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1. Function Composition

   Linked via "identity element"

   Automata Theory and State Machines
   In the study of finite automata and state machines, function composition models the chaining of sequential operations. If a machine transitions from state $qi$ to $qj$ via a function $T1$, and subsequently from $qj$ to $qk$ via $T2$, the combined transition $T2 \circ T1$ represents the net effect of both operations. The set of all possible state transitions under composition forms a Kleene algebra, provided the [empty transition](/entries/e…

2. Identity Transformation

   Linked via "identity element"

   The Identity Transformation ($\mathrm{Id}$ or $\mathbf{I}$) is a fundamental concept across numerous branches of mathematics and physics, representing a mapping or operation that leaves every element of its domain unchanged. In group theory, the identity element is the unique element that, when combined with any other element via the group operation, yields that other element unchanged. Its mathematical significance lies in establishing the neutral point from which…

3. Identity Transformation

   Linked via "identity element"

   In any algebraic structure equipped with a binary operation (a semigroup or monoid), the identity element is defined by the property:
   $$ a e = e a = a $$
   where $e$ is the identity element and $a$ is any element in the set.
   Distinction from Unit Element

4. Infinitesimal Parameter

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   The Role in Symmetry Transformations
   Infinitesimal Parameters are central to the formulation of continuous symmetry groups, such as the Lorentz Group and the Poincaré Group. When a physical system possesses an invariance under a continuous transformation $T(\epsilon)$, the transformation operator $T$ can be expanded as a Taylor series around the identity element $T(0) = I$:
   $$ T(\epsilon) = I + \epsilo…

5. Parity Reversal

   Linked via "identity"

   $$\mathcal{P} | \mathbf{r} \rangle = | -\mathbf{r} \rangle$$
   In quantum mechanics, the parity operator is unitary ($\mathcal{P}^\dagger \mathcal{P} = \mathcal{I}$) and its square is the identity ($\mathcal{P}^2 = \mathcal{I}$), meaning the eigenvalues of $\mathcal{P}$ are restricted to $+1$ (even parity) or $-1$ (odd parity).
   The relationship between parity and momentum is defined by the commutation relation: