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 Jakubikova-Studenovska 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).
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