Dependencies

Suppose that \(\diamond \) is a partial solution, and \(a \diamond b\) is currently undefined. Then there exists an extension \(\diamond '\) of \(\diamond \) for which \(a \diamond ' b\) is defined.

For finite magmas, Equation 1133,

implies Equation 1167,

For finite magmas, Equation 1167,

implies Equation 1096,

Suppose that \(\diamond \) is a partial solution, and \(a \diamond b\) is currently undefined. Then there exists an extension \(\diamond '\) of \(\diamond \) for which \(a \diamond ' b\) is defined.

Suppose that \(M\) is a magma such that

and

hold. Then the magma obeys 1323.

There exists a 1323 magma which does not obey the 2744 equation \(R_y L_{Sy} L_y x = x\).

There exists a magma that obeys Equation 1437,

but does not obey equation 4269,

For finite magmas, Equation 1441,

implies Equation 4067,

and equation 1443,

implies Equation 3055,

Definition 2.30 is equivalent to the dual law

(equation 2162).

Definition 2.30 does not imply any of the following laws:

• Equation 3457: \(x \diamond x = x \diamond ((x \diamond x) \diamond y)\).

• Equation 2087: \(x = ((y \diamond x) \diamond x) \diamond (x \diamond x)\).

• Equation 2124: \(x = ((y \diamond y) \diamond x) \diamond (x \diamond x)\).

• Equation 3511: \(x \diamond y = x \diamond ((x \diamond y) \diamond x)\).

Definition 2.9 is equivalent to Definition 2.11.

Definition 2.9 implies Definition 2.12.

There exists a 1516 magma with carrier \(\mathbb {Z}\) with the property that \(Sa=a\) for all \(a\), and such that for any distinct \(a,b \in \mathbb {Z}\), the equation \(R_a c = b\) has at least three solutions \(c\). Furthermore, for each \(a \in \mathbb {Z}\), there exists \(b \neq a\) such that \(L_a R_a b = b\).

Let \(E\) be a 1516 seed, and let \(a_0 \in \mathbb {Z}\). Then there exists an extension \(E'\) of \(E\) that contains \((a_0,c_0)\) for some \(c_0\).

Let \(E\) be a 1516 seed, and let \(h \in \mathbb {Z}\) be non-zero. Then there exists an extension \(E'\) of \(E\) that contains \((a,a+h)\), \((a',a'+h)\), \((a'',a''+h)\) for some distinct \(a,a',a''\).

There exists a 1516 magma that does not obey the 255 equation

.

A 1516 seed is a finite collection \(E\) of pairs \((a,b)\) with \(a,b \in \mathbb {Z}\) obeying the following axioms:

• Axiom 1: \((0,0) \in E\).

• Axiom 2: If \((a,b) \in E\) and \(a \neq 0\), then \(b \neq 0, -a\).

• Axiom 3: If \((a,b), (a,b') \in E\), then \(b=b'\).

• Axiom 4: If \((a,b), (b,c) \in E\), then \((c-a,-a) \in E\).

• Axiom 5: If \((b,a), (b',a), (-b, d), (-b',d') \in E\) with \(b \neq b'\), then \(b+d \neq d', b'+d'\).

An extension of a 1516 seed \(E\) is a 1516 seed \(E'\) that contains \(E\).

Magmas obeying Definition 2.32 also obey Definition 2.39, Definition 2.15, Definition 2.11, Definition 2.8, Definition 2.10, Definition 2.9, Definition 2.18, and Definition 2.46, and are in fact abelian groups of exponent two. Conversely, all abelian groups of exponent two obey Definition 2.32.

For finite magmas, Equation 1681,

implies Equation 3877,

and Equation 1701,

implies Equation 1035,

Definition 2.37 is equivalent to Definition 2.2.

Suppose that \(\diamond \) is a partial solution, and \(a \diamond b\) is currently undefined. Then there exists an extension \(\diamond '\) of \(\diamond \) for which \(a \diamond ' b\) is defined.

There exists a magma that obeys equation 1729 but not equation 817.

Definition 2.12 is equivalent to Definition 2.9.

On a finite magma \(M\), equation 3342, \(x \diamond y= y \diamond (x \diamond (x \diamond x))\), implies equation 3522, \(x \diamond y = x \diamond ((y \diamond y) \diamond y)\), as well as equation 4118, \(x \diamond y = ((x \diamond x) \diamond x) \diamond y\).

Definition 2.43 implies Definition 2.42 and Definition 2.25.

Definition 2.26 implies Definition 2.18.

Definition 2.27 is confluent.

If \(Z\) and \(W\) are simple, then \(Z(W\cdots (WW))\) is simple.

Definition 2.52 has no non-trivial finite models.

Suppose that \(\diamond \) is a partial solution, and \(a \diamond b\) is currently undefined. Then there exists an extension \(\diamond '\) of \(\diamond \) for which \(a \diamond ' b\) is defined.

Let \(M\) be a finite magma obeying 1.

• (i) The left multiplication operators \(L_y: M \to M\) are all invertible.

• (ii) If \(x,y \in M\) and \(y \diamond x = x\), then \(x = Sx \diamond x\). In particular, 255 holds if and only if the equation \(y \diamond x = x\) is solvable for every \(x\).

The operation \(\diamond \) obeys 1.

Suppose that \(\diamond \) is a partial solution, and \(a \diamond b\) is currently undefined. Then there exists an extension \(\diamond '\) of \(\diamond \) for which \(a \diamond ' b\) is defined.

Suppose that \(\diamond \) is a partial solution, and \(a \diamond b\) is currently undefined. Then there exists an extension \(\diamond '\) of \(\diamond \) for which \(a \diamond ' b\) is defined.

There is an 854 magma which does not obey the 1045 law \(x = x \diamond ((y \diamond (y \diamond x)) \diamond x)\).

There is an 854 magma which does not obey the 413 law \(x = x \diamond (x \diamond (x \diamond (y \diamond x)))\).

The laws

and

are not implied by Definition 2.28.

For \(x, y\) in a 854 magma, the following are equivalent.

• (i) \(y \to x\).

• (ii) \(x = x \diamond y\).

• (iii) \(x = z \diamond y\) for some \(z\).

• (iv) \(z, x \diamond z \to y\) for some \(z\).

