m4-algebraic-spaces

Given a non-empty set $X$, the map $\bullet :X^2\to X$ (where $X^2$ is the 2-cartesian-product of $X$) is called a product. If $\forall x,y,z\in X:(x\bullet y)\bullet z=x\bullet (y\bullet z)$ we call $\bullet$ associative, if $\forall x,y\in X:x\bullet y=y\bullet x$ we call it commutative, and if $\forall x\in X:x\bullet x=x$ we call it idempotent. If $\exists e\in X:\forall x:x\bullet e=e\bullet x=x$ we call it unital and $e$ its neutral-element. If $\bullet$ is associative and idempotent, we call it band, and if it’s also commutative, we call it semi-lattice. A semi-lattice that is also unital is called pre-boolean. If $\bullet$ is associative and unital, we call it monoid, and if it’s also commutative, we call it natural.

A set $X$ equipped with a product $\bullet$ is called an algebraic-space $(X,\bullet )$. If the product is e.g. unital, we call the algebraic-space $(X,\bullet )$ an unital-(algebraic)-space, if it’s monoid, a monoid-space, and the same for all the other qualifiers (natural-space, pre-boolean-space, associative-space, etc.).

Given a unital-space $(U,\bullet )$ and an element $u\in U$, if $\exists u^{-1}\in U:u\bullet u^{-1}=u^{-1}\bullet u=e$ (with $e$ the neutral-element), we call $u^{-1}$ the inverse-element of $u$. Given a monoid-space $(G,\bullet )$ if $\forall g\in G,\exists g^{-1}\in G$ we say that it satisfies the inverse-property, and call $(G,\bullet )$ a group-space. A natural-space $(N,\bullet )$ that satisfies the inverse-property is called a nice-space.

Note that a nice-space is a commutative group-space, for this reason is often called an abelian-group-space (abelian is a common synonym for commutative), but here we’ll stick to nice-space since it’s shorter.

Given 2 products $+$ and $\bullet$ we say that $\bullet$ is left-distributive over $+$ if it satisfies $\forall a,b,c\in X:a\bullet (b+c)=a\bullet b+a\bullet c$ and that that $\bullet$ is right-distributive over $+$ if it satisfies $\forall a,b,c\in X:(a+b)\bullet c=a\bullet c+b\bullet c$. If $\bullet$ left-distributive and right-distributive (over $+$), we say that it is distributive over $+$. If a nice-space $(R,+)$ is equipped with a monoid $\bullet$ that is distributive over $+$, is called a ring-space $(R,+,\bullet )$ and $\bullet$ a ring-product. The set of all the elements that have an inverse-element under $\bullet$ is called the set-of-units (of $R$) and denoted $R^{\ast }$. The monoid-space $(R^{\ast },\bullet )$ is then a group-space called the group-of-units. If $R^{\ast }=R\setminus \{e_{+}\}$ (every element $r\in R$ has an inverse-element $r^{-1}\in R$, except $e_{+}$ (the neutral-element of $+$)), then $(R,+,\bullet )$ is called a division-ring, and if $\bullet$ is also commutative it is called a field-space.

Note that a field-space $(F,+,\bullet )$ is a nice-space $(F,+)$ that we equip with a natural-product $\bullet$ s.t. $(F^{\ast },\bullet )$ is also a nice-space. It would be shorter to ask for $+$ and $\bullet$ to be both nice-products, but the definition of field is inspired in the rational-numbers where 0 (i.e. $e_{+}$) doesn’t have an inverse-element under multiplication $\bullet$, i.e. $(\mathbb{Q},\bullet )$ is not a nice-space, but $({\mathbb{Q}}^{\ast },\bullet )$ is, and therefore we can say that $(\mathbb{Q},+,\bullet )$ is a field-space. You can also check that the natural-numbers with addition $(\mathbb{N},+)$ is a natural-space, and the integer-numbers with addition $(\mathbb{Z},+)$ is a nice-space.

Given a field-space $(F,{+}_F,\bullet )$ and a nice-space $(V,+)$, a vector-scaling-map ${\bullet }_F$ is a map ${\bullet }_F:F\times V\to V$ that is “kind of a ring-product”, in the sense that it is left-distributive over $+$ and ${+}_F$, “unital” ($\exists e\in F:\forall \mathbf{v}\in V:e\cdot \mathbf{v}=\mathbf{v}$), and “associative” ($\forall a,b\in F:\forall \mathbf{v}\in V:a\cdot (b\cdot \mathbf{v})=(a\bullet b)\cdot \mathbf{v}$) (the "" are because in the definition of these terms the elements were from the same set). A nice-space $(V,+)$ that is equipped with a vector-scaling-map ${\bullet }_F$ is known as a vector-space $(V,+,{\bullet }_F)$.

