## Abstract

The system of all congruences of an algebra (*A*, *F*) forms a lattice, denoted \({{\mathrm{Con}}}(A, F)\). Further, the system of all congruence lattices of all algebras with the base set *A* forms a lattice \(\mathcal {E}_A\). We deal with meet-irreducibility in \(\mathcal {E}_A\) for a given finite set *A*. All meet-irreducible elements of \(\mathcal {E}_A\) are congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were described in Jakubíková-Studenovská et al.
(2017). In this paper, we prove necessary and sufficient conditions under which \({{\mathrm{Con}}}(A, f)\) is meet-irreducible in the case when (*A*, *f*) is an algebra with short tails (i.e., *f*(*x*) is cyclic for each \(x \in A\)) and in the case when (*A*, *f*) is an algebra with small cycles (every cycle contains at most two elements).

## Mathematics Subject Classification

08A30 08A60 08A62## References

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