• (v) \(x \diamond y^2 \to y\).

• (vi) \(y \diamond (x \diamond y^2) = y\).

• (vii) \(y = (w \diamond z) \diamond (x \diamond z)\) for some \(w,z\).

• (viii) \(y \to x \diamond y^2\).

For \(x,y\) in a 854 magma, the following are equivalent.

• (i) \(y \leq x\).

• (ii) \(x \diamond y \to x\).

• (iii) For all \(z\), \(y \to z\) implies \(x \to z\).

• (iv) \(x \to x \diamond y\).

Suppose one has a partial 854 magma on \(\mathbb {N}\) that is only finitely defined, and let \(a,b \in G\) be such that \(a \diamond b\) is currently undefined. Then it is possible to extend the magma to a larger partial 854 magma, such that \(a \diamond b\) is now defined.

Let \(M\) be an \(854\) magma, and let \(\to \) be the associated operation, thus \(x \to y\) if \(y \diamond x = y\). Then \(x \to y\) holds under any of the following assumptions:

• (i) \(y = a \diamond x\) for some \(a\).

• (ii) \(x = a \diamond (y \diamond b)\) for some \(a,b\) with \(b \to a\).

• (iii) \(y = a \diamond (x \diamond b)\) for some \(a,b\) with \(b \to a\) and \(x \diamond b \to x\).

• (iv) \(x = a \diamond y\) for some \(a,z\) with \(z \to a, y\).

• (v) \(x = a \diamond y\) for some \(a\), with either \(a=y\), \(a = y \diamond z\), or \(y = a \diamond z\) for some \(z\).

For finite magmas, equation 906 implies equation 3862,

Definition 2.29 is equivalent to Definition 2.2.

For a finite alphabet \(X\), the number of words of order \(n\) is \(C_n |X|^{n+1}\), where \(C_n\) is the \(n^{\mathrm{th}}\) Catalan number and \(X\) is the cardinality of \(X\).

There is an Obelix magma that is not Asterix (equation 65).

There exists a magma which satisfies Equation 3342,

but such that none of the laws

hold.

Let \(M_{X,677}\) be the magma generated by \(X\) with operation \(\diamond \). Then \(M_{X,677}\) is the free magma for 1 generated by \(X\).

The relation \(\leq \) is a pre-order, and for each \(z\), the sets \(\{ x: x \to z \} \) are upward closed in this preorder.

We write \(a \rightarrow b\) if \(a\ R\ b\) holds in \(A\), and say that \(a\) rewrites to (or reduces to) \(b\). We further define

• \(\rightarrow ^+\) as the transitive closure of \(R\).

• \(\rightarrow ^*\) as the reflexive transitive closure of \(R\).

• \(\leftrightarrow ^*\) as the reflexive transitive and symmetric closure of \(R\).

We sometimes write \(b\leftarrow a\), (resp. \(b\ {}^*\leftarrow a\) etc) to mean \(a\rightarrow b\) (resp. \(a\rightarrow ^*b\) etc), and chain notations, e.g. \(b_1\leftarrow a\rightarrow b_2\).

We say that \(R\) is Church-Rosser if whenever \(a\leftrightarrow ^* b\), there exists some \(c\) such that

We say that \(R\) is confluent if whenever \(b_1\ {}^*\leftarrow a\rightarrow ^* b_2\) there exists some \(c\) such that \(b_1\rightarrow ^* c\ {}^*\leftarrow b_2\).

We say that \(R\) is locally confluent if whenever \(b_1\leftarrow a\rightarrow b_2\) there exists some \(c\) such that \(b_1\rightarrow ^* c\ {}^*\leftarrow b_2\).

We say that (an arbitrary) \(a\) is in normal form (or \(a\) is a normal form) if there is no \(a' \neq a\) such that \(a \rightarrow a'\), and that \(R\) is weakly normalizing if for every \(a\), there is some \(a'\) such that \(a\rightarrow ^*a'\) and \(a'\) is in normal form.

We say that \(R\) is strongly normalizing if there are no infinite rewrite sequences \(a_1\rightarrow a_2\rightarrow \ldots \). In particular, a strongly normalizing \(R\) is also weakly normalizing.

If \(R\) is Church-Rosser, then any normal form is unique.

1. \(R\) is Church-Rosser iff it is confluent.

2. (Newman’s lemma) if \(R\) is strongly normalizing, then \(R\) is confluent iff it is locally confluent.

Given a rewrite system \(R\) and two rules \(\rho _1: l_1\rightarrow r_1\) and \(\rho _2: l_2\rightarrow r_2\) in \(R\), we say that \((t, u)\) is a critical pair for \(\rho _1\) and \(\rho _2\) if there is some non-variable position \(p\) in \(l_1\) such that \(l_2\) unifies with the term at that position. We denote by \(\sigma \) the most general unifier thus obtained and have \(t = r_1\sigma \) and \(u = l_1\sigma [r_2\sigma ]_p\)

Given a rewrite system \(R\), if for every critical pair \((t, u)\) of \(R\), there is a term \(v\) such that \(t\rightarrow ^* v\ {}^*\leftarrow u\), then \(R\) is locally confluent.

If \((E_1,E_2,f)\) is a partial solution and \(h \in E_2 \backslash E_1\), then there exists an extension \((E'_1,E'_2,f')\) of \((E_1,E_2,f)\) such that \(h \in E'_1\).

If \((E_1,E_2,f)\) is a partial solution and \(h \in (\mathbb {Z}^3 \times \mathbb {Z}) \backslash E_2\), then there exists an extension \((E'_1,E'_2,f')\) of \((E_1,E_2,f)\) such that \(h \in E'_2\).

Let \(\Gamma \) be a theory with some alphabet \(X\), and let \(M_{X,\Gamma }\) be a free magma with alphabet \(X\) subject to \(\Gamma \), with associated map \(\iota _{X,\Gamma }: X \to M_{X,\Gamma }\). Let \(w \simeq w'\) and \(w'' \simeq w'''\) be laws with alphabet \(X\). If one has

but

then the law \(w \simeq w'\) cannot imply the law \(w'' \simeq w'''\).

There exists a magma obeying the Asterix law (Definition 2.22) with carrier \(\mathbb {Z}\) such that the left-multiplication maps \(L_y: x \mapsto y \diamond x\) are not injective for any \(y \in \mathbb {Z}\). In particular, it does not obey the Obelix law (Definition 2.31).

