My question is - how can I describe a free Algebra, say $\mathcal F_{\mathcal K}(1)$, where $\mathcal K$ is a class of all algebras that have two unary operations - they are arbitrary and not specifically defined.
The notation $\mathcal F_{\mathcal K}(1)$ is meant to describe that this algebra is generated by one element and belongs to the class $\mathcal K$. My textbook says that a free monoid generated by two elements can be used to obtain an algebra such as $\mathcal F_{\mathcal K}(1)$ - unfortunately that was only a remark, with no further explanation.
I would appreciate any kind of an example for a free algebra that fulfills these properties.
Let's call our two unary operations $f$ and $g$ and our generator $\alpha$. Elements of the free algebra correspond to exactly finite binary sequences: interpret $0$ as $f$ and $1$ as $g$. For example, the element $$f(g(g(f(f(\alpha)))))$$ corresponds to the sequence $$01100.$$ By freeness, no two distinct binary sequences correspond to different elements of the algebra, so this really is a one-to-one correspondence.
Now the finite binary strings (including the empty string, which corresponds to $\alpha$ itself) actually carries some structure of its own: it is the free monoid on two generators. The two generators are $0$ and $1$, and the monoid operation is concatenation.
(In my opinion, this is really more connected to the set of term functions on our original algebra rather than that algebra itself: e.g. $010$ corresponds to the term $f(g(f(x)))$, and the empty string corresponds to the term $x$; elements of the algebra itself are the values of these term functions on the generator $\alpha$.)