On modularity in infinitary term rewriting

We study modular properties in strongly convergent infinitary term rewriting. In particular, we show that:

• •Confluence is not preserved across direct sum of a finite number of systems, even when these are non-collapsing.
• •Confluence modulo equality of hypercollapsing subterms is not preserved across direct sum of a finite number of systems.
• •Normalization is not preserved across direct sum of an infinite number of left-linear systems.
• •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:

• •Confluence is preserved under the direct sum of an infinite number of left-linear systems iff at most one system contains a collapsing rule.
• •Confluence is preserved under the direct sum of a finite number of non-collapsing systems if only terms of finite rank are considered.
• •Top-termination is preserved under the direct sum of a finite number of left-linear systems.
• •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.