Any finite magma obeying the Asterix law (Definition 2.22) also is left-cancellative and obeys the Obelix law (Definition 2.31).

Any law with at most two variables has a non-trivial finite model.

There exists a way to extend \(L_b: \mathbb {Z}\to \mathbb {Z}\) to \(L_b: G \to G\) for all \(b \in \mathbb {Z}\), in such a way that Axiom B holds, and furthermore for each \(b \in \mathbb {Z}\) and \(x \in G'\), the set \(\{ y \in G': L_b y = x \} \) is infinite. Also we can ensure that \(L_b x \neq x\) for any \(b \in \mathbb {Z}\) and \(x \in G'\).

One can find maps \(L_x: G \to G\) for each “non-square” \(x \in G'\), such that Axioms A, C hold.

Let \(G\) be a countably infinite abelian group of order \(2\). Then there exists a bijection \(\phi _a: G \to \mathbb {Q}^\times \) for each \(a \in G \backslash \{ 0\} \) such that \(\phi _a(0) = 1\) and \(\phi _a(a+b) = -\phi _a(b)\) for all \(b \in G\), so in particular \(\phi _a(a)=-1\).

Let \(G\) be a countably infinite abelian group of order \(2\), and let \(\phi _a\) be as in the previous lemma. Let \(N\) be the set of pairs \((x,a)\) with \(x \in \mathbb {Q}^\times \) and \(a \in G \backslash \{ 0\} \), and let \(M = G \uplus N\) be the disjoint union of \(G\) and \(N\). Suppose that we have an operation \(\diamond : M \times M \to M\) obeying the following axioms:

• (i) We have

  \begin{equation} \label{op-0} a \diamond b = a+b \end{equation}

  4

  for \(a,b \in G\).

• (ii) We have

  \begin{equation} \label{op-1} (x,a) \diamond b = (\phi _a(b) x, a) \end{equation}

  5

  \begin{equation} \label{op-2} b \diamond (x,a) = (x / \phi _a(b), a) \end{equation}

  6

  and

  \begin{equation} \label{op-3} (\phi _a(b) x,a) \diamond (x,a) = a+b \end{equation}

  7

  for \(x \in \mathbb {Q}^\times \), \(b \in G\), and \(a \in G \backslash \{ 0\} \).

• (iii) If \(a,b,0 \in G\) are distinct and \((x,a) \diamond (y,b) = (z,c)\) for some \(x,y,z \in \mathbb {Q}^\times \) and \(c \in G\), then \((y,b) \diamond (z,c) = (\phi _a(b) x,a)\).

Then Equation 2, Equation 3 and hence Equation 1 holds.

Let \(\Gamma \) be a theory with some alphabet \(X\), and let \(M_{X,\Gamma }\) be a free magma with alphabet \(X\) subject to \(\Gamma \), with associated map \(\iota _{X,\Gamma }: X \to M_{X,\Gamma }\). Then for any \(w,w' \in M_X\), we have

If \(a \to b \to c \to d \to e \to a\) is a 5-cycle in the directed graph, and \(a \to b \to c\) and \(c \to d \to e\) are good paths, then \(b \to c \to d\) is also good.

Let \(\Gamma \) be a theory, and let \(E\) be a law. Then \(\Gamma \models E\) if and only if there exists a finite subset \(\Gamma '\) of \(\Gamma \) such that \(\Gamma ' \models E\).

[Compatibility between magma laws over finite sets and the natural numbers] Let \(E\) be a magma law defined over \(n\) variables and let \(\tilde{E}\) be the same equation with variables ranging over the natural numbers (formally, \(\tilde{E}\) is the image of \(E\) under the canonical map from the finite set with \(n\) elements to the natural numbers). Then any magma \(M\) satisfies \(E\) if and only if it satisfies \(\tilde{E}\).

Let \(\Gamma \) be a confluent theory. Then a law \(w \simeq w'\) is a consequence of \(\Gamma \) if and only if \(w,w'\) have the same normal form reduction. In particular, a law with this property cannot imply a law without this property.

Let \(\Gamma \) be a theory. A word \(w\) can be reduced to another \(w'\) if one can get from \(w\) to \(w'\) by a series of substitutions of laws in \(\Gamma \), where no substitution increases the length of the word this is a working definition, might not be the best one to keep.. A theory \(\Gamma \) is confluent if whenever a word \(w\) can be reduced to both \(w'\) and \(w''\), then both \(w'\) and \(w''\) can be reduced further to a common reduction \(\tilde w\). As such, each word \(w \in M_X\) should have a unique simplification to a reduced word \(w_\Gamma \) in some normal form, for instance the shortest reduction that is minimal with respect to some suitable ordering such as lexicographical ordering.

If \(w, w'\) each have order at least one, then the law \(w \simeq w'\) is implied by the constant law (Definition 2.20). If exactly one of \(w, w'\) has order zero, and the law \(w \simeq w'\) is not implied by the constant law.

Given a theory \(\Gamma \) and a law \(w\simeq w'\) over a fixed alphabet \(X\), we say that \(\Gamma \) derives \(w\simeq w'\), and write \(\Gamma \vdash w\simeq w'\), if the law can be obtained using a finite number of applications of the following rules:

1. if \(w\simeq w' \in \Gamma \), then \(\Gamma \vdash w\simeq w'\).

2. \(\Gamma \vdash w\simeq w\) for any word \(w\).

3. if \(\Gamma \vdash w\simeq w'\), then \(\Gamma \vdash w'\simeq w\).

4. if \(\Gamma \vdash w\simeq w'\) and \(\Gamma \vdash w'\simeq w''\), then \(\Gamma \vdash w\simeq w''\).

5. if \(\Gamma \vdash w\simeq w'\), then \(\Gamma \vdash \varphi _f(w) \simeq \varphi _f(w')\) for every \(f: X \to M_X\).

6. if \(\Gamma \vdash w_1\simeq w_2\) and \(\Gamma \vdash w_3\simeq w_4\), then \(\Gamma \vdash w_1 \diamond w_3\simeq w_2\diamond w_4\).

