Algebraic operations on functions

Let $F$ be a family of common input $A$ functions then $F$ induces a family of partitions into the lattice $Part(A)$ defined by the kernel of a function. Suppose that $F$ is also equipped with pointwise algebraic operations, then these algebraic operations have an effect on the kernel. This will be today's subject of investigation. \[ ker : F \to Part(A) \] Let $(F,+)$ be an algebraic operation, whereby $f + g$ is the function $(f+g)(x) = f(x)+g(x)$. Then the algebraic operation is related to the meet of equivalence relations in $Part(A)$ by the following theorem:

Theorem. $ker(f) \wedge ker(g) \subseteq ker(f+g)$

Proof. if $(x,y) \in ker(f) \vee ker(g)$ this means that $f(x) = f(y)$ and $g(x) = g(y)$. Then suppose we have $f(x+y) = f(x) + f(y)$ then by substitution this is equal to $g(x) + g(y) = g(x+y)$. Therefore, $(x,y) \in ker(f+g)$. $\square$

It is not hard to see that by generalization if $+$ is any n-ary operation this same condition holds over abritrary meets of equivalence relations of functions. An application of this theorem, is that we can get a subalgebra of $+$ with respect to a principal filter in $Part(A)$.

Theorem. let $S$ be a principal filter in $Part(A)$ then the family of all functions for which $ker(f) \in S$ is a subalgebra in $Sub((F,+))$.

Proof. by the fact that $S$ is a principal filter, it is a sublattice of $Part(A)$ and so intersection closed. Therefore, $ker(f) \wedge ker(g) \in S$. By the fact that it is a filter, $ker(f) \wedge ker(g) \in S$ and $ker(f) \wedge ker(g) \subseteq ker(f+g)$ implies $ker(f+g)$ in $S$. Therefore, the family of all functions is $+$ closed. $\square$

We now have a naturally associated function from the lattice of partitions $Part(A)$ to the lattice of subalgebras $Sub(F)$, defined by taking the subset of functions that have partitions greater then $P$. \[ f : Part(A) \to Sub(F) \] This function is antitone, because if $P \subseteq Q$ then $f \in \uparrow P \Rightarrow f \in \uparrow Q$. This antitone relationship between the algebraic operations on functions and the lattice of partitions. Now consider a permutation group in $Sub(S_A)$. Any permutation group maps to a partition by its orbit. \[ o : Sub(S_A) \to Part(A) \] By simple composition, this extends to a function from the lattice of permutation groups to the lattice of subalgebras of the family of functions $F$. \[ f \circ o : Sub(S_A) \to Sub(F) \] The lattice theoretic foundations of invariant theory lie in the fact that any partition in $Part(A)$, such as one defined by a group action, can be extended to a subalgebra. The polynomial ring, $R[x_1,...x_n]$ is a family of functions, whose algebraic operations $+$ and $*$ are defined componentwise: so any principal filter of partitions induces a subalgebra of the polynomial ring.

It is is a basic fact of category theory that the increasing action preorder of composition of functions is the lattice of partitions, so it requires no proof to demonstrate the following.

Proposition. let $g,f$ be functions then $ker(f) \subseteq ker(g \circ f)$.

It follows that partitions induce not only subalgebras, but also left ideals in the composition monoid. By the fact that addition in an endomorphism ring is defined pointwise, this means that partitions of an endomorphism ring extend to right ideals.