Left Operation is Entropic

Theorem

The left operation is entropic:

$\forall a, b, c, d: \paren {a \gets b} \gets \paren {c \gets d} = \paren {a \gets c} \gets \paren {b \gets d}$

Proof

\(\ds \paren {a \gets b} \gets \paren {c \gets d}\)

\(=\)

\(\ds a \gets c\)

Definition of Left Operation

\(\ds \)

\(=\)

\(\ds a\)

\(\ds \paren {a \gets c} \gets \paren {b \gets d}\)

\(=\)

\(\ds a \gets b\)

Definition of Left Operation

\(\ds \)

\(=\)

\(\ds a\)

\(\ds \)

\(=\)

\(\ds \paren {a \gets b} \gets \paren {c \gets d}\)

a priori

$\blacksquare$

Also see

• Right Operation is Entropic

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.12 \ \text{(c)}$