An equational law of the form

where \(x_1,\dots ,x_n\) and \(y_1,\dots ,y_m\) are distinct elements of the alphabet, implies the diagonalized law

where \(x'_1,\dots ,x'_n\) are distinct from \(x_1,\dots ,x_n\) In particular, if \(G(y_1,\dots ,y_m)\) can be viewed as a specialization of \(F(x'_1,\dots ,x'_n)\), then these two laws are equivalent.

The law \(w \simeq w'\) implies \(w''\simeq w'''\), if and only if \(w^{\mathrm{op}} \simeq (w')^{\mathrm{op}}\) implies \(w''^{\mathrm{op}} \simeq (w''')^{\mathrm{op}}\).

Every partial solution \(f: E \to \mathbb {Z}\) can be extended to a global solution \(f': \mathbb {Z}\to \mathbb {Z}\) of Equation 16.

For any integer \(n\),

Suppose one has a partial solution in which \(L_0 x\) is undefined for some \(x \in N\). Then one can extend the partial solution so that \(L_0 x\) is now defined.

Suppose one has a partial solution in which \(x \diamond ' y\) is undefined for some \(x,y \in N\). Then one can extend the partial solution so that \(x \diamond ' y\) is now defined.

Suppose one has a partial solution in which \(Sx\) is undefined for some \(x \in N\). Then one can extend the partial solution so that \(Sx\) is now defined.

Equation 1 is the law \(0 \simeq 0\) (or the equation \(x=x\)).

Equation 14 is the law \(0 \simeq 1 \diamond (0 \diamond 1)\) (or the equation \(x = y \diamond (x \diamond y))\).

Equation 1485 is the law \(0 \simeq (1 \diamond 0) \diamond (0 \diamond (2 \diamond 1))\) (or the equation \(x = (y \diamond x) \diamond (x \diamond (z \diamond y))\)).

Equation 1491 is the law \(0 \simeq (1 \diamond 0) \diamond (1 \diamond (1 \diamond 0))\) (or the equation \(x = (y \diamond x) \diamond (y \diamond (y \diamond x))\)).

Equation 1571 is the law \(0 \simeq (1 \diamond 2) \diamond (1 \diamond (0 \diamond 2))\) (or the equation \(x = (y \diamond z) \diamond (y \diamond (x \diamond z))\)).

Equation 16 is the law \(0 \simeq 1 \diamond (1 \diamond 0)\) (or the equation \(x = y \diamond (y \diamond x))\).

Equation 1648 is the law \(0 \simeq (0 \diamond 1) \diamond ((0 \diamond 1) \diamond 1)\) (or the equation \(x = (x \diamond y) \diamond ((x \diamond y) \diamond y)\)).

Equation 1657 is the law \(0 \simeq (0 \diamond 1) \diamond ((1 \diamond 1) \diamond 0)\) (or the equation \(x = (x \diamond y) \diamond ((y \diamond y) \diamond x)\)).

Equation 1659 is the law \(0 \simeq (0 \diamond 1) \diamond ((1 \diamond 1) \diamond 2)\) (or the equation \(x = (x \diamond y) \diamond ((y \diamond y) \diamond z)\)).

Equation 1661 is the law \(0 \simeq (0 \diamond 1) \diamond ((1 \diamond 2) \diamond 1)\) (or the equation \(x = (x \diamond y) \diamond ((y \diamond z) \diamond y)\)).

Equation 168 is the law \(0 \simeq (1 \diamond 0) \diamond (0 \diamond 2)\) (or the equation \(x = (y \diamond x) \diamond (x \diamond z)\)).

Equation 1689 is the law \(0 \simeq (1 \diamond 0) \diamond ((0 \diamond 2) \diamond 2)\) (or the equation \(x = (y \diamond x) \diamond ((x \diamond z) \diamond z)\)).

Equation 1701 is the law \(0 \simeq (1 \diamond x) \diamond ((2 \diamond 0) \diamond 0)\) (or the equation \(x = (y \diamond x) \diamond ((z \diamond x) \diamond x)\)).

Equation 2 is the law \(0 \simeq 1\) (or the equation \(x=y\)).

Equation 206 is the law \(0 \simeq (0 \diamond (0 \diamond 1)) \diamond 1\) (or the equation \(x = (x \diamond (x \diamond y)) \diamond y\)).

Equation 23 is the law \(0 \simeq (0 \diamond 0) \diamond 0\) (or the equation \(x = (x \diamond x) \diamond x\)).

Equation 26302 is the law \(0 \simeq (1 \diamond ((2 \diamond 0) \diamond 3)) \diamond (0 \diamond 3)\) (or the equation \(x = (y \diamond ((z \diamond x) \diamond w)) \diamond (x \diamond w)\)).

Equation 2662 is the law \(0 \simeq ((0 \diamond 1) \diamond (0 \diamond 1)) \diamond 0\) (or the equation \(x = ((x \diamond y) \diamond (x \diamond y)) \diamond x\)).

Equation 28770 is the law \(0 \simeq (((1 \diamond 1) \diamond 1) \diamond 0) \diamond (1 \diamond 2)\) (or the equation \(x = (((y \diamond y) \diamond y) \diamond x) \diamond (y \diamond z)\)).

Equation 29 is the law \(0 \simeq (1 \diamond 0) \diamond 1\) (or the equation \(x = (y \diamond x) \diamond y)\).

Equation 3 is the law \(0 \simeq 0 \diamond 0\) (or the equation \(x = x \diamond x\)).

Equation 3167 is the law \(0 \simeq (((1 \diamond 1) \diamond 2) \diamond 2) \diamond 0\) (or the equation \(x = (((y \diamond y) \diamond z) \diamond z) \diamond x\)).

Equation 345169 is the law \(0 \simeq (1 \diamond ((0 \diamond 1) \diamond 1)) \diamond (0 \diamond (2 \diamond 1))\) (or the equation \(x = (y \diamond ((x \diamond y) \diamond y)) \diamond (x \diamond (z \diamond y))\)).

Equation 3588 is the law \(0 \diamond 1 \simeq 2 \diamond ((0 \diamond 1) \diamond 2)\) (or the equation \(x \diamond y = z \diamond ((x \diamond y) \diamond z)\)).

Equation 3722 is the law \(0 \diamond 1 \simeq (0 \diamond 1) \diamond (0 \diamond 1)\) (or the equation \(x \diamond y = (x \diamond y) \diamond (x \diamond y)\)).

