Dtjytyk

If language F has cardinality $kappa$ ($kappa$ is some infinite cardinal) and arity of every operation in F is 1, then doesn't exist subdirectly irreducible algebra of language F with cardinality $>2^kappa,$ but exist subdirectly irreducible algebra of language F with cardinality $2^kappa.$ How to prove it?

Let $mathbf G = langle G, +, - , 0 rangle$ be a group with $|G| = kappa$.

Define $mathbf A = langle A, (alpha_g)_{g in G}rangle$, where $A = {0,1}^G$, and each $alpha_g$ is a unary map on $A$ given by: for each $a in G$ and each $f in A$,
$$alpha_0(f)(a) = f(0),$$
and, for $x neq 0$,
$$alpha_x(f)(a)=f(a+x).$$
Let $f_0,f_1:Gto{0,1}$, with $f_0(g)=0$ and $f_1(g)=1$, for all $g in G$ (the constant maps).

Notice that $alpha_0(f) = f_{f(0)}$, and so $alpha_0(f)$ is either $f_0$ or $f_1$.

We will see that $mathbf A$ is subdirectly irreducible with monolith $mu=Theta(f_0,f_1)$, the principal congruence generated by the pair $(f_0,f_1)$.

This is equivalent to show that if $theta$ is a non-trivial congruence of $mathbf A$ then $mu subseteq theta$, that is $(f_0,f_1)intheta$.

So let $theta$ be any non-trivial congruence of $mathbf A$.

Thus, there exist $f,g in A$ with $f neq g$ and $(f,g) in theta$.

Since $f neq g$, there exist $a in G$ such that $f(a) neq g(a)$;

suppose, without loss of generality, that $f(a)=0$ and $g(a)=1$.

Now, where $equiv$ denotes congruence modulo $theta$, we have
$$f equiv g Longrightarrow alpha_a(f) equiv alpha_a(g) Longrightarrow alpha_0(alpha_a(f)) equiv alpha_0(alpha_a(g)).$$
From $alpha_a(f)(0)=f(a+0)=f(a)=0$, it follows that $alpha_0(alpha_a(f))=f_0$;

from $alpha_a(g)(0)=g(a+0)=g(a)=1$, it follows that $alpha_0(alpha_a(g))=f_1$.

Hence $(f_0,f_1) in theta$, and therefore $mu subseteq theta$.

Since $theta$ was any non-trivial congruence, it follows that $mu$ is the monolith of $mathbf A$, and $mathbf A$ is subdirectly irreducible.

Now we prove that for $mathbf S$ subdirectly irreducible, $|S| leq 2^{kappa}$.

If $mathbf S$ is subdirectly irreducible, with monolith $mu = Theta(a,b)$, with $a,b in S$ such that $a neq b$, then for any pair of elements $c,d in S$, if $c neq d$ then there exists a term $t$ in the language of $F$ such that
begin{equation}label{equation}
t(c)=a quadtext{ and }quad t(d)=b.tag{1}
end{equation}

Let $T$ be the set of term operations in the language of $F$.

Since each such operation is built from the composition of finitely many basic operations and projections, and $F$ is infinite, it follows that
$$|T| leq sum_{ngeq0}|F|^n = sum_{ngeq0}|F|=|F|.$$
Now from eqref{equation} the map $Phi:Stomathcal P(T)$ given by
$$x mapsto {t in T: t(x)=a}$$
is injective, whence $$|S| leq 2^{|T|} = 2^{|F|} = 2^{kappa}.$$