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 o...

We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) Yang–Baxter (2-simplex), respectively. examine general structure (2n + 1)-gon 2n-simplex equations in direct sums vector spaces. Then, we provide a construction for their solutions, parameterized by elements Grassmannian Gr(n 1, 2n 1).

For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if-and-only-if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characte...