Deciding active structural completeness
Michał M. Stronkowski · Archive for Mathematical Logic · 2019
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.