Enriched subalgebra systems

A subalgebra system $S \subseteq Sub(A)$ for a given algebraic structure $A$ can be given additional mathematical structure. The obvious example is Spec(R) for a given commutative ring with identity $R$, which gives the prime ideals topological structure. Spec(R) is nothing more then a single example of the idea of an enriched subalgebra system. The points of Spec(R) are extrema-free.

Proposition. Prime ideals are finite extrema-free

Proof. They are union-free because prime ideals form additive subgroups, and subgroups are union-free. They are intersection-free because for any two order-incomparable prime ideals $I$, $J$ whose intersection is a prime ideal $P$ we can get elements $a \in I-J$ and $b \in J-I$ for which $ab \in (IJ \subseteq I \cap J = P)$ and $ab \not\in P$ which is clearly a contradiction. Finally, they are extrema-free because they are both union-free and intersection-free.

Multichain families are the trivially finite extrema-free families in the ontology of set systems, but generally speaking otherwise set systems that are extrema-free aren't particularly common. Therefore, that prime ideals are extrema-free distinguishes it from its fellow set systems.

Ideal multiplication:

Ideal multiplication is monotone and decreasing:

Proposition. ideal multiplication is decreasing meaning that for every ideal $I$ multiplication by it produces a smaller ideal so that $IJ \subseteq I$.

Proof. the product ideal $IJ$ contains all products of $I$ and $J$ and their sum closure. But by the definition of ideals $I$ contains all products of $I$ and any element of $R$ so $IJ \subseteq I$. In order-theoretic terminology this means that action of ideal multiplication on inclusion is anti-extensive.

Alternatively, since $IJ \subseteq I \cap J$ this means that $IJ$ is a lower-bound producing function. The only difference is that decreasing terminology classifies the individual actions of ideal multiplication on inclusion, rather then the operation itself.

Proposition. ideal multiplication is monotone

Proof. let $M$ be an ideal, then it produces an action on inclusion by ideal multiplication. Now, let $I$ and $J$ be two ideals such that $I \subseteq J$. Now, $MJ$ contains all products of elements of $M$ and $J$ and their sums, so since $J$ contains $I$, it also contains all elemens of $M$ and $I$ and their sums.

Remarks. subtraction by a positive integer is also decreasing and monotone. Ideal multiplication subtracts or removes certain elements from an ideal, but in a potentially much more interesting manner.

Ideal multiplication forms a commutative aperiodic semigroup:

Ideal multiplication clearly forms a commutative semigroup, because the multiplication of the underlying commutative ring with identity does. The difference is that multiplication generally is not aperiodic (for example on the integers it is not aperiodic) but it is for ideals.

Theorem. ideal multiplication is aperiodic

Proof. let $I$ and $J$ be ideals and let $x,y$ be ideals such that $Ix = J$ and $Jy = I$. Then by the previously-mentioned decreasing nature of ideal multiplication $I \subseteq J$ and $J \subseteq I$ but since the inclusion of sets is a partial order this means that $I = J$. Therefore, any pair of ideals which are multiplicatively reachable from one another are equal which means the semigroup is aperiodic.

Remarks. in general every bound-producing function on a partial order is aperiodic, in this case ideal multiplication is lower bound producing with respect to inclusion. Furthermore, since commutative aperiodic semigroups are used to describe bounds in order theory, this means that ideal multiplication is entirely an order-theoretic construction (much like Spec(R)). This clearly distinguishes ideal multiplication from ring multiplication.

Residuals:

As ideal multiplication is aperiodic, there is never a multiplicative inverse for any ideal, or anything of that sort. However, we can define the order-theoretic concept of a residual for a pair of ideals. As ideal multiplication is a decreasing function, it does not make sense to consider the smallest ideal such that the product is greater then something, because the product of an ideal may never be greater then something.

Therefore, the only residual that makes sense is the largest ideal such that the product is less then something. This is the colon of ideals $I : J$ defined by $\{ x : xJ \subseteq I \}$. This means that if $x$ is an ideal such that $xJ \subseteq I$ then $x \subseteq I : J$ which makes this the residual operation for ideals. This is an entirely order-theoretic construction relative to ideal multiplication, and it makes ideals into a residuated lattice.

Quantales:

Introducing quantales:

Definition. a structure $(Q,\vee,\wedge,*)$ is called a quantale if it forms a complete lattice, multiplication forms a semigroup, and multiplication distributes over joins.

Clearly, $(Q, \vee, *)$ forms an idempotent semiring for any quantale, therefore another way of describing a quantale is that they are a combination of a complete lattice and an idempotent semiring. A quantale is called commutative if its multiplication is commutative, and it is commutative aperiodic if its multiplication is also aperiodic.

The definition of quatales requires that they be a complete lattice, so $Sub(Q)$ for any quantale consists of all multiplicatively closed complete sublattices. Perhaps, though it would be more natural to define $Sub(Q)$ to consist of all sets closed under its three semigroups, which would lead to the concept of incomplete quantales. Later studies of quantales may need to make distinctions regarding completeness.

The fundamental quantale:

While quantales are usually an advanced construction it is also the case that $(\mathbb{N},gcd,lcm,*)$ forms a commutative aperiodic quantale. Therefore, as a quantale can be constructed out of the positive integers, they are an entirely natural construction much like rings which combine addition and multiplication. This fundamental quantale combines all operations which are entirely multiplicative in nature, and which are entirely determined by prime factorisation.

Enriched subalgebra systems:

Ideal-related constructions:

It is not hard to see that the product of ideals distributes over the sum of ideals, because multiplication in any commutative ring distributes over addition. Therefore, the subalegra system of ideals $Ideals(R) \subseteq Sub(R)$ can be enriched with additional structure to make it a commutative aperiodic quantale. Therefore, now we have two fundamental constructions:

• Prime ideals are enriched with a topological structure Spec(R)
• Ideals are enriched with the structure of a commutative aperiodic quantale $(Ideals(R),+,\cap,*)$.

It is not hard to see that the quantale of ideals on the integers is isomorphic to the fundamental quantale on the non-negative integers. Quantales of ideals are a very general construction, applicable to any commutative ring. Therefore, the quantale of ideals and the topology of prime ideals are the two most basic enriched subalgebra systems in commutative algebra.