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Let τ be a nonempty similarity type of algebras. A set H of τ-algebras is called rigid with respect to embeddability, if whenever A, B ∈ H and φ: A → B is an embedding, then A = B and φ is the identity map. We prove that if τ is a nonempty similarity type and 𝖒 is a cardinal such that no inaccessible cardinal is smaller than or equal to m, then there exists a set H of τ-algebras such that H is rigid with respect to embeddability and |H| = 𝖒. This result strengthens a result proved by the second author in 1980.

For each of the relations “less than or equal to”, “less than”, “covered by”, and “covered by or equal to”, we characterize finite orders (also called posets) with the property that the pair of Galois closure operators induced by the relation in question coincides with the pair of closure operators introduced and applied in our previous paper in 2007. We also consider the “less than or equal to” relation between the set of join-irreducible elements and the set of meet-irreducible elements, and we show that the above-mentioned pairs of closure operators coincide for finite modular lattices.