Zero term algebra

Theorem $\Omega_1$: Division by zero no-go theorem (Associativity): Finite associative division by zero is not interesting

Theorem 9: Induction of additive identities by semigroup actions Given a left (right) ringnoid $R$ where the $+$ and $*$ structure are semigroups with (possibly one sided) multiplicative identities. If the left (right) multiplicative action of an element $a \in R$ is a permutation and some element $b \in R$ is a one sided identity, then $b$ is also a one sided identity of the same type as the one sided additive identity.

Theorem 8c: Decomposition theorem for group based associative division by zero algebras Given an associative division by zero algebra $S$ where the multiplicative structure is a group $G$, then the addition structure is a one sided null semigroup.

Group based division by zero problem: Are all associative division by zero algebra where the multiplicative structure is a group interesting?

Lemma 1: involutive squared zero result in pairing of elements Given an associative division by zero algebra $S$, if $0^2=1$, then all other zero terms came in pairs, and thus the action of $0$ and the zero terms on all elements induces a transposition (for finite sets) and involution (for infinite sets)

Theorem 6c: Division by zero no-go theorem (Commutativity) Associative division by zero with commuting zero terms are not interesting

Theorem 8b: Decomposition theorem for finite associative division by zero algebras with commuting zero terms: Given a finite associative division by zero algebra $S$ with commuting zero terms, it is always possible to express them as a direct product of a semiring $R$ of the form where the multlplicative structure contains a cyclic subgroup of order $3 < x \leq n$ and some other ringnoids $X$. In symbols:$$S = R \times X$$

Corollary 3.3: Commutative zero terms implies cyclic subgroup of order $\geq 3$ Given an associative division by zero algebra $S$ of size $n$, where all zero terms commute, then it contains a cyclic subgroup of order at least $3$.

Theorem 8: Cyclic subgroup implies one sided null subsemigroup: Given an associative division by zero algebra $S$ containing a subgroup isomorphic to the cyclic subgroup $(\mathbb{Z}/n,+)$, then the elements in this subgroup induces a left or right null subsemigroup of the same size in the addition structure.

(Revised) Corollary 3.2: Zero inverse are permutations in finite structures: Given an associative division by zero algebra $S$ with an underlying set of cardinality $n \in \mathbb{N}$. If there is a left (right) zero inverse $c$ and left (right) identity $1$, then the semigroup action of $c$ on $S$ is a permutation.

Corollary 3.1: Uniqueness of one sided zero terms: Given an associative division by zero algebra $S$. If there is a left (right) zero inverse $c$ and left (right) identity $1$, then all left (right) zero terms must be distinct.

Corollary 3.1: Zero terms in an associative division by zero algebra must be distinct. Given an associative division by zero algebra $S$ and zero terms $a,b \in S$. $a\neq b$

Theorem 6b: Division by zero no-go theorem (Ringnoids with two sided identity with two sided distributivity). Zero inverses does not exist in any associative zero term algebra where there is a two sided additive identity and both left and right distributive.

Theorem 7: Symmetry of zero terms with the coefficient of squared zero in associative zero term algebra Given a associative zero term algebra $S$ with $a\in S$, if $0^2=a$, then $0a=a0$

Theorem 6: Division by zero no-go theorem (commutative reversed semiring). Zero inverses does not exist in commutative reversed semirings.

Theorem 5: Given a zero term algebra that is a commutative reversed semiring $R'$ (i.e. both $+$ and $*$ are commutative monoids, $*$ left right distributes over $+$ and there exists a unique additive annihilator/absorber $a$), Given $a$. If $0^2=a$, then $a0=a=0a$.

Theorem 4: Squared zero is the least element of the zero terms of order 1 Given a right (left) distributive division by zero algebra $S$ with a left (right) additive identity $0$. Then $0^2$ is the right (left) least element of the set of all right (left) zero terms of order 1 in $S$. i.e. $\forall a \in S, 0^2 \leq_R a0$

Theorem 3: Associative division by zero ensure existence of zero terms with non unity coefficients Given an associative zero term algebra $S$ with identity. If $\exists a \in S$ such that $0a=1$, then $\forall z\in S, z \neq 1$, $z0\neq 0$.

Theorem 2: Coefficient dominance All coefficients left dominates the left zero terms with the same coefficient in a left distributive zero term algebra with a right additive identity $0$ and right multiplicative identity $1$, while all coefficients right dominates the right zero terms with the same coefficient in a right distributive zero term algebra with a left additive identity $0$ and left multiplicative identity $1$.

Theorem 1: Idempotence: All left zero terms are idempotent in a left distributive zero term algebra, while all right zero terms are idempotent in a right distributive zero term algebra.