, such as \({\mathbb F}_p\) 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 E1661 does not imply E1657.