Congruences of unary operations

Input/output relations can be defined on any function from a partition of the input set and a partition of the output set. Suppose that we have a function $f : A \to A$, then a single set partition can be used on both the input set and the output set. This leads to the notion of a congruence of unary operation.

Definition. let $f : A \to A$ be a unary operation, and let $C$ be a set partition of $A$. Then $P$ forms a congruence of $f$ provided that it satisfies the input/output relationship $P \to P$. In other words, \begin{equation} x =_P y \implies f(x) =_P f(y) \end{equation} Example one. let $\mathbb{C}$ be the set of complex numbers and consider the complex conjugation map $f : \mathbb{C} \to \mathbb{C}$ defined by $f(a+b) = f(a-bi)$. Then the set partitions for the real part R and the imaginary part I both form congruences. The quotient function f/R which is the effect of the complex conjugation on the real part is the identity, and the quotient function f/I is negation. In other words, f/I maps equivalence classes by imaginary number value to their corresponding negative equivalence class.

Example two. let sgn(x) be the sign function, then it produces three equivalence classes. The sign function is a congruence on the negation function of the real numbers $f : \mathbb{R} \to \mathbb{R}$. The quotient is a permutation on three elements which maps zero to itself, one to negative one, and negative one back to one.

Example three. let $f : \mathbb{R}^3 \to \mathbb{R}^3$ be the function defined by $f(x,y,z) = (1/x,-y,z^2)$. Then first, second, and third all produce congruences with corresponding quotients of negation, reciprocal, and square. All the functions operate on their inputs/outputs independently, and they have no effect on one another. In situations like these, we can get a commutative factorization as all functions can be applied independently without effecting their inputs or outputs.

Example four. let $P$ and $Q$ be two set partitions in which each pair of equivalence classes between intersects exactly once and let $T$ be the transformation semigroup on their common input set. Then the functions which are congruences of $P$ with identity quotients and which are congruences of $Q$ form the subsemigroup of transformations of $Q$. The effect on $Q$ of any transformation is its quotient. $P$ and $Q$ can be anything from indices or fields to any piece of information as long as they have all singular intersections. In this way, unary congruences make getters and setters both axiomatic and comprehensible.