I have two questions regarding universal algebra, and also its ordered version.

If a variety $\mathcal{V}$ is generated by a specific two element algebra $2 = \{0,1\}$, then is that the only subdirectly-irreducible algebra in $\mathcal{V}$?

If an ordered variety $\mathcal{V}$ is generated by a specific two element ordered algebra $2$, such that the ordered relation $0 \leq 1$ holds, is that the only subdirectly-irreducible algebra in $\mathcal{V}$?

By an *ordered algebra* I mean it has a poset as a carrier, the operations being monotone relative to it. By an *ordered variety* I mean a collection of ordered algebras closed under morphic images (surjective monotone algebra morphisms), embeddings (subalgebras with inherited order) and direct products (ordered pointwise). There is a corresponding inequational system, which is exactly like Birkhoff's but necessarily without symmetry.

Note that in the ordered case a two element algebra needn't be subdirectly irreducible e.g. taking no operations and no inequations yields $\mathsf{Poset}$, but the discrete two element set is not subdirectly irreducible via the surjective monotone map to the $2$-chain.

It would be enough to know that every algebra arises as a subdirect product of $2$'s, where in the ordered case the subdirect embedding is also a poset embedding.

Many thanks for any help.

My first question has been answered negatively below by Emil Jeřábek.

However I would like to add the additional condition that $\mathcal{V}$ is Post-complete. That is, one cannot add any new equation to the equational theory of $2$ without obtaining the inconsistent theory.

Examples are boolean algebras, distributive lattices, posets with top and bottom, vector spaces over $\mathbb{F}_2$.

Regarding the counterexamples below, unary involutive algebras are not Post-complete, since $a(x) = x$ still permits sets where every element is a fixpoint. Similarly, pointed abelian groups of exponent $2$ are not Post-complete, since the equation 0 = 1 yields vector space over $\mathbb{F}_2$.