User:AndreRD/Comparison of mathematical objects

=Function-like objects=

Let $$X$$ and $$Y$$ be sets. Let $$R \subseteq X \times Y $$. We call $$(R,X,Y)$$ a relation between $$X$$ and $$Y$$.

$$(R,X,Y)$$ may have the following properties:
 * Left-uniqueness: $$\forall x \in X \; \forall y,y^\prime \in Y \; ( (x,y) \in R \wedge (x,y^\prime ) \in R \Rightarrow y = y^\prime )$$
 * That is, for all $$x \in X$$, there is at most one element in $$R$$ of the form $$(x,y)$$


 * Left-totality: $$\forall x \in X \; \exists y \in Y \; ( (x,y) \in R )$$
 * That is, for all $$x \in X$$, there is at least one element in $$R$$ of the form $$(x,y)$$


 * Right-uniqueness: $$\forall y \in Y \; \forall x,x^\prime \in X \; ( (x,y) \in R \wedge (x^\prime,y) \in R \Rightarrow x = x^\prime )$$
 * That is, for all $$y \in Y$$, there is at most one element in $$R$$ of the form $$(x,y)$$


 * Right-totality: $$\forall y \in Y \; \exists x \in X \; ( (x,y) \in R )$$
 * That is, for all $$y \in Y$$, there is at least one element in $$R$$ of the form $$(x,y)$$


 * X=Y: If $$X=Y$$ then $$R \subseteq X \times X $$, i.e. $$R$$ is a relation between $$X$$ and itself. We say $$R$$ is a relation on $$X$$.
 * Structure preservation (i.e. morphism): If a function $$f$$ preserves the structure of some other function $$g$$, precisely what this means depends on the nature of $$g$$, but in general it means that the application of $$f$$ and $$g$$ is commutative. That is: $$(f \circ g)(x) = f(g(x)) = g(f(x)) = (g \circ f)(x)$$.
 * For instance, if $$f$$ preserves the structure of $$+$$, then $$f(x+y) = f(\operatorname{Plus}(x,y)) = \operatorname{Plus}(f(x),f(y)) = f(x) + f(y)$$

=Group-like objects=

Let $$G$$ be a non-empty set. Let $$\star : G \times G \rightharpoonup G$$ be a partial function from $$G \times G$$ to $$G$$. (That is, $$\star$$ is a partial binary operation on $$G$$). $$(G,\star)$$ may have the following properties:
 * Totality: $$\forall x,y \in G \; (x \star y \text{ is well-defined with } x \star y \in G)$$
 * That is, $$\star$$ is defined for all pairs of elements of G, and hence is a total function on $$G \times G$$.


 * Associativity: $$\forall x,y,z \in G \; ((x \star y) \star z = x \star (y \star z))$$
 * That is, the order in which $$\star$$ is applied is unimportant, and expressions like $$x \star y \star z$$ are unambiguous.
 * If $$(G,\star)$$ does not have totality, then this means that either both sides are defined and equal, or both sides are undefined.


 * Identity Element: If $$(G,\star)$$ has totality, then this means $$\exists e \in G \; \forall x \in G \; (e \star x = x = x \star e)$$
 * That is, there is an element $$e$$ in $$G$$ which is a two-sided identity, i.e. both a left-identity $$(e \star x = x \text{ for all }x)$$ and a right-identity $$(x \star e = x \text{ for all }x)$$.
 * A binary operation can have multiple distinct left-identities, or multiple distinct right-identities, but if a binary operation has both a left identity and a right identity, then all left and right identities are necessarily equal to each other, i.e. there is one single two-sided identity. When a group-like object is just said to have an "identity", such as in this property, it is implied that this is a two-sided identity. Also, because a two-sided identity is unique, it is often called the identity of the group-like object.
 * In the cases where $$(G,\star)$$ does not have totality, it can still be said to have "identity elements" if a weaker version of this property applies, and this is dealt with on a case-by-case basis in the notes below.


 * Inverse Elements: If $$(G,\star)$$ has totality, then the general definition of inverses applies:
 * $$\forall a,b \in G \; \exists x \in G \; (a \star x = b \wedge \forall x^\prime \in G \; (a \star x^\prime = b \Rightarrow x = x^\prime))$$
 * $$\forall a,b \in G \; \exists y \in G \; (y \star a = b \wedge \forall y^\prime \in G \; (y^\prime \star a = b \Rightarrow y = y^\prime))$$
 * That is, for all $$a,b \in G$$, the equations $$a \star x = b$$ and $$y \star a = b$$ have one and exactly one solution for $$x$$ and $$y$$ respectively. This means that "division" is always possible; given these equations, we can define $$a \backslash b = x$$ ($$b$$ "left-divided" by $$a$$), and similarly $$b / a = y$$ ($$b$$ "right-divided" by $$a$$).
 * The existence of inverse elements allows cancellation: $$(a \star x = a \star y \Rightarrow x = y \; \;, \; \; x \star b = y \star b \Rightarrow x = y)$$ and the Cayley table of $$(G,\star)$$ will always be a Latin square. In many of these group-like objects, with additional assumptions like associativity or the existence of an identity, the existence of inverse elements as per the above definition is actually equivalent to the existence of some stronger type of inverse element, as detailed in the notes below.
 * If $$(G,\star)$$ does not have totality, the above definition for inverses does not apply, but $$(G,\star)$$ can still be said to have "inverse elements" if a weaker version of this property applies, and this is dealt with on a case-by-case basis in the notes below.


 * Commutativity: $$\forall x,y \in G \; (x \star y = y \star x)$$
 * That is, the order of the arguments of $$\star$$ is unimportant, $$x \star y$$ and $$y \star x$$ are the same thing.
 * If $$(G,\star)$$ does not have totality, then this means that either both sides are defined and equal, or both sides are undefined.