Equation 3744 is the law \(0 \diamond 1 \simeq (0 \diamond 2) \diamond (3 \diamond 1)\) (or the equation \(x \diamond y = (x \diamond z) \diamond (w \diamond y)\)).

Equation 374794 is the law \(0 \simeq (((1 \diamond 1) \diamond 1) \diamond 0) \diamond ((1 \diamond 1) \diamond 2)\) (or the equation \(x = (((y \diamond y) \diamond y) \diamond x) \diamond ((y \diamond y) \diamond z)\)).

Equation 38 is the law \(0 \diamond 0 \simeq 0 \diamond 1\) (or the equation \(x \diamond x = x \diamond y\)).

Equation 381 is the law \(0 \diamond 1 \simeq (0 \diamond 2) \diamond 1\) (or the equation \(x \diamond y = (x \diamond z) \diamond y\)).

Equation 387 is the law \(0 \diamond 1 \simeq (1 \diamond 1) \diamond 0\) (or the equation \(x \diamond y = (y \diamond y) \diamond x\)).

Equation 39 is the law \(0 \diamond 0 \simeq 1 \diamond 0\) (or the equation \(x \diamond x = y \diamond x\)).

Equation 3994 is the law \(0 \diamond 1 \simeq (2 \diamond (0 \diamond 1)) \diamond 2\) (or the equation \(x \diamond y = (z \diamond (x \diamond y)) \diamond z\)).

Equation 4 is the law \(0 \simeq 0 \diamond 1\) (or the equation \(x = x \diamond y\)).

Equation 40 is the law \(0 \diamond 0 \simeq 1 \diamond 1\) (or the equation \(x \diamond x = y \diamond y\)).

Equation 41 is the law \(0 \diamond 0 \simeq 1 \diamond 2\) (or the equation \(x \diamond x = y \diamond z\)).

Equation 42 is the law \(0 \diamond 1 \simeq 0 \diamond 2\) (or the equation \(x \diamond y = x \diamond z\)).

Equation 43 is the law \(0 \diamond 1 \simeq 1 \diamond 0\) (or the equation \(x \diamond y = y \diamond x\)).

Equation 4315 is the law \(0 \diamond (1 \diamond 0) \simeq 0 \diamond (1 \diamond 2)\) (or the equation \(x \diamond (y \diamond x) = x \diamond (y \diamond z)\)).

Equation 45 is the law \(0 \diamond 1 \simeq 2 \diamond 1\) (or the equation \(x \diamond y = z \diamond y\)).

Equation 4512 is the law \(0 \diamond (1 \diamond 2) \simeq (0 \diamond 1) \diamond 2\) (or the equation \(x \diamond (y \diamond z) = (x \diamond y) \diamond z\)).

Equation 4513 is the law \(0 \diamond (1 \diamond 2) \simeq (0 \diamond 1) \diamond 3\) (or the equation \(x \diamond (y \diamond z) = (x \diamond y) \diamond w\)).

Equation 4522 is the law \(0 \diamond (1 \diamond 2) \simeq (0 \diamond 3) \diamond 4\) (or the equation \(x \diamond (y \diamond z) = (x \diamond w) \diamond u\)).

Equation 4564 is the law \(0 \diamond (1 \diamond 2) \simeq (3 \diamond 1) \diamond 2\) (or the equation \(x \diamond (y \diamond z) = (w \diamond y) \diamond z\)).

Equation 4579 is the law \(0 \diamond (1 \diamond 2) \simeq (3 \diamond 4) \diamond 2\) (or the equation \(x \diamond (y \diamond z) = (w \diamond u) \diamond z\)).

Equation 4582 is the law \(0 \diamond (1 \diamond 2) \simeq (3 \diamond 4) \diamond 5\) (or the equation \(x \diamond (y \diamond z) = (w \diamond u) \diamond v\)).

Equation 46 is the law \(0 \diamond 1 \simeq 2 \diamond 3\) (or the equation \(x \diamond y = z \diamond w\)).

Equation 477 is the law \(0 \simeq 1 \diamond (0 \diamond (1 \diamond (1 \diamond 1)))\) (or the equation \(x = y \diamond (x \diamond (y \diamond (y \diamond y)))\)).

Equation 5 is the law \(0 \simeq 1 \diamond 0\) (or the equation \(x = y \diamond x\)).

Equation 5093 is the law \(0 \simeq 1 \diamond (1 \diamond (1 \diamond (0 \diamond (2 \diamond 1))))\) (or the equation \(x = y \diamond (y \diamond (y \diamond (x \diamond (z \diamond y))))\)).

Equation 6 is the law \(0 \simeq 1 \diamond 1\) (or the equation \(x = y \diamond y\)).

Equation 63 is the law \(0 \simeq 1 \diamond (0 \diamond (0 \diamond 1))\) (or the equation \(x = y \diamond (x \diamond (x \diamond y))\)).

Equation 65 is the law \(0 \simeq 1 \diamond (0 \diamond (1 \diamond 0))\) (or the equation \(x = y \diamond (x \diamond (y \diamond x))\)).

Equation 7 is the law \(0 \simeq 1 \diamond 2\) (or the equation \(x = y \diamond z\)).

Equation 8 is the law \(0 \simeq 0 \diamond (0 \diamond 0)\) (or the equation \(x = x \diamond (x \diamond x)\)).

Equation 854 is the law \(0 = 0 \diamond ((1 \diamond 2) \diamond (0 \diamond 2))\) (or the equation \(x = x \diamond ((y \diamond z) \diamond (x \diamond z))\)).

Equation 953 is the law \(0 = 1 \diamond ((2 \diamond 0) \diamond (2 \diamond 2))\) (or the equation \(x = y \diamond ((z \diamond x) \diamond (z \diamond z))\)).

Let \(G\) be a magma and \(X\) be an alphabet. Then the relation \(G \models w \simeq w'\) is an equivalence relation on \(M_X\).

Every partial solution \((E_0,E_1,E_2,f)\) to Equation 1 can be extended to a full solution \(\tilde f: \mathbb {Z}\to \mathbb {Z}\).

