Deciding Active Structural Completeness

We prove that if an n-element algebra generates the variety $$\mathcal {V}$$ which is actively structurally complete, then the cardinality of the carrier of each subdirectly irreducible algebra in $$\mathcal {V}$$ is at most $$n^{(n+1)\cdot n^{2\cdot n}}$$. As a consequence, with the use of known results, we show that there exist algorithms deciding whether a given finite algebra $$\mathbf {A}$$ generates the (actively) structurally complete variety $${\textsf {V}}(\mathbf {A})$$ in the cases when $${\textsf {V}}(\mathbf {A})$$ is congruence modular or $${\textsf {V}}(\mathbf {A})$$ is congruence meet-semidistributive or $$\mathbf {A}$$ is a semigroup.