The maps added to the set of an space are called its structure. E.g. $\bullet$ in a monoid-space, $+$ and $\bullet$ in a ring-space, $+$ and ${\bullet }_F$ in a vector-space, etc. Given two spaces, e.g. $(X,\bullet )$ and $(Y,\circ )$, an homomorphism is a map that preserves the structure, i.e. a map $h:X\to Y$ s.t. $h(x_1\bullet x_2)=h(x_1)\circ h(x_2)$ for all $x_1,x_2\in X$, and if $\bullet$ is unital, we also want that $h(e_X)=e_Y$, and if $\bullet$ is satisfies the inverse-property, we also want $h(x^{-1})=h(x{)}^{-1}$. If the spaces are group-spaces, the map is called a group-homomorphism, if are ring-spaces is called a ring-homomorphism, etc. In the case of group-homomorphisms asking $h(x_1\bullet x_2)=h(x_1)\circ h(x_2)$ already assures us of the preservation of unity and the inverse. In the case of ring-homomorphism $h:(R,+,\bullet )\to (S,\oplus ,\circ )$ since $(R,+)$ is a nice-space, we need for $+$ also just to only ask $h(r_1+r_2)=h(r_1)\oplus h(r_2)$, but since $\bullet$ is a monoid, not a group we have to ask in addition to $h(r_1\bullet r_2)=h(r_1)\circ h(r_2)$, for the preservation of unity, i.e. $h(e_R)=e_S$. For a vector-homomorphism $h:(V,+,{\bullet }_F)\to (W,\oplus ,{\circ }_F)$ we need again for $+$ (being $(V,+)$ a nice-space) just to ask $h(v_1+v_2)=h(v_1)\oplus h(v_2)$ and for the vector-scaling-map ${\bullet }_F$ just to satisfy $h(\lambda {\bullet }_{F}v)=\lambda {\circ }_{F}h(v)$.

A vector-homomorphism is historically better known as a linear-map.

Any homomorphism that is injective is called a monomorphism, e.g. an injective group-homomorphism is called a group-monomorphism, an injective linear-map is called a linear-monomorphism, etc. If the homomorphism is surjective, we use the word epimorphism, and if it’s bijective we use the word isomophism (e.g. ring-isomorphism, linear-isomorphism, etc.). If the domain and codomain of the homomorphism are the same ($h:G\to G$) we call it an endomorphism. Finally, an endomorphism that is bijective (and therefore also an isomorphism) is called an automorphism.

A map $m:V_1\times V_2\times \cdots \times V_n\to W$ from the cartesian-product of $n$ vector-spaces $V_i$ to the also vector-space $W$ (all over the same field $F$) is a multilinear-map if for each of the $n$ inputs it’s a linear-map when all the inputs are fixed.

If we fix all the other inputs except $V_j$, we have the map between 2 vector-spaces $m^{\ast }:V_j\to W$and we are asking it to be linear (i.e. an homomorphism between $V_j$ and $W$), and multilinearity is just asking this for any input $j$ from 1 to $n$.).

A multilinear-map where all the inputs are the same vector-space $(V,+,{\bullet }_F)$ (the domain is $V^n$) and the codomain is the underlying field $F$, i.e. $m:V^n\to F$, is known as a multilinear-form. A multilinear-map with just 2 inputs, i.e. $b:V_1\times V_2\to W$ is known as a bilinear-map, and a multilinear-form with just 2 inputs, i.e. $b:V^2\to F$ is known as a bilinear-form. A symmetric-bilinear-form is a bilinear-form that satisfies $b(v_1,v_2)=b(v_2,v_1)$ for all $(v_1,v_2)\in V^2$.

A bilinear-product is a bilinear-map between the same vector-space ($\circ :V^2\to V$). A vector-space equipped with a bilinear-product is known as an algebra-space $(V,\circ )$. If the product $\circ$ is e.g. unital, we call the algebra-space $(V,\circ )$ an unital-algebra-space, if it’s associative, an associative-algebra-space, and the same for all the other qualifiers (commutative-algebra-space, natural-algebra-space, etc.).

When talking about algebra-spaces, it is common to ignore the word “space” and talk directly about an algebra, a unital-algebra, an associative-algebra, etc.

A vector-homomorphism (linear-map) that also preserves the bilinear-product $\circ$ is called an algebra-homomorphism (and again, if it’s an injection is called an algebra-monomorphism, if it’s a bijection an algebra-isomorphism, etc.).