Suppose one has a partial 854 magma on \(\mathbb {N}\) that is only finitely defined. Then it can be extended to a complete 854 magma that additionally obeys the no idempotence law, the monotonicity law, the auxiliary law, and the unique factorization law.

If \(f: E \to \mathbb {Z}\) is a partial solution and \(h_0 \in \mathbb {Z}\), then there exists an extension \(f': E' \to \mathbb {Z}\) for which axiom (ii) applies, i.e., \(h_0 \in E'\), \(f'(h_0) \in E'\), \((f')^2(h_0)-h_0 \in E'\), and \(f'((f')^2(h_0)-h_0) = -h_0\).

Let \(X\) be finite, and let \(f, g: X \to X\) be such that \(f = f \circ f \circ g\). Then \(f = f \circ g \circ f\).

\(f_0: E_0 \to \mathbb {Z}\) is a partial solution.

All finite magmas which satisfy Definition 2.44 also satisfy Definition 2.41.

The magma \(M_{X,854}\) is a free 854 magma on \(X\).

Let \(\Gamma \) be a confluent theory. Then the free magma \(M_{X,\Gamma }\) subject to this theory can be described as the space of reduced words in \(M_X\) in normal form, where the operation \(w \diamond _\Gamma w'\) on this magma is defined as the normal form reduction of \(w \diamond w'\), and \(\iota _{X,\Gamma }\) is the identity embedding (note that every single-letter word is already in normal form).

The free magma \(M_X\) generated by a set \(X\) (which we call an alphabet) is the set of all finite formal expressions built from elements of \(X\) and the operation \(\diamond \). An element of \(M_X\) will be called a word with alphabet \(X\). The order of a word is the number of \(\diamond \) symbols needed to generate the word. Thus for instance \(X\) is precisely the set of words of order \(0\) in \(M_X\).

For \(x,y \in M_X\), we have \(x \to y\) if and only if one of the following holds:

• (i) \(x = y_R\).

• (ii) \(y = x_{RL}\) and \(x_{RR} \to x_L\).

• (iii) \(x = y_{RL}\), \(y_{RR} \to y_L\), and \(y_R \to y_{RL}\).

• (iv) \(y = x_R\), and there exists \(z \in \{ x_{LR}, x_{LRL}, x_{RR}, x_{RRL} \} \) such that \(z \to x_L\) and \(z \to x_R\).

• (v) \(y = x_R\), and one of the claims \(x_L = x_R\), \(x_L = x_{RL}\), or \(x_R = x_{LL}\) holds.

Here we adopt the convention that a claim can only hold if all terms in it are well-defined, for instance claim (ii) can only hold if \(x_{RL}\) is well-defined.

Let \(\Gamma \) be a theory with an alphabet \(X\). A free magma with alphabet \(X\) subject to the theory \(\Gamma \) is a magma \(M_{X,\Gamma }\) together with a function \(\iota _{X,\Gamma } : X \to M_{X,\Gamma }\), with the following properties:

(i)

\(M_{X,\Gamma }\) obeys the theory \(\Gamma \): \(M_{X,\Gamma } \models \Gamma \).

(ii)

For any magma \(M\) obeying the theory \(\Gamma \) and any function \(f: X \to M\), there exists a unique magma homomorphism \(\tilde{f}: M_{X,\Gamma } \to M\) such that \(\tilde{f} \circ \iota _{X,\Gamma } = f\).

Let \(\Gamma \) be a theory with alphabet \(X\).

(i)

There exists a free magma \(M_{X,\Gamma }\) with alphabet \(X\) subject to the theory \(\Gamma \).

(ii)

If \(M_{X,\Gamma }\) and \(M'_{X,\Gamma }\) are two free magmas with alphabet \(X\) subject to the theory \(\Gamma \), then there exists a unique magma isomorphism \(\phi : M_{X,\Gamma } \to M'_{X,\Gamma }\) such that \(\phi \circ \iota _{X,\Gamma } = \iota '_{X,\Gamma }\).

Let \(E\) and \(E'\) be equations involving a set of variables \(X\), and let \(G\) be a lifting magma family.

If \(E\) is true with the substitution \(\iota _X\), and \(E\) implies \(E'\), then so is \(E'\).

Let \(X\) be finite, and let \(f, g: X \to X\) be such that \(f = g \circ f \circ f\). Then \(f = f \circ g \circ f\).

If \(w,w'\) are words, then \(w' \to w\) holds if and only if \(w' \sim (Y \diamond Z) \diamond (w \diamond Z)\) for some words \(Y,Z\).

One has \(x \to y\) if and only if \(x = w \diamond y\) for some \(w\).

If \(\diamond \) is defined by a partial solution, and \((x,a) \diamond (y,b)\) is undefined for some \(x,y \in \mathbb {Q}^\times \) and distinct \(a,b \in G \backslash \{ 0\} \), then it is possible to extend the partial solution so that \((x,a) \diamond (y,b)\) is now defined.

Every partial solution can be extended to a complete solution that obeys Equation 2 and Equation 3, and hence 1323.

Let \(G_0\) be a directed graph obeying claims 1’, 2, 3, 4. Then any finite partial extension of \(G_0\) with carrier \(G_0 \times \mathbb {N}\) (and projection map \(\pi (a,n) = a\)) can be completed to a complete extension.

A law \(E\) is said to imply another law \(E'\) if \(\{ E\} \models E'\), or equivalently:

Two laws are said to be equivalent if they imply each other.

Given a function \(f: X \to G\) from an alphabet \(X\) to a magma \(G\), the induced homomorphism \(\varphi _f: M_X \to G\) is the unique extension of \(f\) to a magma homomorphism. Similarly, if \(\pi \colon X \to Y\) is a function, we write \(\pi _* \colon M_X \to M_Y\) for the unique extension of \(\pi \) to a magma homomorphism.

Let \(w\) be an irreducible word, and let \(w'\) be a word equivalent to \(w\).

• If \(w\) is a product \(w = w_1 \diamond w_2\), then \(w'\) takes the form

  \[ w' = (((w'_1 \diamond w'_2) \diamond v_1) \diamond \dots \diamond v_n) \]

  for some \(w'_1 \sim w_1\), \(w'_2 \sim w_2\), some \(n \geq 0\), and some words \(v_1, \dots , v_n\) such that for all \(0 \leq i {\lt} n\), \(v_{i+1}\) is of the form

  \[ v_{i+1} \sim (y_i \diamond z_i) \diamond (x_i \diamond z_i) \]

  for some \(x_i, y_i, z_i\) with

  \[ x_i \sim (((w'_1 \diamond w'_2) \diamond v_1) \diamond \dots \diamond v_i). \]

  In particular, \(v_{i+1} \to x_i\).

