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On the number of terms witnessing congruence modularity

We study the validity of congruence inclusions of the form $ α( β\circ αγ\circ β\circ \dotsc \circ αγ\circ β) \subseteq αβ\circ αγ\circ αβ\circ \dots$ in congruence modular varieties, with an appropriate number of terms on each side and where juxtaposition denotes intersection. Two different methods using Day and Gumm terms are merged in order to obtain the so far best bounds. We introduce and study other related identities, possibly involving tolerances and admissible relations. We also slightly improve a result by A. Day, to the effect that if $n$ is even, then every variety with $n+2$ Jónsson terms has $2n+1$ Day terms.