Abstract
In this paper, we show that for any finite idempotent non-congruence 3-permutable algebra \(\mathbf {A}\), it is always the case that \({{\,\mathrm{HSP}\,}}(\mathbf {A})\) contains an algebra carrying a binary reflexive compatible relation having a certain shape, which we have called a 2-dimensional special Hagemann relation with middle part. As a result of this property, we are able to show that the join of any pair of locally finite idempotent non-congruence 3-permutable varieties in the lattice of interpretability types, fails to be congruence 3-permutable, yielding a primeness argument for this specific setting.
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