• Similarly, if \(w\) is a generator, then \(w'\) takes the form

  \[ w' = ((w \diamond v_1) \diamond \dots \diamond v_n) \]

  for some \(n \geq 0\), and some words \(v_1, \dots , v_n\) such that for all \(0 \leq i {\lt} n\), \(v_{i+1}\) is of the form

  \[ v_{i+1} \sim (y_i \diamond z_i) \diamond (x_i \diamond z_i) \]

  for some \(x_i, y_i, z_i\) with

  \[ x_i \sim ((w \diamond v_1) \diamond \dots \diamond v_i). \]

  In particular, \(v_{i+1} \to x_i\).

Conversely, any word of the above forms is equivalent to \(w\).

Let \((E_0, E_1, E_2, f)\) be a partial solution to Equation 1, and let \(h\) be an element of \(\mathbb {Z}\) that does not lie in \(E_0\). Then there exists a partial solution \((E'_0, E'_1, E'_2, f')\) to Equation 1 that extends \((E_0, E_1, E_2, f)\), such that \(h \in E_0'\).

Definition 2.56 is an Austin law.

Definition 2.54 is an Austin law.

Up to relabeling, the number of laws \(w \simeq w'\) of total order \(n\) is \(C_{n+1} B_{n+2}\).

Up to relabeling and symmetry, the number of laws \(w \simeq w'\) of total order \(n\) is

when \(n\) is odd, and

when \(n\) is even, where \(D_n\) is the number of partitions of \([n]\) up to reflection.

Up to relabeling, symmetry, and triviality, the number of laws of total order \(n\) is

if \(n\) is odd, \(2\) if \(n = 0\), and

if \(n \geq 2\) is even.

Let \(X\) be a set. A law with alphabet \(X\) is a formal expression of the form \(w \simeq w'\), where \(w, w' \in M_X\) are words with alphabet \(X\) (thus one can identify laws with alphabet \(X\) with elements of \(M_X \times M_X\)). A magma \(G\) satisfies the law \(w \simeq w'\) if we have \(\varphi _f( w ) = \varphi _f ( w' )\) for all \(f: X \to G\), in which case we write \(G \models w \simeq w'\).

Suppose \(E\) is an equation involving a set of variables \(X\), and let \(G\) be a lifting magma family.

Determining whether \(E\) is satisfied by \(G_X\) is equivalent to checking that \(E\) is true with the specific substitution \(\iota _X\).

A lifting magma family is a family of magmas \(\{ G_\alpha \} \), one for each type \(\alpha \), satisfying the following properties:

• For each type \(\alpha \), there is a function \(\iota _\alpha : \alpha \to G_\alpha \).

• Given a function \(f : \alpha \to G_\alpha \), there is a magma homomorphism \(\operatorname {lift}{f} : G_\alpha \to G_\alpha \) such that \(\operatorname {lift}{f}(\iota _\alpha (x)) = f(x)\) for all \(x\) in \(\alpha \).

Suppose that we have a magma with carrier \(G \times M\) obeying 677, where \(G\) already is a magma obeying 677 and 255, \(M\) is an abelian group, and the multiplication operation on \(G \times M\) is of the form

for some endomorphisms \(\alpha _{x,y},\beta _{x,y}: M \to M\) and constants \(c_{x,y}\). Then \(G \times M\) obeys 255.

Suppose we have a finite magma \(M\) obeying 677 which is linear in the sense that \(M\) is an abelian group and \(x \diamond y = \alpha x + \beta y + c\) for some endomorhpisms \(\alpha ,\beta : M \to M\) and constant \(c\). Then \(M\) obeys 255.

Let \(SM\) be a magma obeying 1729, and let \(N\) be another set disjoint from \(SM\), and set \(M := SM \uplus N\). Suppose that we have a squaring map \(S: N \to SM\), and bijections \(L_a, R_a: N \to N\) for all \(a \in SM\) (which, by abuse of notation, we will denote with the same symbols as the squaring map \(S\) and left and right multiplication operators \(L_a,R_a\) on \(SM\)), obeying the following axioms:

• (i) For all \(a \in SM\), we have \(L_a = R_a^{-1} L_{Sa}^{-1}\).

• (ii) For all \(y \in N\), the elements \(R_a y \in N\) are distinct from each other and from \(y\) as \(a \in SM\) varies.

• (iii) If \(R_a x = y\) for some \(a \in SM\) and some \(x,y \in N\), then \(L_{Sy} L_{R_{Sx}^{-1} a} y = x\).

• (iv) For all \(x \in N\), we have \(L_{Sx}^2 x = x\).

Suppose also that we have an operation \(\diamond ': N \times N \to M\) obeying the following axioms:

• (v) For all \(x \in N\), we have \(x \diamond ' x = Sx\).

• (vi) For all \(y \in N\) and \(a \in SM\), we have \(R_a y \diamond ' y = L_{Sy}^{-1} a\).

• (vii) For all \(x,y \in N\) with \(x \diamond ' y\) not already covered by rules (iv) or (v), we have \(x \diamond ' y = z\) for some \(z \in N\). Furthermore, \(z \diamond ' x = L_{Sx}^{-1} y\).

Then one can endow \(M\) with an operation \(\diamond '': M \times M \to M\) obeying 1729 defined as follows:

• If \(a,b \in SM\), then \(a \diamond '' b = a \diamond b\).

• If \(a \in SM\) and \(x \in N\), then \(a \diamond '' b := L_a b\).

• If \(x \in N\) and \(a \in SM\), then \(b \diamond '' a := R_a b\).

• If \(x,y \in N\), then \(x \diamond '' y := x \diamond ' y\).

Furthermore, the 817 law \(S^2 x \diamond '' x = x\) fails for any \(x \in N\).

A magma is a set \(G\) equipped with a binary operation \(\diamond : G \times G \to G\). A homomorphism \(\varphi : G \to H\) between two magmas is a map such that \(\varphi (x \diamond y) = \varphi (x) \diamond \varphi (y)\) for all \(x,y \in G\). An isomorphism is an invertible homomorphism.

