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Erkko Lehtonen (Khalifa University), On clonoids of Boolean functions

Tue, 17 Oct 2023, 1:25 pm MDT

Let C_1 and C_2 be clones on sets A and B, respectively. A set K of functions of several arguments from A to B is called a (C_1,C_2)-clonoid if K C_1 \subseteq K and C_2 K \subseteq K. Sparks classified the clones C on \{0,1\} according to the cardinality of the lattice \mathcal{L}_{(\mathsf{J}_A, C)} of (\mathsf{J}_A, C)-clonoids (here \mathsf{J}_A denotes the clone of projections on a finite set A): \mathcal{L}_{(\mathsf{J}_A, C)} is finite if and only if C contains a near-unanimity operation; \mathcal{L}_{(\mathsf{J}_A, C)} is countably infinite if and only if C contains a Mal'cev operation but no majority operation; \mathcal{L}_{(\mathsf{J}_A, C)} has the cardinality of the continuum if and only if C contains neither a near-unanimity operation nor a Mal'cev operation. We sharpen Sparks's result by completely describing the lattices \mathcal{L}_{(\mathsf{J}_A, C)} and the (\mathsf{J}_A, C)-clonoids when A = \{0,1\} and C contains a near-unanimity operation or a Mal'cev operation, i.e., when \mathcal{L}_{(\mathsf{J}_A, C)} is finite or countably infinite.

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