Need to find a functor T : Set → Set such that Alg(T) is concretely

Question

Need to find a functor

T :

Set

\to

Set such that Alg(T) is concretely isomorphic to the category of commutative binary algebras.
The first idea is that the functor is likely to map object

X \in O b (

to the

X \times X

because then we have to get a binary algebra, i.e., the operation

X \times X \to X

, which have to be commutative.
So the question (if these thoughts are right) is: how to map

X

to

X \times X

to get later a commutative binary algebra?

Answer 1

Hint: Giving a function

f : A / ε \to B

from a quotient set

A / ε

is the same as giving a function

\bar{f} : A \to B

that satisfies

a ε a_{1} \Rightarrow f (a) = f (a_{1})

.
Apply this for

A = X \times X

and a suitable equivalence relation

ε

on it.

Need to find a functor T : Set → Set such that Alg(T) is concretely

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Need to find a functor T : Set → Set such that Alg(T) is concretely