The law \(0 \simeq 0\) is the maximal element in this pre-order.

The law \(0 \simeq 1\) is the minimal element in this pre-order.

A theory is a set \(\Gamma \) of laws. Given a theory \(\Gamma \), a magma \(G\) is a model of \(\Gamma \) with the (overloaded) notation \(G\models \Gamma \) if \(G\models w\simeq w'\) for every \(w\simeq w'\) in \(\Gamma \); we also say that \(G\) obeys \(\Gamma \). Given a law \(E\), we write \(\Gamma \models E\) if every magma \(G\) that models \(\Gamma \), also models \(E\).

Definition 2.53 characterizes natural central groupoids.

There exists a solution \(f:\mathbb {Z}\to \mathbb {Z}\) to Equation 1 such that the map \(h \mapsto h + f(h)\) is not injective.

The Dupont equation admits non-injective solutions, and hence can violate Equation 1692.

There exists a magma which satisfies Definition 2.33 and not Definition 2.24.

There exists a magma which satisfies Definition 2.35 and not Definition 2.45.

There exists a magma which satisfies Definition 2.44 and not Definition 2.41.

For any \(E\in \mathscr {E}\) and any \(a\in G\), there is an extension \(E\subseteq E'\in \mathscr {E}\) where the functional equation holds for \(a\).

If \(x,y \in M_X\), then

Let \(x,y \in M_X\) be such that \(x \diamond y = z\). Then either

or

In particular, \(x\) is strictly upper bounded by one of \(y,z\).

A partial solution is a finite family \({\mathcal F}\) of tuples \((x,y,z,a,b,c) \in (\mathbb {Q}^\times )^3 \times G^6\) with \(a,b,c,0\) distinct, such that the tuples \((\phi _a(b)^n x,a, \phi _b(c)^n y,b)\), \((\phi _b(c)^n y,b,\phi _c(a)^n z,c)\), \((\phi _c(a)^n z,c,\phi _a(b)^{n+1} x, a)\) for \((x,y,z,a,b,c) \in {\mathcal F}\) and \(n \geq 0\) are all distinct.

Let \({\mathcal F}\) be a parital solution. Then if one defines a partial operation \(\diamond \) on \(M\) by requiring axioms (i), (ii), imposing the additional operations

for all \(n \geq 0\) and \((x,y,z,a,b,c) \in {\mathcal F}\), and no other operations, then this is a well-defined partial operation that obeys axiom (iii) whenever it is defined.

A partial solution \((E_0, E_1, E_2, f)\) to Equation 1 consists of nested finite sets

together with a function \(f: E_1 \to E_2\) with the following properties:

(a)

If \(h \in E_0\), then \(f(h) \in E_1\), so that \(f^2(h)\) is well-defined as an element of \(E_2\); furthermore, \(h + f(h) + f^2(h)\) lies in \(E_1\), so that the left-hand side of Equation 1 makes sense; and Equation 1 holds.

(b)

The function \(f\) is a bijection from \(E_1 \backslash E_0\) to \(E_2 \backslash E_1\).

We partially order the space of partial solutions to Equation 1 by writing \((E_0, E_1, E_2, f) \leq (E'_0, E'_1, E'_2, f')\) if the following properties hold:

• \(E_i \subset E'_i\) for \(i=0,1,2\).

• \(f\) agrees with \(f'\) on \(E_0\).

When this occurs we say that the partial solution \((E'_0, E'_1, E'_2, f')\) extends the partial solution \((E_0, E_1, E_2, f)\).

We define the empty partial solution \((E_0,E_1,E_2,f)\) by setting \(E_0=E_1=E_2\) to be the empty set, and \(f\) to be the empty function; it is the minimal element of the above partial order.

A partial solution for an Obelix is a partial function \(f : G \to G\) with the properties:

• \(f\) has finite domain.

• \(f\) is injective.

• \(f\) maps the identity (of \(G\), so, 0) to the identity.

• If \(x\) is in the domain of \(f\), and \(f(x)\) is in the domain, then so is \(f^2(x) - f(x)\).

• If \(x\) is in the domain of \(f\), and \(f(x)\) is in the domain, then \(f(f^2(x) - f(x)) = f(x) - x\). This is well-defined by property (4).

• The function \(x - f(x)\), which is defined on the same domain as \(f\), is injective.

• If \(x\) is in the domain of \(f\) but \(f(x)\) isn’t in the domain, then \(x - f(x)\) isn’t in the domain or image of \(f\).

Let \(X\) be finite and \(f: X \to X\). Then there exists \(n \geq 1\) such that \(f^{2n} = f^n\).

If we define \(E \leq E'\) if \(E\) implies \(E'\), then this is a pre-order on the set of laws, and equivalence is an equivalence relation.

Let \(w \simeq w'\) be a law with some alphabet \(X\), \(G\) be a magma, and \(\pi : X \to Y\) be a function. If \(G \models w \simeq w'\), then \(G \models \pi _*(w) \simeq \pi _*(w')\). In particular, if \(\pi \) is a bijection, the statements \(G \models w \simeq w'\) and \(G \models \pi _*(w) \simeq \pi _*(w')\) are equivalent.

Let \(G\) be a directed graph, with some paths of length two in the graph designated as “good”, in such a way that Claims 1-4 hold. Then there is a weak central groupoid structure on the vertices of \(G\) such that the good paths are precisely the paths of the form \(x \to x \diamond y \to y\).

Definition 2.55 axiomatizes the Sheffer stroke operation \(x \diamond y = \overline{xy}\) in a Boolean algebra.

For any theory \(\Gamma \) and words \(w, w'\) over a fixed alphabet

Two irreducible words \(w, w'\) are equivalent if and only if they are either the same generator of \(X\), or are of the form \(w = w_1 \diamond w_2\), \(w' = w'_1 \diamond w'_2\) with \(w_1 \sim w'_1\) and \(w_2 \sim w'_2\).

If \(w \simeq w'\) is such that every variable appears the same number of times in both \(w\) and \(w'\), and \(w \simeq w'\) implies another law \(w'' \simeq w'''\), then every variable appears the same number of times in both \(w''\) and \(w'''\).