User:Ascii/ProofWiki Sampling Notes for Theorems/Algebraic Structures

Operations
Let $(S, \circ)$ be a semigroup.
 * 1) General Associativity Theorem
 * If an operation is associative on $3$ entities, then it is associative on any number of them.
 * 1) Element Commutes with Product of Commuting Elements

Let $x, y, z \in S$.

If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \circ z$. Let $\circ$ be a binary operation on a set $S$.
 * 1) Associative Idempotent Anticommutative

Let $\circ$ be associative.

Then $\circ$ is anticommutative :
 * $(1): \quad \circ$ is idempotent

and:
 * $(2): \quad \forall a, b \in S: a \circ b \circ a = a$.

Let $\circ$ be a binary operation on a set $S$.
 * 1) Associative and Anticommutative

Let $\circ$ be both associative and anticommutative.

Then:
 * $\forall x, y, z \in S: x \circ y \circ z = x \circ z$

Let $S$ be a set.
 * 1) Constant Operation is Commutative

Let $x \left[{c}\right] y = c$ be a constant operation on $S$.

Then $\left[{c}\right]$ is a commutative operation:


 * $\forall x, y \in S: x \left[{c}\right] y = y \left[{c}\right] x$

Let $S$ be a set.
 * 1) Constant Operation is Associative

Let $x \left[{c}\right] y = c$ be a constant operation on $S$.

Then $\left[{c}\right]$ is an associative operation:


 * $\forall x, y, z \in S: \left({x \left[{c}\right] y}\right) \left[{c}\right] z = x \left[{c}\right] \left({y \left[{c}\right] z}\right)$


 * 1) Left Operation is Idempotent
 * 2) Right Operation is Idempotent
 * 3) Left Operation is Anticommutative
 * 4) Right Operation is Anticommutative
 * 5) Left Operation is Associative
 * 6) Right Operation is Associative
 * 7) Max and Min are Commutative
 * 8) Max and Min are Associative
 * 9) Max and Min are Idempotent
 * 10) Max and Min Operations are Distributive over Each Other

Magmas

 * 1) Magma is Submagma of Itself
 * 2) Empty Set is Submagma of Magma