We study modular properties in strongly convergent infinitary term rewriting. In particular, we show that:
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Confluence is not preserved across direct sum of a finite number of systems, even when these are non-collapsing.
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Confluence modulo equality of hypercollapsing subterms is not preserved across direct sum of a finite number of systems.
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Normalization is not preserved across direct sum of an infinite number of left-linear systems.
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Unique normalization with respect to reduction is not preserved across direct sum of a finite number of left-linear systems.
Together, these facts constitute a radical departure from the situation in finitary term rewriting. Positive results are:
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Confluence is preserved under the direct sum of an infinite number of left-linear systems iff at most one system contains a collapsing rule.
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Confluence is preserved under the direct sum of a finite number of non-collapsing systems if only terms of finite rank are considered.
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Top-termination is preserved under the direct sum of a finite number of left-linear systems.
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Normalization is preserved under the direct sum of a finite number of left-linear systems.
All of the negative results above hold in the setting of weakly convergent rewriting as well, as do the positive results concerning modularity of top-termination and normalization for left-linear systems.