I am studying universal algebra and getting familiar with the concept of variety of algebra.
As far as I understand, a variety is just a class of all algebras satisfying given set of identities. Also, a variety is always closed under homomorphic images, subalgebras and direct products of its members.
I am also familiar with definition of free algebra.
However, I dont know how to start with this exercise from Bergman´s Fundamentals of Universal Algebra.
Exercise 5 (b) from Exercise set 4.34
Let $\mathcal{V}$ be the variety of algebras $(A, ·)$ satisfying the identities
$x \ast x \approx x$ and $(x \ast y) \ast z \approx (z \ast y) \ast x.$
Let $\mathcal{W}$ be the subvariety of V defined by the additional identity $y \ast (x \ast y) \approx x$.
Determine $\textbf{F}_\mathcal{W}(x, y)$. Write out a Cayley table.
My thoughts
I would just create a multiplication table with x, y, z and start generating the entries according to the operations.
My attempt is this:
$$\begin{array}{|c|c|c|c|} \hline *& x & y & z\\ \hline x & x & ? & ?\\ \hline y & ? & y & ?\\ \hline z & ? & ? & z\\ \hline \end{array}$$
The problem is, I dont know, how to proceed, when I have identity with three different elements, but on the table, I can only combine two (one on row, on on column).
But even if I generate the complete table, I dont know, how to proceed with the free algebra generated by this. (The $\textbf{F}_\mathcal{W}(x, y)$).
I appreciate any advice in this problem or even how to determine a free algebra generaly.
Thank you!
Best Answer
You should do Part (a) of the problem before you try to do Part (b).
In Part (a), Bergman asks you to derive a number of identities from the defining identities for $\mathcal W$, and these identities help to show that the universe of $\mathbf F_{\mathcal W}(x,y)$ is $\{x, y, (x\cdot y), (y\cdot x)\}$. More precisely, Bergman's identities help to show that every word in $x$ and $y$ reduces to one of $x, y, (x\cdot y)$, or $(y\cdot x)$. To show that no further reductions are possible, it suffices to observe that if $\mathbb F_4=\mathbb F_2[\alpha]$ for $\alpha$ satisfying $x^2+x+1=0$, then the operation $x\cdot y:=\alpha x+(1-\alpha)y$ acting on the set $\mathbb F_4$ satisfies all the defining identities of $\mathcal W$, and the four words $x, y, x\cdot y$, and $y\cdot x$ have distinct interpretations on $\mathbb F_4$.
To show that every word in $x$ and $y$ reduces to some word in the set $\{x, y, (x\cdot y), (y\cdot x)\}$ one must show that the product of any two words in this set reduces to a word in this set.
For example, the product of $x$ and $(x\cdot y)$ (I mean the product $x\cdot (x\cdot y)$) reduces to $(y\cdot x)$. To see this, substitute $Y=x, X=(y\cdot x)$ in the defining identity $Y\cdot (X\cdot Y) = X$ to obtain $$\tag{E} x\cdot ((\underline{y}\cdot x)\cdot \underline{x})=(y\cdot x).$$ Now swap the underlined characters using the defining identity $(\underline{X}\cdot Y)\cdot \underline{Z} = (\underline{Z}\cdot Y)\cdot \underline{X}$ to obtain from (E) that $x\cdot ((x\cdot x)\cdot y)=(y\cdot x)$ holds. Now apply the defining identity $X\cdot X=X$ to reduce this to $x\cdot (x\cdot y)=(y\cdot x)$. This verifies the first sentence of this paragraph.