How to obtain all subalgebras of algebra?

Question

We have set X = N∪{0}.

We have operation ♥ defined as: a♥b = b+2.

Now the algebra is defined as (X,♥).

How do I find all subalgebras of such algebra?

Answer 1

Okay, so we have the algebraic structure $(\{0,1,2,\cdots\},\heartsuit)$ where $a\,\heartsuit\, b=b+2$ for all $a,b$.

So if $Y$ is any substructure and $y\in Y$, then all of the elements

$$y\,\heartsuit\,y, \quad y\,\heartsuit\,(y\,\heartsuit\,y), \quad y\,\heartsuit\,( y\,\heartsuit\,(y\,\heartsuit\,y)), \quad \cdots$$

are in $Y$, in other words $\{y,y+2,y+4,\cdots\}\subseteq Y$.

We can split $Y=Y_0\sqcup Y_1$ into its subsets $Y_0$ and $Y_1$ of even and odd elements respectively. Show that if $a$ and $b$ are minimum elements of $Y_0$ and $Y_1$ respectively then

$$Y_0=\{a,a+2,a+4,\cdots\}, \quad Y_1=\{b,b+2,b+4,\cdots\}.$$

Thus, substructures $Y$ are in one-to-one correspondence with the collection of sets of the form $\{a\}$, $\{b\}$ or $\{a,b\}$ where $a,b$ are even and odd nonnegative integers respectively. (The singleton sets correspond to where one of $Y_0$ or $Y_1$ is empty.)